diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzlecp" "b/data_all_eng_slimpj/shuffled/split2/finalzzlecp" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzlecp" @@ -0,0 +1,5 @@ +{"text":"\n\\section{Acknowledgements}\nThis work was supported by a Google Research\nScholar Award, a Google Research Gift, the Yandex Initiative in Machine Learning, the Israel Science Foundation (grant 1780\/21), the European Research\nCouncil (ERC) under the European Unions Horizon 2020\nresearch and innovation programme (grant ERC HOLI\n819080), Len Blavatnik and the Blavatnik Family Foundation, and Amnon\nand Anat Shashua.\n\\section{Theoretical Analysis}\\label{sec:analysis}\n\nWe turn to our theoretical analysis.\n\\Secref{sec:exact_extrapolation} proves that in the setting of Section~\\ref{sec:setup}, convergence of GF to a zero loss solution leads the student to extrapolate, \\emph{irrespective of how large its state space dimension is}.\n\\Secref{sec:approx_extrapolation} extends this result by establishing that, under mild conditions, approximate convergence leads to approximate extrapolation.\nThe results of sections \\ref{sec:exact_extrapolation} and~\\ref{sec:approx_extrapolation} assume that GF emanates from a balanced initialization, which empirically is known to capture near-zero initialization as commonly employed in practice (see \\Secref{sec:setup}).\n\\Secref{sec:relaxed_balanced_assumption} theoretically supports this empirical premise, by showing that with high probability, random near-zero initialization leads to balancedness.\n\n\n\nWe introduce notations that will be used throughout the analysis. \nFor a matrix ${\\bm{Q}}\\in\\mathbb{R}^{m\\times n}$, we let $\\|{\\bm{Q}}\\|_F$, $\\|{\\bm{Q}}\\|_{\\infty}$ and~$\\|{\\bm{Q}}\\|_2$ denote the Frobenius, $\\ell_{\\infty}$ and $\\ell_2$ (spectral) norms, respectively.\nFor a vector ${\\bm{v}} \\in \\mathbb{R}^m$, we use $\\|{\\bm{v}} \\|$ to denote the Euclidean norm and~${v}_i$ to denote its $i^{th}$ entry.\n\n\\subsection{Convergence Leads to Extrapolation} \\label{sec:exact_extrapolation}\n\nTheoretical analyses of implicit generalization often assume convergence to a solution attaining zero loss (see, e.g., \\cite{azulay2021implicit, gunasekar2017implicit, lyu2019gradient, woodworth2020kernel}).\nUnder such an assumption, Theorem~\\ref{thm:exact_extrapolation} below establishes implicit extrapolation, i.e.~that the solution to which GF converges extrapolates, \\emph{irrespective of how large the student's state space dimension~$d$ is}.\nA condition posed by the theorem is that the training sequence length~$k$ is greater than two times the teacher's state space dimension~\\smash{$\\hat{d}$}.\nWe empirically demonstrate the necessity of this condition in \\Secref{sec:experiments}, by showing that smaller values for~$k$ lead to failure in extrapolation.\n\\begin{theorem}\\label{thm:exact_extrapolation}\nAssume that $d > k > 2\\hat{d}$, the teacher parameters~$\\hat{\\Theta}$ are balanced (Definition~\\ref{def:balanced}), and the student parameters~$\\Theta$ are learned by applying GF to the loss~$\\mathcal{L} ( \\cdot )$ (\\Eqref{eq:population_loss}) starting from a balanced initialization.\nThen, if GF converges to a point~$\\Theta^*$ satisfying $\\mathcal{L} ( \\Theta^* ) = 0$, this point extrapolates (Definition~\\ref{def:extrapolation}).\n\\end{theorem}\n\nIn order to prove Theorem~\\ref{thm:exact_extrapolation}, we introduce two lemmas:\nLemma~\\ref{lemma:balanced}, which shows that balancedness is preserved under GF;\nand \nLemma~\\ref{lemma:exact_extrapolation}, which (through a surprising connection to a moment problem from statistics) establishes that a balanced solution attaining zero loss necessarily extrapolates. With Lemmas \\ref{lemma:balanced} and~\\ref{lemma:exact_extrapolation} in place, the proof of Theorem~\\ref{thm:exact_extrapolation} readily follows.\n\n\\begin{restatable}{lemma}{balanced}\n\\label{lemma:balanced}\nLet $\\Theta(\\tau)$, with $\\tau \\geq 0$, be a curve brought forth by applying GF to the loss~$\\mathcal{L} ( \\cdot )$ starting from a balanced initialization.\nThen, $\\Theta(\\tau)$ is balanced for every $\\tau \\geq 0$.\n\\end{restatable}\n\\begin{sproof}[Proof sketch (for complete proof see \\Appref{sec:lemma_balanced_proof}).]\nThe result follows from the symmetric role of ${\\bm{B}}$ and~${\\bm{C}}$ in the loss $\\mathcal{L} ( \\cdot )$.\n\\end{sproof}\n\n\\begin{lemma}\\label{lemma:exact_extrapolation}\nSuppose that $d > k > 2\\hat{d}$, the teacher is balanced, and that the student parameters $\\Theta$ are balanced and satisfy $\\mathcal{L} ( \\Theta ) = 0$. Then $\\Theta$ extrapolates.\n\\end{lemma}\n\\begin{sproof}[Proof sketch (for complete proof see \\Appref{sec:exact_extrapolation_proof}).]\nThe proof is based on a surprising connection to the \\textit{moment problem} from statistics (recovery of a probability distribution from its moments), which has been studied for decades (see, e.g., \\cite{schmudgen2017moment}). \n\nWithout loss of generality, we may assume that \\smash{$\\hat{{\\bm{A}}}$} is diagonal (if this is not the case then we apply an orthogonal eigendecomposition to~\\smash{$\\hat{{\\bm{A}}}$} and subsequently absorb eigenvectors into \\smash{$\\hat{{\\bm{B}}}$} and~\\smash{$\\hat{{\\bm{C}}}$}).\nWe may also assume that \\smash{$\\hat{{\\bm{C}}}\\hat{{\\bm{B}}} = 1$} (otherwise we absorb a scaling factor into ${\\bm{B}}$ and\/or~${\\bm{C}}$).\nSince \\smash{$\\hat{{\\bm{C}}}^\\top=\\hat{{\\bm{B}}}$} (teacher is balanced), we may define a probability vector (i.e.~a vector with non-negative entries summing up to one) \\smash{$\\hat{{\\bm{p}}} \\in \\mathbb{R}^{\\hat{d}}$} via \\smash{$\\hat{p}_i=\\hat{{C}}_i\\hat{{B}}_i$}, \\smash{$i = 1 , \\ldots , \\hat{d}$}.\nWe let \\smash{$\\hat{Z}$} denote the random variable supported on \\smash{$\\{ \\hat{A}_{1, 1} , \\ldots , \\hat{A}_{\\hat{d}, \\hat{d}} \\}$}, which assumes the value \\smash{$\\hat{A}_{i, i}$} with probability~\\smash{$\\hat{p}_i$}, \\smash{$i = 1 , \\ldots , \\hat{d}$}.\nNotice that for every $j \\in \\mathbb{N}$:\n\\[\n\\hat{{\\bm{C}}} \\hat{{\\bm{A}}}^j \\hat{{\\bm{B}}} = \\sum\\nolimits_{i=1}^{\\hat{d}} \\hat{p}_i \\hat{{A}}_{i i} ^j = \\mathbb{E}[\\hat{Z}^j]\n\\text{\\,,}\n\\]\nmeaning that the elements of the teacher's impulse response are precisely the moments of~\\smash{$\\hat{Z}$}.\n\nSimilarly to above we may assume~${\\bm{A}}$ is diagonal, and since $\\mathcal{L} ( \\Theta ) = 0$ it holds that \\smash{${\\bm{C}} {\\bm{B}} = \\hat{{\\bm{C}}}\\hat{{\\bm{B}}} = 1$}.\nWe may thus define a probability vector ${\\bm{p}} \\in \\mathbb{R}^{d}$ via $p_i={C}_i{B}_i$, $i = 1 , \\ldots , d$, and a random variable~$Z$ which assumes the value $A_{i i}$ with probability~$p_i$, $i = 1 , \\ldots , d$.\nFor every $j \\in \\mathbb{N}$:\n\\[\n{\\bm{C}} {\\bm{A}}^j {\\bm{B}} = \\sum\\nolimits_{i=1}^d {p}_i {A}_{i i} ^j = \\mathbb{E}[ Z^j]\n\\text{\\,,}\n\\]\nand so the elements of the student's impulse response are precisely the moments of~$Z$.\n\nThe probabilistic formulation we set forth admits an interpretation of extrapolation as a moment problem.\nNamely, since $\\mathcal{L} ( \\Theta ) = 0$ (i.e.~\\smash{${\\bm{C}} {\\bm{A}}^j {\\bm{B}}=\\hat{{\\bm{C}}} \\hat{{\\bm{A}}}^j \\hat{{\\bm{B}}}$} for $j=0,\\dots,k-1$) the random variables $Z$ and~\\smash{$\\hat{Z}$} agree on their first $k - 1$ moments, and the question is whether they agree on all higher moments as well.\nWe note that this question is somewhat more challenging than that tackled in classic instances of the moment problem, since the support of the random variable whose moments we match~($Z$) is not known to coincide with the support of the random variable we seek to recover~(\\smash{$\\hat{Z}$}).\nLuckily, a recent powerful result allow addressing the question we face~---~\\cite{cohen2011use} showed that the first $2 n$ moments of a discrete random variable~$X$ taking at most $n \\in \\mathbb{N}$ values uniquely define~$X$, in the sense that \\emph{any} discrete random variable agreeing with these $2 n$ moments must be identical to~$X$.\nTranslating this result to our setting, we have that if~$Z$ agrees with~\\smash{$\\hat{Z}$} on its first~\\smash{$2 \\hat{d}$} moments, it must be identical to~\\smash{$\\hat{Z}$}, and in particular it must agree with~\\smash{$\\hat{Z}$} on all higher moments as well. \nThe fact that \\smash{$k - 1 \\geq 2 \\hat{d}$} then concludes the proof.\n\n\n\n\nTo attain some intuition for the result we imported from \\cite{cohen2011use}, consider the simple case where \\smash{$\\hat{d}=1$}.\nThe transition matrix \\smash{$\\hat{{\\bm{A}}}$} is then a scalar \\smash{$\\hat{a} \\in \\mathbb{R}$}, the random variable \\smash{$\\hat{Z}$} is deterministically equal to~\\smash{$\\hat{a}$}, and the teacher's impulse response is given by the moments \\smash{$\\mathbb{E}[\\hat{Z}^j] = \\hat{a}^j$}, $j = 0 , 1 , \\ldots$.\nSince by assumption \\smash{$k > 2 \\hat{d}$}, the fact that $\\mathcal{L} ( \\Theta ) = 0$ means the random variable corresponding to the student, $Z$, agrees with the first two moments of~\\smash{$\\hat{Z}$}.\nThat is, $Z$~satisfies \\smash{$\\mathbb{E}[Z] = \\hat{a}$} and \\smash{$\\mathbb{E}[Z^2] = \\hat{a}^2$}.\nThis implies that $\\mathrm{Var}[Z]=\\mathbb{E}[Z^2] - \\mathbb{E}[Z]^2 = 0$, and therefore $Z$ is deterministically equal to~\\smash{$\\hat{a}$}, i.e.~it is identical to~\\smash{$\\hat{Z}$}.\nThe two random variables thus agree on all of their moments, meaning the impulse responses of the student and teacher are the same.\n\\end{sproof}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:exact_extrapolation}]\nBy Lemma~\\ref{lemma:balanced} (as well as continuity considerations) $\\Theta^*$~is balanced.\nLemma~\\ref{lemma:exact_extrapolation} then implies that it extrapolates.\n\\end{proof}\n\n\\subsection{Approximate Convergence Leads to Approximate Extrapolation} \\label{sec:approx_extrapolation}\n\nTheorem~\\ref{thm:exact_extrapolation} in \\Secref{sec:exact_extrapolation} proves extrapolation in the case where GF converges to a zero loss solution.\nTheorem~\\ref{thm:main_result} below extends this result by establishing that, under mild conditions, approximate convergence leads to approximate extrapolation~---~or more formally~---~for any $\\epsilon > 0$ and $q \\in \\mathbb{N}$, when GF leads the loss to be sufficiently small, the student $\\epsilon$-extrapolates with horizon~$q$.\n\\begin{theorem}\\label{thm:main_result}\nAssume the conditions of Theorem~\\ref{thm:exact_extrapolation}, and that the teacher parameters~$\\hat{\\Theta}$ are stable, i.e. the eigenvalues of~\\smash{$\\hat{{\\bm{A}}}$} are in~$[-1,1]$.\nAssume also that~\\smash{$\\hat{\\Theta}$} are non-degenerate, in the sense that the input-output mapping they realize is not identically zero.\nFinally, assume that the student parameters~$\\Theta$ learned by GF are confined to some bounded domain in parameter space. \nThen, for any $\\epsilon > 0$ and $q \\in \\mathbb{N}$, there exists $\\delta(\\epsilon, q) > 0$ such that whenever $\\mathcal{L} ( \\Theta ) \\leq \\delta(\\epsilon, q)$, the student $\\epsilon$-extrapolates with horizon~$q$.\n\\ignore{Assume the conditions of Theorem~\\ref{thm:exact_extrapolation}, and that the teacher parameters~$\\hat{\\Theta}$ are stable, i.e. the singular values of~\\smash{$\\hat{{\\bm{A}}}$} are in~$[-1,1]$.\nAssume also that~\\smash{$\\hat{\\Theta}$} are non-degenerate, in the sense that the input-output mapping they realize is not identically zero.\nThen, for any $\\epsilon > 0$ and $q \\in \\mathbb{N}$, there exists $\\delta(\\epsilon, q) > 0$ which admits the following:\nwhenever GF leads the student parameters~$\\Theta$ to within (Frobenius) distance $\\delta(\\epsilon,q)$ from a balanced point~$\\Theta^*$ satisfying $\\mathcal{L} ( \\Theta^* ) = 0$, the student $\\epsilon$-extrapolates with horizon~$q$.}\n\\end{theorem}\n\\begin{sproof}[Proof sketch (for complete proof see \\Appref{sec:apdx:approx_extrapolation}).]\nLet $\\delta > 0$ be a constant whose value will be chosen later, and suppose GF reached a point~$\\Theta$ satisfying $\\mathcal{L} ( \\Theta ) \\leq \\delta$.\n\nFollowing the proof of Lemma~\\ref{lemma:exact_extrapolation}, \\smash{$\\hat{\\Theta}$}~is identified with a distribution supported on the eigenvalues of~\\smash{$\\hat{{\\bm{A}}}$}, whose $j$'th moment is \\smash{$\\hat{m}_j := \\hat{{\\bm{C}}}\\hat{{\\bm{A}}}^j\\hat{{\\bm{B}}} ( \\hat{{\\bm{C}}} \\hat{{\\bm{B}}} )^{-1}$} for every $j \\in \\mathbb{N}$.\nSimilarly, $\\Theta$~is identified with a distribution supported on the eigenvalues of~${\\bm{A}}$, whose $j$'th moment is $m_j := {\\bm{C}} {\\bm{A}}^j {\\bm{B}} ( {\\bm{C}} {\\bm{B}} )^{-1}$ for every $j \\in \\mathbb{N}$.\nThe fact that $\\mathcal{L} ( \\Theta ) \\leq \\delta$ implies \\smash{$| {\\bm{C}} {\\bm{B}} - \\hat{{\\bm{C}}} \\hat{{\\bm{B}}} | \\leq \\sqrt{\\delta}$}, and in addition \\smash{$| \\hat{m}_j - m_j | \\leq \\mathcal{O} ( \\sqrt{\\delta} )$} for every $j = 1 , \\ldots , k - 1$. \nTo conclude the proof it suffices to show that\n\\begin{equation}\n\\label{eq:moments_approx}\n| \\hat{m}_j - m_j | \\leq \\mathcal{O} ( \\epsilon ) \\quad \\forall j \\in \\{ 1 , \\ldots , q - 1 \\}\n\\end{equation}\ngiven a small enough choice for~$\\delta$ (this choice then serves as $\\delta ( \\epsilon , q )$ in theorem statement).\n\nWe establish \\Eqref{eq:moments_approx} by employing the theory of Wasserstein distances \\citep{vaserstein1969markov}.\nFor $p \\in \\mathbb{N}$, denote by~$\\mathcal{W}_p$ the $p$-Wasserstein distance between the distributions identified with \\smash{$\\hat{\\Theta}$} and~$\\Theta$.\nSince \\smash{$k > 2 \\hat{d}$}, it holds that \\smash{$| \\hat{m}_j - m_j | \\leq \\mathcal{O} ( \\sqrt{\\delta} )$} for every \\smash{$j = 1 , \\ldots , 2 \\hat{d}$}.\nProposition~2 in \\cite{wu2020optimal} then implies $\\mathcal{W}_1 \\leq \\mathcal{O} ( \\delta^{1 \/ 4 \\hat{d}} )$.\nFor any $p \\in \\mathbb{N}$, $\\mathcal{W}_p \\leq \\mathcal{O} ( \\mathcal{W}_1^{1 \/ p})$ (see Section~2.3 in~\\cite{panaretos2019statistical}) and \\smash{$| \\hat{m}_p - m_p | \\leq \\mathcal{O} ( \\mathcal{W}_p )$} (see Section~1.2 in \\cite{biswas2021bounding}).\nCombining the latter three inequalities, we have that \\smash{$| \\hat{m}_p - m_p | \\leq \\mathcal{O} ( \\delta^{1 \/ 4 \\hat{d} p} )$} for any $p \\in \\mathbb{N}$.\nChoosing \\smash{$\\delta = \\mathcal{O} ( \\epsilon^{4 \\hat{d} ( q - 1 )} )$} therefore establishes \\Eqref{eq:moments_approx}.\n\\ignore{\nIn order to show $\\epsilon$-extrapolation with horizon $q$ we need to show that $\\forall\\epsilon>0, q\\in\\mathbb{N}$, $\\exists \\delta(\\epsilon,q)>0$ such that for any $\\Theta'$ satisfying $||\\Theta'-\\Theta^*||<\\delta$, we have for $n=0,\\dots,q$ that\n\\begin{equation}\n {\\bm{C}}'({\\bm{A}}')^n{\\bm{B}}'-\\hat{{\\bm{C}}}\\hat{{\\bm{A}}}^n\\hat{{\\bm{B}}}=O(\\epsilon).\n\\end{equation}\n\nFirst, by Theorem~\\ref{thm:exact_extrapolation} $\\Theta^*$ is an extrapolating solution. Following the interpretation of discrete random variables of Lemma~\\ref{lemma:exact_extrapolation} proof, we can bound the Wasserstein-1 distance between the extrapolating distribution defined by $\\Theta^*$ and the approximating distribution defined by $\\Theta'$. From Proposition 2 in \\cite{wu2020optimal}, we have that if $|{\\bm{C}}'({\\bm{A}}')^n{\\bm{B}}'-{\\bm{C}}^*({\\bm{A}}^*)^n{\\bm{B}}^*|\\le \\delta$ for $n\\in\\lbrace 0,\\dots,2\\hat{d}\\rbrace$, then\n\\begin{equation}\n \\mathcal{W}_1(\\Theta',\\Theta^*)=O\\left(\\delta^{\\frac{1}{4\\hat{d}}}\\right).\n\\end{equation}\n\nThe second part of the proof ties the error at step $p$ with the $\\mathcal{W}_p$ distance \\citep{biswas2021bounding}:\n\\begin{equation}\n {\\bm{C}}'({\\bm{A}}')^p{\\bm{B}}'-{\\bm{C}}^*({\\bm{A}}^*)^p{\\bm{B}}^*\\le \\mathcal{W}_p.\n\\end{equation}\nThe final step of the proof is based on \\cite{panaretos2019statistical} and bounds $\\mathcal{W}_p$ in terms of $\\mathcal{W}_1$, i.e.,\n\\begin{equation}\n \\mathcal{W}_p(\\Theta,\\Theta^*)= S^{p-1}\\mathcal{W}_1(\\Theta,\\Theta^*),\n\\end{equation}\nwhere $S$ is the difference between the maximal and minimal eigenvalues in the teacher and student support (due to our assumption on convergence, $S$ is bounded).\n\n\nCombining the three steps above, we have that if $|{\\bm{C}}'({\\bm{A}}')^n{\\bm{B}}'-{\\bm{C}}^*({\\bm{A}}^*)^n{\\bm{B}}^*|\\le \\delta$ for $n\\in\\lbrace 0,\\dots,2\\hat{d}\\rbrace$, then $\\forall p>n$ \\begin{equation}\\label{eq:final_inequality}\n {\\bm{C}}'({\\bm{A}}')^p{\\bm{B}}'-{\\bm{C}}^*({\\bm{A}}^*)^p{\\bm{B}}^*\\le S^{p-1}O\\left(\\delta^{\\frac{1}{4\\hat{d}}}\\right).\n\\end{equation}\n\nBy setting $\\delta<\\left(\\frac{\\epsilon}{c_1\\cdot S^{q-1}}\\right)^{4\\hat{d}}$ the RHS of \\Eqref{eq:final_inequality} satisfies $c_1S^{p-1}\\delta^{\\frac{1}{4\\hat{d}}}<\\epsilon$.\n}\n\\end{sproof}\n\n\\ignore{\n\\begin{corollary}\\label{corollary:extrapolation_for_any_horizon}\n\\amirg{not sure this corollary is helpful. Also seems strange that thing will work in the non-stable case so need to check this carefully.}\nIf $\\max_{i} A^*_i-\\min_{j}A^*_j<1$ we have extrapolation for an infinite horizon.\n\\end{corollary}\nThe proof for Corollary \\ref{corollary:extrapolation_for_any_horizon} follows by noting that $\\exists\\delta$ such that if $|\\Theta'-\\Theta^*|<\\delta$, then $S<1$. With $S<1$ the final $\\delta$ term does not depend on $q$ and therefore extrapolates for an infinite horizon.\n}\n\n\\subsection{Balancedness Captures Near-Zero Initialization}\n\\label{sec:relaxed_balanced_assumption}\n\nTheorems \\ref{thm:exact_extrapolation} and~\\ref{thm:main_result} assume that GF emanates from a balanced initialization, i.e.~from a point $\\Theta = ( {\\bm{A}} , {\\bm{B}} , {\\bm{C}} )$ satisfying ${\\bm{B}} = {\\bm{C}}^\\top$.\nIt was shown in~\\cite{cohen2022extrapolation} that theoretical predictions derived assuming balanced initialization faithfully match experiments conducted with near-zero initialization (an initialization commonly used in practice).\nProposition~\\ref{prop:implicit_bias_for_balanced_init} below theoretically supports this finding, establishing that with high probability, random near-zero initialization leads GF to arrive at an approximately balanced point, i.e.~a point $\\Theta = ( {\\bm{A}} , {\\bm{B}} , {\\bm{C}} )$ for which the difference between ${\\bm{B}}$ and~${\\bm{C}}^\\top$ is negligible compared to their size.\n\\begin{proposition}\\label{prop:implicit_bias_for_balanced_init}\nSuppose that:\n\\emph{(i)}~$d > 20$;\n\\emph{(ii)}~the teacher parameters~$\\hat{\\Theta}$ are balanced and are non-degenerate, in the sense that the input-output mapping they realize is not identically zero;\nand \n\\emph{(iii)}~the student parameters are learned by applying GF to the loss~$\\mathcal{L} ( \\cdot )$.\nLet $\\tilde{\\Theta}$ be a random point in parameter space, with entries drawn independently from the standard normal distribution.\nFor $\\epsilon > 0$, consider the case where GF emanates from the initialization~$\\epsilon \\tilde{\\Theta}$, and denote the resulting curve by $\\Theta_\\epsilon ( \\tau ) = ( {\\bm{A}}_\\epsilon ( \\tau ) , {\\bm{B}}_\\epsilon ( \\tau ) , {\\bm{C}}_\\epsilon ( \\tau ) )$, with $\\tau \\geq 0$.\nThen, w.p. at least~$0.75$, for every $\\epsilon > 0$ there exists $\\tau_\\epsilon \\geq 0$ such that:\n\\begin{equation}\n \\lim_{\\epsilon\\rightarrow 0^+}\\frac{||{\\bm{B}}_{\\epsilon}(\\tau_\\epsilon) - {\\bm{C}}_{\\epsilon}^\\top(\\tau_\\epsilon) ||_F}{||{\\bm{B}}_{\\epsilon}(\\tau_\\epsilon) + {\\bm{C}}_{\\epsilon}^\\top(\\tau_\\epsilon) ||_F}=0\n \\text{\\,.}\n\\end{equation}\n\\end{proposition}\n\\begin{sproof}[Proof sketch (for complete proof see \\Appref{sec:apdx:bias_to_symmetry}).]\nThe idea behind the proof is as follows.\nAssume $\\epsilon$ is sufficiently small.\nThen, when the entries of $\\Theta = ( {\\bm{A}} , {\\bm{B}} , {\\bm{C}})$ are on the order of~$\\epsilon$, we have \\smash{$\\frac{\\partial}{\\partial {\\bm{B}}} \\mathcal{L} ( \\Theta ) \\approx -2 \\hat{{\\bm{C}}} \\hat{{\\bm{B}}} \\cdot {\\bm{C}}^\\top$} and \\smash{$\\frac{\\partial}{\\partial {\\bm{C}}} \\mathcal{L} ( \\Theta ) \\approx -2 \\hat{{\\bm{C}}} \\hat{{\\bm{B}}} \\cdot {\\bm{B}}^\\top$}.\nThis implies that during the first part of the curve~$\\Theta_\\epsilon ( \\tau )$ it holds that\n$\\tfrac{d}{d \\tau} ( {\\bm{B}}_\\epsilon ( \\tau ) - {\\bm{C}}_\\epsilon^\\top ( \\tau ) ) \\approx - 2 \\hat{{\\bm{C}}} \\hat{{\\bm{B}}} \\cdot ( {\\bm{B}}_\\epsilon ( \\tau ) - {\\bm{C}}_\\epsilon^\\top ( \\tau ) )$\nand similarly\n$\\tfrac{d}{d \\tau} ( {\\bm{B}}_\\epsilon ( \\tau ) + {\\bm{C}}_\\epsilon^\\top ( \\tau ) ) \\approx ~~~ 2 \\hat{{\\bm{C}}} \\hat{{\\bm{B}}} \\cdot ( {\\bm{B}}_\\epsilon ( \\tau ) + {\\bm{C}}_\\epsilon^\\top ( \\tau ) )$.\nSince $\\hat{{\\bm{C}}} \\hat{{\\bm{B}}} > 0$ (follows from the teacher parameters being balanced and non-degenerate), the entries of ${\\bm{B}}_\\epsilon ( \\tau ) - {\\bm{C}}_\\epsilon^\\top ( \\tau )$ shrink exponentially fast while those of ${\\bm{B}}_\\epsilon ( \\tau ) + {\\bm{C}}_\\epsilon^\\top ( \\tau )$ grow at the same rate.\nThis exponential shrinkage\/growth leads $\\| {\\bm{B}}_\\epsilon ( \\tau ) - {\\bm{C}}_\\epsilon^\\top ( \\tau ) \\| \\big\/ \\| {\\bm{B}}_\\epsilon ( \\tau ) + {\\bm{C}}_\\epsilon^\\top ( \\tau ) \\|$ to become extremely small, more so the smaller $\\epsilon$ is.\n\\end{sproof}\n\\section{Further Experiments}\\label{sec:apdx:additional_experiments}\nIn this section we provide additional experiments that are not included in the main manuscript due to space constraints.\n\n\\begin{figure}[H]\n \\centering\n \\subfigure[Balanced teacher and general (unbalanced) student]{\\includegraphics[width=0.49\\textwidth]{figures\/extrapolation_as_func_of_k_non_balanced_init.png}}\n \\subfigure[Random (unbalanced) teacher and general (unbalanced) student]{\\includegraphics[width=0.49\\textwidth]{figures\/extrapolation_as_func_of_k_non_sym_teacher_init_1e-06.png}} \n \\caption{\n Extrapolation error as a function of the training sequence length $k$. (a) a balanced teacher with state dimensions $\\hat{d}=5$ and a general (unbalanced and non-diagonal) student with $d=40$. (b) a random unbalanced teacher (see \\Secref{sec:apdx:unbalanced_teacher}) with dimension $\\hat{d}=5$, and a student that has a non-diagonal transition matrix and is trained with standard (small) initialization, with state dimension $d=50$. In both plots results are averaged over 3 seeds.}\n \\label{fig:apdx_additional_experiments}\n\\end{figure}\n\n\\subsection{Balanced Teacher}\\label{sec:apdx:balanced_teacher}\nIn \\Secref{sec:exp:sym_teacher}, we have experimented with our proposed theoretical setup. In this section we provide additional figures and experiments.\n\n\\subsubsection{Unbalanced Student}\\label{sec:apdx:unbalanced_student}\n\nIn this experiment we use the same balanced teacher with $\\hat{d}=5$ as done in \\Secref{sec:exp:sym_teacher}. Instead of the diagonal student with balanced initialization, we use a general (non-diagonal) student with weights sampled from a Gaussian with scale $10^{-5}$ and $d=40$. Results are depicted in \\figref{fig:apdx_additional_experiments}(a). A similar phase transition phenomenon to the one in \\figref{fig:phase_transition_v2} is found also here.\n\n\n\\subsubsection{Effect of the Initialization Scale}\\label{sec:largeinit}\n\nProposition~\\ref{prop:implicit_bias_for_balanced_init} provides theoretical support for the fact that under near-zero initialization, the learned RNN tends to balancedness, which according to theorems \\ref{thm:exact_extrapolation} and~\\ref{thm:main_result} guarantees extrapolation.\nBelow we empirically explore the impact of varying the initialization scale. We use the same setting as in \\Secref{sec:apdx:unbalanced_student}, and repeat the experiment with different initialization scales for the students' weights.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.75\\textwidth]{figures\/init_breaks2.png}\n\\caption{\nExtrapolation error as a function of training sequence length $k$ for different initialization scales. Extrapolation error increases along with the scale of initialization.\n}\n\\label{fig:init_breaks}\n\\end{figure}\n\nAs can be seen in \\Figref{fig:init_breaks}, the extrapolation deteriorates for larger initialization scale, in the sense that it requires longer training sequences for getting good extrapolation error. This suggests that the condition of small initialization required by our theory is not an artifact of our proof technique, but rather a necessary condition for extrapolation to occur.\n\n\\subsection{Unbalanced Teacher}\\label{sec:apdx:unbalanced_teacher}\nIn \\Secref{sec:exp:step_func}, we have tested the extrapolation with respect to a specific unbalanced teacher and have observed a similar phase transition as predicted by the theory of \\Secref{sec:analysis} and empirical evaluation of \\Secref{sec:exp:sym_teacher}. Here we show that the phase transition is not limited to the specific teacher discussed by testing with respect to a randomly generated unbalanced (non-diagonal) teacher (see \\Secref{sec:apdx:unbalanced_teacher_generation}). The teacher is set to $\\hat{d}=5$ and student to $d=50$. Results are presented in \\figref{fig:apdx_additional_experiments}~(b). Here too we can observe the phase transition phenomena.\n\n\n\\subsection{Impulse Response Figures}\nIn \\Secref{sec:experiments} we have presented the extrapolation performance in different settings. In order to better convey the meaning of extrapolating vs non-extrapolating solutions we present here figures of the impulse response of different models.\n\nWe start with the impulse response corresponding to the experiment described in \\Secref{sec:exp:sym_teacher}. The figure depicts the balanced teacher with $\\hat{d}=5$ and two selected students (with $d=40$), one trained with $k=10$ and the other with $k=20$.\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=0.8\\textwidth]{figures\/balanced_impulse_responses.png}\n \\caption{\\textbf{Balanced teacher and student impulse response}. Students trained with: $k=10,20$ with respect to the balanced teacher described in \\Secref{sec:apdx:balanced_teacher_generation}. As can be seen, both students track the teacher up to the $k$ used in training, for $k=10$ there is no extrapolation for larger values of $k$, whereas $k=20$ tracks the teacher well beyond the sequence length used in training.}\n \\label{fig:apdx:balanced_ir}\n\\end{figure}\n\nWe can see that the student trained with $k=10$ tracks the teacher several steps beyond the $10th$ time step and then decays to zero. For $k=20$ we can see near perfect extrapolation for the horizon evaluated.\n\nNext we turn to \\Secref{sec:exp:step_func} and depict the average impulse responses of the ``delay teacher'' and the students trained with respect to the mentioned teacher.\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=0.8\\textwidth]{figures\/non_sym_impulse_responses.png}\n \\caption{\\textbf{Unbalanced teacher (delay) and student impulse response}. Students trained with: $k=8,18,20$ with respect to the unbalanced delay teacher described in \\Secref{sec:apdx:unbalanced_teacher_generation}. We can see that for $k=18$ the student diverges for longer sequences while $k=20$ which is trained for merely two additional time steps extrapolates and tracks the teacher almost perfectly.}\n \\label{fig:apdx:delay_teacher_ir}\n\\end{figure}\n\nSince the teacher here has $\\hat{d}=10$, a model trained with $k=8$ is trained with respect to the zero impulse response (see \\Secref{sec:apdx:unbalanced_teacher_generation} for details on delay teacher), and as expected results with the `zero' solution. we can see that for $k=18$ the student diverges from the teacher shortly after the $18th$ time step. For $k=20$ we can see near perfect extrapolation up to the horizon considered.\n\n\\section{Implementation Details}\\label{sec:apdx:impl_details}\n\nAll the experiments are implemented using PyTorch.\n\n\\subsection{Optimization}\nIn \\Secref{sec:exp:sym_teacher} we optimize the population loss, which entails minimizing \\Eqref{eq:population_loss} with respect to the parameters of the learned model. We use 15K optimization steps with Adam optimizer and a learning rate of $10^{-3}$. In this experiment, the results were not sensitive to the initialization scale of the (balanced) student.\nIn \\Secref{sec:exp:step_func} and \\Secref{sec:exp:non_linear_teacher} in the experiments that involve minimizing the empirical loss, we use 50K optimization steps with early stopping (most experiments required less than 10K steps). The batch size is set to $100$, data is sampled from a Gaussian with zero mean and scale of $1$. Experiments were not sensitive to most hyper-parameters other than learning rate and initialization scale. The examination of the effect of initialization scale presented in \\Secref{sec:largeinit} is done with learning rate scheduler \\verb|torch.optim.lr_scheduler.MultiStepLR| using milestones at $[5000,10000,15000,30000]$ and a decaying factor of $\\gamma=0.1$.\n\n\n\\subsection{Teacher Generation}\\label{sec:apdx:teacher_gen}\nOne of the main challenges in empirically evaluating extrapolation is that randomly sampling weights from a Gaussian distribution may result with an RNN of lower effective rank (i.e. the resulting RNN may be accurately approximated with another RNN with a smaller hidden dimension). We will now describe the teacher generation scheme for the different experiments.\n\n\\subsubsection{Balanced Teacher Generation}\\label{sec:apdx:balanced_teacher_generation}\nA balanced teacher consists of $d$ entries corresponding to the diagonal teacher and $d$ entries representing $\\hat{{\\bm{B}}}=\\hat{{\\bm{C}}}^\\top$. In order to avoid cases of rapid decay in the impulse response on the one hand, and exponential growth on the other, we set the eigenvalues to distribute uniformly between $0.6$ and $1.05$. The values of $\\hat{{\\bm{B}}}$ and $\\hat{{\\bm{C}}}$ are randomly sampled from a Gaussian around 0.5 and scale 1 and then normalized such that $\\hat{{\\bm{C}}}\\hat{{\\bm{B}}}=1$.\n\n\\subsubsection{Unbalanced Teacher Generation}\\label{sec:apdx:unbalanced_teacher_generation}\nIn this experiment, the teacher has a general (non-symmetrid) matrix $\\hat{{\\bm{A}}}$ and $\\hat{{\\bm{B}}}\\neq\\hat{{\\bm{C}}}^\\top$. We set the weights as described next.\n\n\\paragraph{Delay Teacher}\nA `delay' teacher has an impulse response of $1$ at time step $i=\\hat{d}-1$, that is, the teacher has an impulse response of $(0,\\dots,0,1,0,\\dots)$. In order to generate the mentioned impulse response we set the weights as follows,\n\n\\begin{equation}\\label{eq:step_func}\n {\\bm{A}}=\\begin{pmatrix}\n 0 & 1 & & 0 \\\\\n & & \\ddots & \\\\\n 0 & 0 & & 1 \\\\\n 0 & 0 & & 0\n \\end{pmatrix},\\; {\\bm{B}}=\\begin{pmatrix}\n 0 \\\\\n \\vdots\\\\\n 0\\\\\n 1\n \\end{pmatrix}, \\text{ and } {\\bm{C}}^\\top=\\begin{pmatrix}\n 1 \\\\\n 0 \\\\\n \\vdots\\\\\n 0\n \\end{pmatrix}.\n\\end{equation}\n\nNote that ${\\bm{B}},{\\bm{C}}$ above are set to extract the last entry of the first row of ${\\bm{A}}^i$ and ${\\bm{A}}$ is a Nilpotent shift matrix. It is straightforward to verify that ${\\bm{C}}{\\bm{A}}^i{\\bm{B}}=1$ for $i=\\hat{d}-1$ and $0$ otherwise.\n\n\\paragraph{Random Unbalanced Teacher}\nThe second unbalanced teacher is randomly generated. In order to avoid the caveats mentioned in \\Secref{sec:apdx:balanced_teacher}, we randomly sample the diagonal (from a Gaussian with zero mean and scale $0.1$) and super diagonal (from a Gaussian with mean $0.7$ and scale $0.1$) of $A$. We set $B,C$ as in \\eqref{eq:step_func}. The structure of ${\\bm{A}}$ ensures similar properties to that of the delayed teacher, specifically, that the first entries of the impulse response is zero and the teacher is `revealed' only after $\\hat{d}$ time steps.\n\n\\subsubsection{Non-Linear Teacher Generation}\\label{sec:apdx:gru_teacher_generation}\nAs opposed to the linear teacher discussed in previous sections, when the teacher is a Gated Recurrent Units (GRU), it is unclear how to generate a non-trivial teacher. When randomly generating a teacher GRU the result is either a trivial model that quickly decays to zero or a teacher with an exploding impulse response (depending on the scale of the initialization). In order to produce a teacher with interesting extrapolation behaviour, we initialize a model with an initialization scale of $10^{-6}$ and train for $1000$ step the model to mimic an arbitrarily chosen impulse response. The result of the mentioned procedure is a teacher GRU with non-trivial behaviour. \\figref{fig:gru_extrapolation}(b) shows that we get with this non-trivial teacher the phase transition phenomena as described in \\Secref{sec:exp:non_linear_teacher}.\n\n\\subsection{Extrapolation Error}\\label{sec:apdx:extrapolation_error}\nThe concept of extrapolation is very intuitive, and yet it does not admit any standard error measure. A proper extrapolation error measure should: (a) capture fine differences between two models with good extrapolation behaviour; and on the other hand, (b) be insensitive to the scale in which two non-extrapolating model explode. \nA natural approach which we take here is to report the $\\ell_{\\infty}$ norm difference on the tail of the impulse response. A model is considered non-extrapolating if the extrapolation error is worse than the extrapolation error of a trivial solution which has an impulse response of zeros.\n\\section{Deferred Proofs}\\label{apdx:a}\nHere we provide complete proofs for the results in the paper.\n\n\\subsection{Auxilary Proofs}\nIn this section we provide missing proofs from the main paper and additional lemmas to be used in the main proofs.\n\\subsubsection{Population Loss}\n\n\\begin{lemma}[\\textbf{Proof of \\Eqref{eq:population_loss}}]\\label{lemma:expected_loss}\nAssume ${\\bm{x}}\\sim \\mathcal{D}$ such that $\\mathbb{E}_{{\\bm{x}}\\sim\\mathcal{D}}[{\\bm{x}}]=0, \\mathbb{E}_{{\\bm{x}}\\sim\\mathcal{D}}[{\\bm{x}}\\vx^\\top]={\\bm{I}}_k \\in \\mathbb{R}^{k,k}$, where ${\\bm{I}}_k$ is the identity matrix. $y$ is given by $y=\\widehat{RNN}({\\bm{x}})$ where $\\widehat{RNN}(\\cdot)$ denotes the output of a teacher RNN, $\\hat{\\Theta}=(\\hat{{\\bm{A}}}, \\hat{{\\bm{B}}}, \\hat{{\\bm{C}}})$. Denote $w_i=\\hat{{\\bm{C}}}\\hat{{\\bm{A}}}^i\\hat{{\\bm{B}}}$, the loss for the student RNN satisfies:\n\n\\begin{equation}\\label{eq:apdx:pop_loss}\n \\mathbb{E}_{{\\bm{x}}\\sim\\mathcal{D}}\\left[\\ell\\left( RNN\\left({\\bm{x}}\\right), y\\right)\\right] = \\sum_{i=0}^{k-1} \\left( {\\bm{C}}{\\bm{A}}^i{\\bm{B}} - w_i \\right)^2.\n\\end{equation}\n\n\\end{lemma}\n\n\\begin{proof}[\\textbf{Proof of Lemma~\\ref{lemma:expected_loss}}]\nThe population loss for training with sequences of length $k$ is\n\n\\begin{equation}\n \\mathbb{E}_{{\\bm{x}}\\sim\\mathcal{D}}\\left[\\ell\\left( RNN\\left({\\bm{x}}\\right), y\\right)\\right]=\\mathbb{E}_{{\\bm{x}}\\sim\\mathcal{D}}\\left[ \\left(\\sum_{i=0}^{k-1}{\\bm{C}}{\\bm{A}}^{k-1-i}{\\bm{B}} {x}_i - \\sum_{j=0}^{k-1}w_{k-1-j} {x}_j\\right)^2 \\right].\n\\end{equation}\n\nReversing the order of summation, expanding the terms,\n\n\\begin{align}\n \\mathbb{E}_{{\\bm{x}}\\sim\\mathcal{D}}&\\left[\\ell\\left( RNN\\left({\\bm{x}}\\right), y\\right)\\right]=\\mathbb{E}_{{\\bm{x}}\\sim\\mathcal{D}}\\left[ \\left(\\sum_{i=0}^{k-1}{\\bm{C}}{\\bm{A}}^{i}{\\bm{B}} {x}_{k-1-i} - \\sum_{j=0}^{k-1}w_{j} {x}_{k-1-j}\\right)^2 \\right] \\\\ &= \\sum_{i, j=0}^{k-1} \\left[ {\\bm{C}}{\\bm{A}}^i{\\bm{B}} {\\bm{C}}{\\bm{A}}^j{\\bm{B}} -2{\\bm{C}}{\\bm{A}}^i{\\bm{B}} w_j + w_i w_j \\right] \\mathbb{E}_{{\\bm{x}}\\sim\\mathcal{D}}\\left[{x}_{k-1-i} {x}_{k-1-j}\\right] \\\\\n &= \\sum_{i, j=0}^{k-1} \\left[ {\\bm{C}}{\\bm{A}}^i{\\bm{B}} {\\bm{C}}{\\bm{A}}^j{\\bm{B}} -2{\\bm{C}}{\\bm{A}}^i{\\bm{B}} w_j + w_i w_j \\right] \\bm{1}_\\mathrm{k-1-i = k-1-j} \\\\ &= \\sum_{i, j=0}^{k-1} \\left[ {\\bm{C}}{\\bm{A}}^i{\\bm{B}} {\\bm{C}}{\\bm{A}}^j{\\bm{B}} -2{\\bm{C}}{\\bm{A}}^i{\\bm{B}} w_j + w_i w_j \\right] \\bm{1}_\\mathrm{i = j}\n \\\\ &= \\sum_{i =0}^{k-1} \\left[ ({\\bm{C}}{\\bm{A}}^i{\\bm{B}})^2 -2{\\bm{C}}{\\bm{A}}^i{\\bm{B}} w_i + w_i^2 \\right] = \\sum_{i=0}^{k-1} \\left( {\\bm{C}}{\\bm{A}}^i{\\bm{B}} - w_i \\right)^2.\n\\end{align}\nwhere the transition from the second to third rows is by our assumption that $\\mathbb{E}_{{\\bm{x}}\\sim\\mathcal{D}}[{x}_i{x}_j]=\\bm{1}_\\mathrm{i=j}$.\nTherefore we have,\n\n\\begin{equation}\n \\mathbb{E}_{{\\bm{x}}\\sim\\mathcal{D}}\\left[ \\ell\\left( RNN\\left({\\bm{x}}\\right), y\\right)\\right]= \\sum_{i=0}^{k-1} \\left( {\\bm{C}}{\\bm{A}}^i{\\bm{B}} - w_i \\right)^2.\n\\end{equation}\nconcluding the proof.\n\\end{proof}\n\n\\subsubsection{Perfect Generalization and Failed Extrapolation}\\label{sec:apdx:perfect_generalization_failed_extrapolation}\n\n\\begin{proposition}[Proposition~\\ref{prop:symmetric_lds_expressivity} in main paper]\nAssume $d > k$, and let $\\epsilon \\geq 0$ and $q \\in \\{ k + 1 , k + 2 , \\ldots\\}$.\nThen, for any teacher parameters \\smash{$\\hat{\\Theta}$}, there exist student parameters $\\Theta$ with which the population loss in \\Eqref{eq:population_loss} equals zero, and yet the student does \\emph{not} $\\epsilon$-extrapolate with horizon $q$.\n\\end{proposition}\n\n\\begin{proof}\n Consider a student, $\\Theta$, such that ${\\bm{A}}$ is symmetric (and therefore has an orthogonal eigendecomposition). Denote ${\\bm{A}}={\\bm{U}}{\\bm{\\Lambda}} {\\bm{U}}^\\top$. The impulse response at time step $i$ can be expressed as ${\\bm{C}}{\\bm{A}}^i{\\bm{B}}={\\bm{C}}{\\bm{U}}{\\bm{\\Lambda}}^i {\\bm{U}}^\\top {\\bm{B}}$. The latter can be written compactly in matrix form as ${\\bm{V}}{\\bm{g}}$ where ${\\bm{V}}$ is the Vandermonde matrix with $diag({\\bm{\\Lambda}})$ as its values,\n \\begin{equation*}\n {\\bm{V}}=\\begin{pmatrix}\n 1 & 1 & \\dots & 1\\\\\n \\lambda_1 & \\lambda_2 & \\dots & \\lambda_d\\\\\n \\lambda_1^2 & \\lambda_2^2 & \\dots & \\lambda_d^2\\\\\n \\vdots & \\vdots & & \\vdots \\\\\n \\lambda_1^{d-1} & \\lambda_2^{d-1} & \\dots & \\lambda_d^{d-1}\\\\\n \\end{pmatrix},\n \\end{equation*}\n and ${\\bm{g}}$ is defined as ${\\bm{g}}\\equiv ({\\bm{C}}{\\bm{U}})^\\top\\odot {\\bm{U}}^\\top {\\bm{B}}$.\\footnote{Here $\\odot$ denotes the Hadamard (elementwise) product.} A known result on square Vandermonde matrices is that they are invertible if and only if $\\lambda_i\\neq \\lambda_j,\\; \\forall i\\neq j$. Given a fixed set of distinct values $(\\lambda_1,\\dots,\\lambda_d)$ and an arbitrary impulse response ${\\bm{r}}\\in\\mathbb{R}^d$, in order for the student to generate the impulse response ${\\bm{r}}$ (i.e. ${\\bm{V}}{\\bm{g}}={\\bm{r}}$), one can set the coefficient vector, ${\\bm{g}}={\\bm{V}}^{-1}{\\bm{r}}$ and end up with a symmetric student with ${\\bm{r}}$ as its impulse response of length $d$.\n \n Consider a teacher RNN, $\\hat{\\Theta}=\\left({\\bm{A}},{\\bm{B}},{\\bm{C}}\\right)$, we can set and the first $k$ entries of ${\\bm{r}}$ to $ {r}_i=\\hat{{\\bm{C}}}\\hat{{\\bm{A}}}^{i-1}\\hat{{\\bm{B}}},\\; \\forall i=\\{1,\\dots,k\\}$. We are therefore left with $d-k$ degrees of freedom which yields many different students that correspond to the first $k$ entries of the teacher while fitting arbitrary values beyond the $k$ considered.\n\\end{proof}\n\n\\subsubsection{Equivalence Between Balanced RNNs with Symmetric and Diagonal Transition Matrices}\n\\begin{lemma}\\label{lemma:equiv_diagonal_rnn}\nA balanced RNN, $\\Theta=({\\bm{A}},{\\bm{B}},{\\bm{C}})$, with a symmetric transition matrix (i.e. ${\\bm{B}}={\\bm{C}}^\\top$ and ${\\bm{A}}={\\bm{A}}^\\top$) has an equivalent (i.e. generating the same impulse response) RNN, $\\Theta'=({\\bm{A}}',{\\bm{B}}',{\\bm{C}}')$, which is balanced and its transition matrix is diagonal.\n\\end{lemma}\n\nLemma~\\ref{lemma:equiv_diagonal_rnn} allows alternating between systems with symmetric and diagonal matrices. This is useful to simplify the analysis in \\Secref{sec:analysis}.\n\n\\begin{proof}[\\textbf{Proof of Lemma~\\ref{lemma:equiv_diagonal_rnn}}]\n Any symmetric matrix admits an orthogonal eigendecomposition with real (non-imaginary) eigenvalues. Denote ${\\bm{A}}={\\bm{U}}{\\bm{\\Lambda}}{\\bm{U}}^\\top$. We can define\n\\begin{equation*}\n \n {\\bm{A}}'={\\bm{\\Lambda}},\n \\quad\n {\\bm{B}}'={\\bm{U}}^\\top {\\bm{B}}\n \\quad\n \\text{ and }\n \\quad\n {\\bm{C}}'={\\bm{C}}{\\bm{U}},\n\\end{equation*}\nThe $i^{th}$ index of the impulse response is given by \n\\begin{equation*}\n {\\bm{C}}{\\bm{A}}^i{\\bm{B}}={\\bm{C}}{\\bm{U}}{\\bm{\\Lambda}}^i{\\bm{U}}^\\top{\\bm{B}}={\\bm{C}}'\\left({\\bm{A}}'\\right)^i{\\bm{B}}'\n\\end{equation*}\nconcluding that $\\Theta$ and $\\Theta'$ have the same impulse response of any length.\n\\end{proof}\n\n\\subsubsection{Gradient Derivation}\\label{sec:grad_derivation}\nFor completeness and \\Secref{sec:apdx:bias_to_symmetry}, we compute the gradients for the general setting.\n\\begin{lemma} \\label{lemma:gradients}\nGiven the population loss\n\\begin{equation}\n\\mathcal{L}({\\bm{A}},{\\bm{B}},{\\bm{C}})=\\sum_{j=0}^{k-1} \\left({\\bm{C}}{\\bm{A}}^j {\\bm{B}}-\\hat{{\\bm{C}}}\\hat{{\\bm{A}}}^j\\hat{{\\bm{B}}}\\right)^2\n \\text{\\,.}\\tag{\\ref{eq:population_loss} revisited}\n\\end{equation}\nDenote $\\nabla \\ell_i = {\\bm{C}}{\\bm{A}}^i{\\bm{B}}-w_i$, the derivatives of the loss with respect to ${\\bm{B}}$, and ${\\bm{C}}$ satisfy:\n\n\n\\begin{equation}\n \\frac{\\partial \\mathcal{L}}{\\partial {\\bm{B}}} = \\sum_{i=0}^{k-1} \\nabla \\ell_i ({\\bm{A}}^i)^\\top {\\bm{C}}^\\top,\n\\end{equation}\n\n\\begin{equation}\n \\frac{\\partial \\mathcal{L}}{\\partial {\\bm{C}}} = \\sum_{i=0}^{k-1} \\nabla \\ell_i {\\bm{B}}^\\top \\left({\\bm{A}}^i\\right)^\\top.\n\\end{equation}\n\n\n\\end{lemma}\n\n\\begin{proof}[\\textbf{Proof of Lemma~\\ref{lemma:gradients}}]\nHere, we will compute the gradient of the population loss.\n\nNote that for $j\\ge 0$, the derivative of ${\\bm{C}}{\\bm{A}}^j{\\bm{B}}$ with respect to to ${\\bm{B}}$ is given by\n\\begin{equation}\\label{eq:apdx:dB}\n \\frac{\\partial ({\\bm{C}}{\\bm{A}}^j{\\bm{B}})}{\\partial {\\bm{B}}}=({\\bm{A}}^j)^\\top{\\bm{C}}^\\top.\n\\end{equation}\nSimilarly, the derivative of ${\\bm{C}}{\\bm{A}}^j{\\bm{B}}$ with respect to to ${\\bm{C}}$ is given by\n\\begin{equation}\\label{eq:apdx:dC}\n \\frac{\\partial ({\\bm{C}}{\\bm{A}}^j{\\bm{B}})}{\\partial {\\bm{C}}}={\\bm{B}}^\\top({\\bm{A}}^j)^\\top.\n\\end{equation}\nUsing these derivatives, we can calculate the derivative of the population loss, (assigning $w_i=\\hat{\\bm{B}}\\hat{{\\bm{A}}}^i\\hat{{\\bm{C}}}$),\n\\begin{equation}\n \\mathcal{L}({\\bm{A}},{\\bm{B}},{\\bm{C}}) = \\mathbb{E}_{{\\bm{x}}\\sim\\mathcal{D}}\\left[ \\ell\\left( RNN\\left({\\bm{x}}\\right), y\\right)\\right]= \\sum_{i=0}^{k-1} \\left( {\\bm{C}}{\\bm{A}}^i{\\bm{B}} - w_i \\right)^2.\n\\end{equation}\nDenoting $\\nabla \\ell_i = {\\bm{C}}{\\bm{A}}^i{\\bm{B}}-w_i$, and noting that $w_i$ is constant (depends on \\smash{$\\hat{\\Theta}$}), we have for ${\\bm{X}} \\in \\{{\\bm{B}}, {\\bm{C}}\\}$:\n\\begin{equation}\n \\frac{\\partial \\mathcal{L}}{\\partial {\\bm{X}}} = \\sum_{i=0}^{k-1} \\frac{\\partial\\left( {\\bm{C}}{\\bm{A}}^i{\\bm{B}} - w_i \\right)^2}{\\partial {\\bm{X}}} = \\sum_{i=0}^{k-1} \\nabla \\ell_i \\frac{\\partial\\left( {\\bm{C}}{\\bm{A}}^i{\\bm{B}} - w_i \\right)}{\\partial {\\bm{X}}} = \\sum_{i=0}^{k-1} \\nabla \\ell_i \\frac{\\partial\\left( {\\bm{C}}{\\bm{A}}^i{\\bm{B}} \\right)}{\\partial {\\bm{X}}}.\n\\end{equation}\n\nPlugging in \\Eqref{eq:apdx:dB} and \\Eqref{eq:apdx:dC}, we have:\n\\begin{equation}\n \\frac{\\partial \\mathcal{L}}{\\partial {\\bm{B}}} = \\sum_{i=0}^{k-1} \\nabla \\ell_i \\frac{\\partial\\left( {\\bm{C}}{\\bm{A}}^i{\\bm{B}} \\right)}{\\partial {\\bm{B}}} = \\sum_{i=0}^{k-1} \\nabla \\ell_i ({\\bm{A}}^i)^\\top {\\bm{C}}^\\top,\n\\end{equation}\n\n\\begin{equation}\n \\frac{\\partial \\mathcal{L}}{\\partial {\\bm{C}}} = \\sum_{i=0}^{k-1} \\nabla \\ell_i \\frac{\\partial\\left( {\\bm{C}}{\\bm{A}}^i{\\bm{B}} \\right)}{\\partial {\\bm{C}}} = \\sum_{i=0}^{k-1} \\nabla \\ell_i {\\bm{B}}^\\top({\\bm{A}}^i)^\\top,\n\\end{equation}\n\n\n\\end{proof}\n\n\n\\subsubsection{Lemma~\\ref{lemma:balanced} (Conservation of Balancedness)}\\label{sec:lemma_balanced_proof}\n\n\\begin{lemma} \\textbf{[Lemma \\ref{lemma:balanced} in main paper]}\nWhen optimizing \\eqref{eq:population_loss} with GF emenating from a balanced initialization $\\Theta(0)$, the parameters $\\Theta(\\tau)$ are balanced for all $\\tau \\in \\mathbb{R}_+$.\n\\end{lemma}\n\nWe prove the above result by first showing it for GD and then translating the result to GF. The GD result is stated below, and generalizes a result that was shown in \\cite{cohen2022extrapolation} for the memoryless case.\n\n\\begin{lemma}\\label{lemma:gd_invariance}\nWhen optimizing \\eqref{eq:population_loss} with GD with balanced initial conditions, then $\\forall t\\in\\mathbb{N}$, $\\Theta$ has a balanced weight configuration, i.e. ${\\bm{B}}_t={\\bm{C}}_t^\\top$.\n\\end{lemma}\n\n\\begin{proof}[\\textbf{Proof of Lemma~\\ref{lemma:gd_invariance}}]\nWe prove by induction. By our assumption, the condition holds for $t=0$. Assume ${\\bm{B}}_t={\\bm{C}}_t^\\top$, our goal is to show the conditions hold for $({\\bm{B}}_{t+1},{\\bm{C}}_{t+1})$.\nIn order to show that ${\\bm{B}}_{t+1}={\\bm{C}}_{t+1}^\\top$, we only need to show that $\\frac{\\partial \\mathcal{L}}{\\partial {\\bm{B}}_t}=\\left( \\frac{\\partial \\mathcal{L}}{\\partial {\\bm{C}}_t} \\right)^\\top$. Writing the gradients (Lemma~\\ref{lemma:gradients}), we have\n\\begin{equation}\n \\left(\\frac{\\partial\\mathcal{L}}{\\partial {\\bm{C}}_t}\\right)^\\top = \\sum_{i=0}^{k-1}\\nabla\\ell_i {\\bm{A}}_t^i{\\bm{B}}_t =\\sum_{i=0}^{k-1}\\nabla\\ell_i ({\\bm{A}}_t^\\top)^i {\\bm{C}}_t^\\top= \\frac{\\partial\\mathcal{L}}{\\partial {\\bm{B}}_t},\n\\end{equation}\nwhere the inequality follows from the induction assumption and the symmetric structure of ${\\bm{A}}_t$. To conclude, the gradients at time $t$ are the same and ${\\bm{B}}_t={\\bm{C}}_t^\\top$ by the induction assumption, arriving at\n\\begin{equation}\n {\\bm{B}}_{t+1}={\\bm{B}}_t-\\eta\\frac{\\partial\\mathcal{L}}{\\partial {\\bm{B}}_t}={\\bm{C}}_t^\\top - \\eta \\left(\\frac{\\partial\\mathcal{L}}{\\partial {\\bm{C}}_t}\\right)^\\top={\\bm{C}}_{t+1}^\\top\n\\end{equation} \n\\end{proof}\n\n\nThe proof of Lemma \\ref{lemma:balanced} follows from Lemma \\ref{lemma:gd_invariance} and the fact that for sufficiently small step size GD approximates GF with arbitrary precision \\citep[see Theorem 3 in][]{elkabetz2021continuous}.\n\n\\subsubsection{Conservation of difference of norms}\\label{sec:apdx:conservation_of_norm_diff}\n\n\\Appref{sec:lemma_balanced_proof} shows that if weights are initialized to be balanced, this property is conserved throughout optimization. Here we show under standard initialization schemes, the difference between the norms of ${\\bm{B}}$ and ${\\bm{C}}$ is also conserved.\n\n\\begin{lemma}\\label{lemma:preserve_norm_diff}\nWhen optimizing \\eqref{eq:population_loss} with GF the difference between the norms of ${\\bm{B}}$, ${\\bm{C}}$ is conserved throughout GF, i.e.,\n\\begin{equation}\n \\frac{d}{dt}\\left(\\|{\\bm{B}}\\|_F^2 - \\|{\\bm{C}}\\|_F^2 \\right) = 0.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}[\\textbf{Proof of Lemma~\\ref{lemma:preserve_norm_diff}}]\nWe wish to prove that the difference between the norms is conserved over time.\nConsider the following expression:\\footnote{The last equality follows since in the SISO setup, ${\\bm{B}}^\\top{\\bm{B}}$ and ${\\bm{C}}\\mC^\\top$ are scalars and therefore the trace operator can be omitted.}\n\\begin{equation}\n \\alpha \\equiv \\|{\\bm{B}}\\|_F^2-\\|{\\bm{C}}\\|_F^2 = Tr({\\bm{B}}^\\top {\\bm{B}}) - Tr({\\bm{C}}\\mC^\\top)={\\bm{B}}^\\top {\\bm{B}} - {\\bm{C}}\\mC^\\top.\n\\end{equation}\nWith this notation, we just need to prove that $\\dot{\\alpha} = 0$.\nThe derivative of ${\\bm{B}}$, ${\\bm{C}}$ with respect to time is given by,\n\\begin{equation}\n\\label{eq:b_dot_def}\n \\dot{{\\bm{B}}}=-\\sum_{i=0}^{k-1}\\nabla \\ell_i ({\\bm{A}}^\\top)^i {\\bm{C}}^\\top,\n\\end{equation}\n\\begin{equation}\n\\label{eq:c_dot_def}\n\\dot{{\\bm{C}}} = -\\sum_{i=0}^{k-1}\\nabla \\ell_i {\\bm{B}}^\\top ({\\bm{A}}^\\top)^i.\n\\end{equation}\nUsing the interchangeability of derivative and transpose, we have:\n\\begin{equation}\n \\dot{\\alpha} = \\dot{{\\bm{B}}^\\top}{\\bm{B}} + {\\bm{B}}^\\top \\dot{{\\bm{B}}} - \\dot{{\\bm{C}}}{\\bm{C}}^\\top - {\\bm{C}} \\dot{{\\bm{C}}^\\top} = 2{\\bm{B}}^\\top \\dot{{\\bm{B}}} -2\\dot{{\\bm{C}}} {\\bm{C}}^\\top.\n\\end{equation}\nPlugging \\eqref{eq:b_dot_def} and \\eqref{eq:c_dot_def}, we get\n\\begin{align}\n\\dot{\\alpha} = 2 {\\bm{B}}^\\top \\left(- \\sum_{i=0}^{k-1} \\nabla \\ell_i ({\\bm{A}}^\\top)^i {\\bm{C}}^\\top \\right) -2\\left(-\\sum_{i=0}^{k-1}\\nabla \\ell_i {\\bm{B}}^\\top ({\\bm{A}}^\\top)^i\\right) {\\bm{C}}^\\top \\\\\n = -2\\left[ {\\bm{B}}^\\top \\left( \\sum_{i=0}^{k-1} \\nabla \\ell_i ({\\bm{A}}^\\top)^i \\right){\\bm{C}}^\\top -{\\bm{B}}^\\top\\left(\\sum_{i=0}^{k-1}\\nabla \\ell_i ({\\bm{A}}^\\top)^i\\right) {\\bm{C}}^\\top \\right] = 0.\n\\end{align}\nestablishing that $\\frac{d}{dt}\\left(\\|{\\bm{B}}\\|_F^2-\\|{\\bm{C}}\\|_F^2\\right)=0$.\n\n\\end{proof}\n\n\\subsection{Lemma~\\ref{lemma:exact_extrapolation} (Exact Extrapolation)}\\label{sec:exact_extrapolation_proof}\n\n\n\\begin{lemma}\\textbf{[Lemma \\ref{lemma:exact_extrapolation} in main paper]} \n\\label{lemma:exact_extrapolation_appendix}\nSuppose that $d > k > 2\\hat{d}$, the teacher is balanced, and that the student parameters $\\Theta$ are balanced and satisfy $\\mathcal{L} ( \\Theta ) = 0$. Then $\\Theta$ extrapolates.\n\\end{lemma}\n\\begin{proof}[\\textbf{Proof of Lemma~\\ref{lemma:exact_extrapolation_appendix}}]\nBy Lemma~\\ref{lemma:equiv_diagonal_rnn}, a balanced RNN with symmetric transition matrix has an equivalent (generating the same impulse response) balanced RNN with a diagonal transition matrix. We will continue under the assumption of diagonal transition matrices. \n\nWithout loss of generality we assume $\\hat{{\\bm{C}}}\\hat{{\\bm{B}}}=1$. Otherwise, the problem can be rescaled by $\\hat{{\\bm{C}}}\\hat{{\\bm{B}}}$, which is equivalent to rescaling the initial conditions, and providing no additional information.\\footnote{The case for which $\\hat{{\\bm{C}}}\\hat{{\\bm{B}}}=0$ is handled separately.}\n\nFrom the balanced assumption, we have $\\hat{{\\bm{C}}}^\\top=\\hat{{\\bm{B}}}$. Denote $\\hat{{\\bm{p}}} = \\hat{{\\bm{C}}}^\\top \\odot \\hat{{\\bm{B}}}=\\hat{{\\bm{B}}}\\odot \\hat{{\\bm{B}}}$, and we get $\\hat{{p}}_i\\geq 0$ and $\\sum_i \\hat{{p}}_i=1$, and therefore $\\hat{{\\bm{p}}}$ may be interpreted as a distribution over a random variable with $\\hat{d}$ possible values. We shall assume that these values are $\\hat{{A}}_{1,1},\\ldots,\\hat{{A}}_{\\hat{d},\\hat{d}}$, and denote the corresponding random variable by $\\hat{Z}$.\n\nFurthermore, we can also interpret elements of the impulse response of $\\hat{\\Theta}$ as moments of this distribution.\nLet us write the $n^{th}$ element of the impulse response as:\n\\begin{equation}\n\\hat{{\\bm{C}}} \\hat{{\\bm{A}}}^n \\hat{{\\bm{B}}} = \\sum_i \\hat{{p}}_i \\hat{a}_i ^n = \\mathbb{E}_{\\hat{{\\bm{p}}}}[\\hat{Z}^n],\n\\end{equation}\nwhere $\\mathbb{E}_{{\\bm{p}}}[Z]$ is the expected value of a random variable $Z$ under the distribution ${\\bm{p}}$.\nIn the same way, we can define for the learned model $\\Theta$, a distribution $p_i = {C}_i {B}_i$, and write the learned impulse response as:\n\\begin{equation}\n{\\bm{C}} {\\bm{A}}^n {\\bm{B}} = \\sum_i {p}_i a_i ^n = \\mathbb{E}_{{\\bm{p}}}[{Z}^n].\n\\end{equation}\nThis view provides us with a moment matching interpretation of the learning problem. Namely, the fact that $\\Theta$ matches the first $k$ elements of the teacher impulse response, is the same as saying they agree on the first $k-1$ moments $\\mathbb{E}_{{\\bm{p}}}[{Z}^j]$ for $j\\in\\{1,\\ldots, k-1\\}$.\\footnote{Equality of the $0^{th}$ moment ensures the student induces a valid probability, i.e. $\\sum_{i}{C}_i{B}_i$=$\\sum_{i}\\hat{{C}}_i\\hat{{B}}_i$=1.} The question of extrapolation is whether equality in the first $k-1$ moments implies an equality in all other moments.\n\n\n In \\cite[Theorem 1]{cohen2011use} and in \\cite[Lemma 4]{wu2020optimal} it is shown that the first $2\\hat{d}$ moments of a discrete random variable taking at most $\\hat{d}$ different values uniquely define this random variable. Therefore, any other discrete random variable identifying with the teacher on $2\\hat{d}$ moments must be the same random variable and therefore identifies on higher moments as well. Since we assumed $k > 2\\hat{d}$, this result immediately implies that equality in the first $k-1$ moments implies equality in all other moments.\n\n For the case $\\hat{{\\bm{C}}}\\hat{{\\bm{B}}}=0$, from our assumption that the teacher is balanced, we have that the condition is met only if $\\hat{{C}}_i=\\hat{{B}}_i=0$ for $i=1,\\dots,\\hat{d}$. Such a teacher has an impulse response of zeros, for $k\\ge 1$, a student minimizing the loss must also satisfy ${\\bm{C}}{\\bm{B}}=0$ and therefore has the zeros as its impulse response (recall the student is balanced) thus extrapolating with respect to the said teacher.\n\n\\end{proof}\n\n\n\\subsection{Theorem~\\ref{thm:main_result} (Approximate Extrapolation)}\\label{sec:apdx:approx_extrapolation}\nThis section is devoted to the proof of Theorem~\\ref{thm:main_result} which ties the approximation error of optimization to that of extrapolation. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{theorem}\\textbf{[Theorem~\\ref{thm:main_result} in main paper]}\n\\label{thm:approx_extrapolation}\nConsider the minimization of \\Eqref{eq:population_loss} and assume: (i) $d>k>2\\hat{d}$; (ii) the teacher is balanced and stable (i.e. the eigenvalues of $\\hat{{\\bm{A}}}$ are in $[-1,1]$); (iii) the teacher is non-degenerate, i.e. the input output mapping they realize is not identically zero; (iv) the student parameters are learned by applying GF to the loss $\\mathcal{L}(\\cdot)$, starting from a balanced initialization; (v) the student parameters $\\Theta$ are bounded.\n\nThen, for any $\\epsilon>0$ and $q\\in\\mathbb{N}$, there exists $\\delta(\\epsilon,q)>0$ such that whenever $\\mathcal{L}(\\Theta)\\le \\delta(\\epsilon,q)$, the student $\\epsilon$-extrapolates with horizon~$q$.\n\n\\end{theorem}\n\n\\begin{proof}[\\textbf{Proof of Theorem~\\ref{thm:approx_extrapolation}}]\n\nLet $\\delta > 0$ be a constant whose value will be chosen later, and suppose GF reached a point~$\\Theta$ satisfying $\\mathcal{L} ( \\Theta ) \\leq \\delta$.\nFollowing the proof of Lemma~\\ref{lemma:exact_extrapolation}, \\smash{$\\hat{\\Theta}$}~is identified with a distribution supported on the eigenvalues of~\\smash{$\\hat{{\\bm{A}}}$}, whose $j$'th moment is \\smash{$\\hat{m}_j := \\hat{{\\bm{C}}}\\hat{{\\bm{A}}}^j\\hat{{\\bm{B}}} ( \\hat{{\\bm{C}}} \\hat{{\\bm{B}}} )^{-1}$} for every $j \\in \\mathbb{N}$.\nSimilarly, $\\Theta$~is identified with a distribution supported on the eigenvalues of~${\\bm{A}}$, whose $j$'th moment is $m_j := {\\bm{C}} {\\bm{A}}^j {\\bm{B}} ( {\\bm{C}} {\\bm{B}} )^{-1}$ for every $j \\in \\mathbb{N}$. \nFrom our assumption that $\\mathcal{L}(\\Theta)\\leq \\delta$,\n\\begin{equation}\\label{eq:loss_smaller_than_delta}\n \\mathcal{L} (\\Theta)=\\sum_{j=0}^{k-1} \\left( {\\bm{C}}{\\bm{A}}^j{\\bm{B}} -\\hat{{\\bm{C}}}\\hat{{\\bm{A}}}^j\\hat{{\\bm{B}}}\\right)^2\\leq \\delta.\n\\end{equation}\nand specifically, each term satisfies $({\\bm{C}}{\\bm{A}}^j{\\bm{B}}-\\hat{{\\bm{C}}}\\hat{{\\bm{A}}}^j\\hat{{\\bm{C}}})^2\\le \\delta$ for $j=0,\\dots,k-1$. In particular, $({\\bm{C}}{\\bm{B}}-\\hat{{\\bm{C}}}\\hat{{\\bm{B}}})^2\\le \\delta$. Denote $\\beta=\\hat{{\\bm{C}}}\\hat{{\\bm{B}}}-{\\bm{C}}{\\bm{B}}$, then $\\beta\\in[-\\sqrt{\\delta},\\sqrt{\\delta}]$. \nNote that $\\hat{{\\bm{C}}}\\hat{{\\bm{B}}}$ is a (positive) constant, multiplying the loss by $(\\hat{{\\bm{C}}}\\hat{{\\bm{B}}})^{-2}$ we have that each term $\\leq\\delta(\\hat{{\\bm{C}}}\\hat{{\\bm{B}}})^{-2}$. We can write for each $j=0,\\dots,k-1$,\n\n\\begin{align}\n \\left( \\frac{{\\bm{C}}{\\bm{A}}^j{\\bm{B}}}{\\hat{{\\bm{C}}}\\hat{{\\bm{B}}}} -\\frac{\\hat{{\\bm{C}}}\\hat{{\\bm{A}}}^j\\hat{{\\bm{B}}}}{\\hat{{\\bm{C}}}\\hat{{\\bm{B}}}}\\right)^2 & = \\left( \\frac{{\\bm{C}}{\\bm{A}}^j{\\bm{B}}}{\\hat{{\\bm{C}}}\\hat{{\\bm{B}}}} - \\frac{{\\bm{C}}{\\bm{A}}^j{\\bm{B}}}{{\\bm{C}}{\\bm{B}}} + \\frac{{\\bm{C}}{\\bm{A}}^j{\\bm{B}}}{{\\bm{C}}{\\bm{B}}} -\\frac{\\hat{{\\bm{C}}}\\hat{{\\bm{A}}}^j\\hat{{\\bm{B}}}}{\\hat{{\\bm{C}}}\\hat{{\\bm{B}}}}\\right)^2\\\\\n & = \\left( {\\bm{C}}{\\bm{A}}^j{\\bm{B}}\\left(\\frac{1}{\\hat{{\\bm{C}}}\\hat{{\\bm{B}}}} - \\frac{1}{{\\bm{C}}{\\bm{B}}}\\right) + \\underbrace{\\frac{{\\bm{C}}{\\bm{A}}^j{\\bm{B}}}{{\\bm{C}}{\\bm{B}}} -\\frac{\\hat{{\\bm{C}}}\\hat{{\\bm{A}}}^j\\hat{{\\bm{B}}}}{\\hat{{\\bm{C}}}\\hat{{\\bm{B}}}}}_{m_j-\\hat{m}_j}\\right)^2\n \n\\end{align}\nWe can further expand the term on the left,\n\\begin{equation}\n \\frac{1}{\\hat{{\\bm{C}}}\\hat{{\\bm{B}}}} - \\frac{1}{{\\bm{C}}{\\bm{B}}}=\\frac{1}{\\hat{{\\bm{C}}}\\hat{{\\bm{B}}}}-\\frac{1}{\\hat{{\\bm{C}}}\\hat{{\\bm{B}}}-\\beta}=\\frac{\\hat{{\\bm{C}}}\\hat{{\\bm{B}}}-\\beta - \\hat{{\\bm{C}}}\\hat{{\\bm{B}}}}{\\hat{{\\bm{C}}}\\hat{{\\bm{B}}}(\\hat{{\\bm{C}}}\\hat{{\\bm{B}}}-\\beta)}=\\frac{\\beta}{\\hat{{\\bm{C}}}\\hat{{\\bm{B}}}(\\beta-\\hat{{\\bm{C}}}\\hat{{\\bm{B}}})}\n\\end{equation}\nPlugging back to the above, we have\n\\begin{align}\n \\delta(\\hat{{\\bm{C}}}\\hat{{\\bm{B}}})^{-2}\\geq\\left( \\frac{{\\bm{C}}{\\bm{A}}^j{\\bm{B}}}{\\hat{{\\bm{C}}}\\hat{{\\bm{B}}}} -\\frac{\\hat{{\\bm{C}}}\\hat{{\\bm{A}}}^j\\hat{{\\bm{B}}}}{\\hat{{\\bm{C}}}\\hat{{\\bm{B}}}}\\right)^2 & =\\left(\\frac{\\beta{\\bm{C}}{\\bm{A}}^j{\\bm{B}}}{\\hat{{\\bm{C}}}\\hat{{\\bm{B}}}(\\beta-\\hat{{\\bm{C}}}\\hat{{\\bm{B}}})} + (m_j-\\hat{m}_j) \\right)^2\\\\\n & = \\beta^2\\kappa^2+2\\beta\\kappa(m_j-\\hat{m}_j) + (m_j-\\hat{m}_j)^2\\\\\n & \\geq 2\\beta\\kappa(m_j-\\hat{m}_j) + (m_j-\\hat{m}_j)^2\\\\\n & \\geq -2|\\delta\\kappa(m_j-\\hat{m}_j)| + (m_j-\\hat{m}_j)^2\n\\end{align}\nwhere $\\kappa\\equiv \\frac{{\\bm{C}}{\\bm{A}}^j{\\bm{B}}}{\\hat{{\\bm{C}}}\\hat{{\\bm{B}}}(\\beta-\\hat{{\\bm{C}}}\\hat{{\\bm{B}}})}$. From assumption (ii), the teacher is stable and therefore $\\hat{m}_j\\le \\hat{{\\bm{C}}}\\hat{{\\bm{B}}}$ for all $j=0,\\dots,k-1$. Similarly, from assumption (v) the student parameters are bounded and therefore ${\\bm{C}}{\\bm{A}}^j{\\bm{B}}$ is bounded by $\\tau^{j+2}$ (where $\\tau\\equiv \\max\\lbrace 1, \\eta \\rbrace$ and $\\eta$ is a bound on the Frobenous norm of ${\\bm{A}},{\\bm{B}},{\\bm{C}}$). $m_j$ is bounded in a similar fashion by $\\tau^j$.\n\nCombining the above, for $\\delta<\\hat{{\\bm{C}}}\\hat{{\\bm{B}}}$ we have,\n\\begin{equation}\n \\frac{\\delta}{(\\hat{{\\bm{C}}}\\hat{{\\bm{B}}})^2}\\geq \\frac{-2\\delta \\tau^{j+2}(\\tau^j+1)}{2(\\hat{{\\bm{C}}}\\hat{{\\bm{B}}})^2}+(m_j-\\hat{m}_j)^2\n\\end{equation}\nSetting $\\delta'<\\frac{\\delta(\\hat{{\\bm{C}}}\\hat{{\\bm{B}}})^2}{1+\\tau^{k+1}(\\tau^{k-1}+1)}$, if $\\mathcal{L}(\\Theta)\\leq\\delta'$ then $|m_j-\\hat{m}_j|\\le \\sqrt{\\delta}$ for $j=1,\\dots,k-1$. Proposition~2 in \\cite{wu2020optimal} then implies $\\mathcal{W}_1(\\Theta,\\hat{\\Theta}) \\leq \\mathcal{O} ( \\delta^{1 \/ 4 \\hat{d}} )$.\\footnote{Here we overload notations and denote the distributions of the teacher and student by $\\Theta$ and $\\hat{\\Theta}$ respectively}\n\nDenote $\\Omega\\equiv \\left(\\bigcup_{i=1}^{d} A_{ii}\\right) \\bigcup \\left(\\bigcup_{j=1}^{\\hat{d}}\\hat{A}_{jj}\\right)$ (the union of the supports of $\\Theta$ and $\\hat{\\Theta}$), from Section~2.3 in~\\cite{panaretos2019statistical}, for $q>p$ the $q^{th}$ and $p^{th}$ Wasserstein distances satisfy $\\mathcal{W}_q^q(\\Theta,\\hat{\\Theta})\\le \\mathcal{W}_p^p(\\Theta,\\hat{\\Theta}) \\gamma^{q-p}$ where $\\gamma=\\max_{x,y\\in\\Omega} |x-y|$. In particular, for $p=1$, $\\mathcal{W}_q(\\Theta,\\hat{\\Theta})\\le \\left(\\mathcal{W}_1(\\Theta,\\hat{\\Theta}) \\gamma^{q-1}\\right)^{1\/q}$.\nNote that $\\gamma$ can is bounded by $\\gamma\\le\\tau+1\\le 2\\tau$ (recall the student is bounded and teacher is stable).\n\n\n\nFinally, $|m_q-\\hat{m}_q|\\le \\mathcal{W}_q(\\Theta,\\hat{\\Theta})$ (see Section~1.2 in \\cite{biswas2021bounding}).\nCombining the steps above, for all $q\\in\\mathbb{N}$,\n\n\\begin{equation}\n |m_q-\\hat{m}_q|\\le \\mathcal{W}_q(\\Theta,\\hat{\\Theta})\\le \\left(\\mathcal{W}_1(\\Theta,\\hat{\\Theta}) (2\\tau)^{q-1}\\right)^{1\/q}\\le \\left(\\rho \\delta^{1\/4\\hat{d}}(2\\tau)^{q-1}\\right)^{1\/q}\n\\end{equation}\nwhere $\\rho$ is a constant satisfying $\\mathcal{W}_1(\\Theta,\\hat{\\Theta})\\le \\rho\\delta^{1\/4\\hat{d}}$. To achieve $|m_j-\\hat{m}_j|<\\epsilon$ for any $\\epsilon>0$, we can set $\\delta(\\epsilon,q)<\\left(\\frac{\\epsilon^q}{\\rho\\gamma^{q-1}}\\right)^{4\\hat{d}}$ concluding the proof.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Proposition~\\ref{prop:implicit_bias_for_balanced_init} (Implicit Bias for Balancedness)}\n\\label{sec:apdx:bias_to_symmetry} \n\nThe proof of Proposition~\\ref{prop:implicit_bias_for_balanced_init} consists of several steps. First, we bound with high probability the norms of ${\\bm{B}}$ and ${\\bm{C}}$ at initialization (Lemma~\\ref{lemma:bound_init}). We then derive bounds on the differential equations of $\\frac{d}{dt}({\\bm{B}}(t)+{\\bm{C}}^\\top(t))$ and $\\frac{d}{dt}({\\bm{B}}(t)-{\\bm{C}}^\\top(t))$ (Lemma~\\ref{lemma:Y_and_W_ode}). We show that when the initialization scale tends to zero, the ratio between the differential equations tends to zero. (Lemma~\\ref{lemma:upper_bound_in_time}).\n\nBefore we turn to prove Proposition~\\ref{prop:implicit_bias_for_balanced_init}, we first need to bound the initial values for a vector ${\\bm{v}}\\in\\mathbb{R}^n$ initialized with $\\mathcal{N}(0,\\frac{\\epsilon^2}{n})$.\n\n\\begin{lemma}\\label{lemma:bound_init}\n Assume a vector ${\\bm{v}} \\in \\mathbb{R}^n$ with $\\mathcal{N}(0,\\frac{\\epsilon^2}{n})$ per coordinate. Then:\n\n\\begin{equation}\\label{eq:init_bound}\n Pr\\left(\\frac{\\epsilon}{2}< \\|\\mathbf{v}\\| < \\frac{3\\epsilon}{2} \\right) \\geq 1- 2 \\exp(-9n\/64 ).\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}[\\textbf{Proof of Lemma~\\ref{lemma:bound_init}}]\nThe proof of \\ref{lemma:bound_init} uses known results on the Chi-square distribution \\cite{laurent2000adaptive}, applied to our specific setting to achieve the desired bounds.\nWe will begin by changing variables, $\\tilde{{v}}_i = {v}_i\\cdot \\frac{\\sqrt{n}}{\\epsilon}$. The entries $\\tilde{{v}}_i$, are standard Gaussian variables. The squared norm of $\\tilde{{\\bm{v}}}$ distributes according to the $\\chi$-squared distribution.\n\nBy \\cite[Lemma 1]{laurent2000adaptive}, in our case (assigning $x= 9n\/64$) the following inequalities hold:\n\n\\begin{align}\n &Pr\\left(\\|\\tilde{{\\bm{v}}}\\|^2 \\geq (1.75 +9\/32) n\\right) \\leq \\exp(-9n\/64),\\\\\n &Pr\\left(\\|\\tilde{{\\bm{v}}}\\|^2 \\leq n\/4\\right) \\leq \\exp(-9n\/64).\n\\end{align}\n\nIn particular,\n\\begin{equation}\n Pr\\left(\\|\\tilde{{\\bm{v}}}\\|^2 \\geq 2.25 n\\right) \\leq \\exp(-9n\/64) \\Longrightarrow Pr\\left(\\|\\tilde{{\\bm{v}}}\\| \\geq 1.5 \\sqrt{n}\\right) \\leq \\exp(-9n\/64).\n\\end{equation}\n\nChanging variables back to ${\\bm{v}}$,\n\\begin{equation}\n Pr\\left(\\|{\\bm{v}}\\| \\geq 1.5 \\epsilon\\right) \\leq \\exp(-9n\/64).\n\\end{equation}\n\nSimilarly, for the second bound:\n\\begin{align}\n Pr\\left(\\|\\tilde{{\\bm{v}}}\\| \\leq \\sqrt{n}\/2\\right) \\leq \\exp(-9n\/64)\n \\Longrightarrow Pr\\left(\\|{\\bm{v}}\\| \\leq \\epsilon\/2\\right) \\leq \\exp(-9n\/64).\n\\end{align}\n\nTaking the complementary probability, we have the desired result of\n\\begin{equation}\nPr\\left(\\frac{\\epsilon}{2}< \\|{\\bm{v}}\\| < \\frac{3\\epsilon}{2}\\right) \\geq 1- 2 \\exp(-9n\/64 ).\n\\end{equation}\n\\end{proof}\n\nNote that for a matrix ${\\bm{X}}\\in\\mathbb{R}^{m\\times p}$, Lemma~\\ref{lemma:bound_init} bounds its Frobenius norm, $Pr\\left(\\frac{\\epsilon}{2}< \\|{\\bm{X}}\\|_F<\\frac{3\\epsilon}{2} \\right)\\ge 1-2exp\\left(-9mp\/64\\right)$. The result is straight forward by applying the lemma to ${\\bm{X}}$'s vectorized form.\n\n\n\\begin{proposition}\\textbf{[Proposition~\\ref{prop:implicit_bias_for_balanced_init} in main paper]} \nSuppose that:\n\\emph{(i)}~$d > 4$;\n\\emph{(ii)}~the teacher parameters~$\\hat{\\Theta}$ are balanced and are non-degenerate, in the sense that the input-output mapping they realize is not identically zero;\nand \n\\emph{(iii)}~the student parameters are learned by applying GF to the loss~$\\mathcal{L} ( \\cdot )$.\nLet $\\tilde{\\Theta}$ be a random point in parameter space, with entries drawn independently from the standard normal distribution.\nFor $\\epsilon > 0$, consider the case where GF emanates from the initialization~$\\epsilon \\tilde{\\Theta}$, and denote the resulting curve by $\\Theta_\\epsilon ( \\tau ) = ( {\\bm{A}}_\\epsilon ( \\tau ) , {\\bm{B}}_\\epsilon ( \\tau ) , {\\bm{C}}_\\epsilon ( \\tau ) )$, with $\\tau \\geq 0$.\nThen, w.p. at least~$0.75$, for every $\\epsilon > 0$ there exists $\\tau_\\epsilon \\geq 0$ such that:\n\\begin{equation}\n \\lim_{\\epsilon\\rightarrow 0^+}\\frac{||{\\bm{B}}_{\\epsilon}(\\tau_\\epsilon) - {\\bm{C}}_{\\epsilon}^\\top(\\tau_\\epsilon) ||_F}{||{\\bm{B}}_{\\epsilon}(\\tau_\\epsilon) + {\\bm{C}}_{\\epsilon}^\\top(\\tau_\\epsilon) ||_F}=0\n \\text{\\,.}\n\\end{equation}\n\n\\end{proposition}\n\nThe consequence of Proposition \\ref{prop:implicit_bias_for_balanced_init} is that as $\\epsilon$ converges to zero, ${\\bm{B}}$ and ${\\bm{C}}$ converge towards each other. \n\nFor convenience, we refer to the mentioned initialization scheme (where every coordinate in a vector is initialized as $\\mathcal{N}\\left(0,\\frac{\\epsilon^2}{d}\\right)$) as \\textit{\\textbf{$\\epsilon$-normal initialization}}.\nIn order to prove the proposition we define a few relevant terms, \n\\begin{equation}\n {\\bm{Y}} = {\\bm{B}} - {\\bm{C}}^\\top,\\qquad {\\bm{W}} = {\\bm{B}} + {\\bm{C}}^\\top,\\qquad w_0=\\hat{{\\bm{C}}}\\hat{{\\bm{B}}}.\n\\end{equation}\n We will in fact prove the stronger, following lemma, for any matrix ${\\bm{A}}$, not necessarily symmetric.\n\n\\begin{lemma}\\label{thm:convergence_to_symmetric_configuration}\nAssume $w_0>0$, and ${\\bm{A}},{\\bm{B}},{\\bm{C}}$ are $\\epsilon$-normally initialized. Then $\\exists t$ such that \n\\begin{equation}\\label{eq:balanced_prop_objective}\n \\lim_{\\epsilon\\rightarrow 0}\\frac{\\|{\\bm{Y}}(t)\\|^2}{\\|{\\bm{W}}(t)\\|^2}=0 ,\\hspace{1cm} \\lim_{\\epsilon\\rightarrow 0}\\frac{\\|{\\bm{A}}(t)\\|_F^2}{\\|{\\bm{W}}(t)\\|^2}=0.\n\\end{equation}\n\\end{lemma}\n\n\nThe proof of Lemma \\ref{thm:convergence_to_symmetric_configuration} follows three steps: (1) establish a time in the optimization for which the norms of all parameters are bounded (Lemma \\ref{lemma:upper_bound_in_time}); (2) derive upper (and lower) bounds for the differential equations describing the evolvement of ${\\bm{Y}},{\\bm{W}}$ and ${\\bm{A}}$. Our approximations are limited to the initial phase of training. Concretely, we show that for $0\\le t\\le \\frac{1}{2w_0}\\ln{\\left(\\frac{1}{\\epsilon^{0.5}} \\right)}$, all norms are bounded. Thus, it is possible to obtain meaningful bounds on the ODEs of ${\\bm{Y}}$ and ${\\bm{W}}$ while ${\\bm{A}}$ remains in the magnitude of initialization (Lemma~\\ref{lemma:Y_and_W_ode}); (3) using the relevant bounds, we show that as the initialization scale tends to zero, so do the limits in \\Eqref{eq:balanced_prop_objective}.\n\n\n\\begin{lemma}\\label{lemma:upper_bound_in_time}\nAssume $d>20$, student parameters are $\\epsilon$-normally initialized, assume also a balanced teacher. Then w.p. at least 0.75, for all $0\\leq t \\leq \\frac{1}{2w_0}\\ln{\\left( \\frac{1}{\\epsilon^{0.5}}\\right)}$, there exist $M_1, M_2$ such that:\n\\begin{equation}\n \\|{\\bm{C}}(t)\\|, \\|{\\bm{B}}(t)\\| < M_1\\epsilon^{0.75}\n\\end{equation}\nand\n\\begin{equation}\n \\|{\\bm{A}}(t)\\|_F < M_2 \\epsilon\n\\end{equation}\n\\end{lemma}\n\nTo prove this, we note that at initialization, ${\\bm{A}},{\\bm{B}}$ and ${\\bm{C}}$ satisfy these bounds. From continuity, there exists a maximal time for which they are satisfied. We bound the rate of their growth, and thus show that for all $t$ as described, we are within this region.\n\n\\begin{lemma}\\label{lemma:Y_and_W_ode}\nAssume $w_0>0$ and assume ${\\bm{A}},{\\bm{B}},{\\bm{C}}$ are $\\epsilon$-normally initialized, we have the following bounds hold for all $0\\le t\\le \\frac{1}{2w_0}\\ln{\\left( \\frac{1}{\\epsilon^{0.5}}\\right)}$ w.p. at least 0.75,\n\\begin{equation}\\label{eq:z_ode}\n {\\bm{Y}}(t)^\\top {\\bm{Y}}(t) \\le c_1\\epsilon^2e^{-2w_0 t} + c_2 \\epsilon^{2.5},\n\\end{equation}\nand\n\\begin{equation}\\label{eq:x_ode}\n {\\bm{W}}(t)^\\top {\\bm{W}}(t) \\ge c_3\\epsilon^2e^{2w_0 t} - c_4 \\epsilon^{2.5}.\n\\end{equation}\n\\end{lemma}\n\nLemma \\ref{lemma:Y_and_W_ode} shows that the growth rate of ${\\bm{W}}(t)^\\top {\\bm{W}}(t)$ and the decay rate of ${\\bm{Y}}(t)^\\top {\\bm{Y}}(t)$ both depend on the sign of $w_0$. In our analysis we assume the teacher is balanced and therefore $w_0>0$, the same analysis applies for $w_0<0$ with opposite roles for ${\\bm{Y}}$ and ${\\bm{W}}$. The proof of Lemma \\ref{lemma:Y_and_W_ode} follows from writing the leading terms of the ODE and bounding the remaining terms by their upper bounds in the time considered.\nUsing these lemmas, we proceed to prove Lemma \\ref{thm:convergence_to_symmetric_configuration}. \n\n\\begin{proof}[\\textbf{Proof of Lemma~\\ref{thm:convergence_to_symmetric_configuration}}]\nConsider the dynamics at time $0\\le t\\le \\frac{1}{2w_0}\\ln{\\left( \\frac{1}{\\epsilon^{0.5}}\\right)}$.\nBy Lemma \\ref{lemma:Y_and_W_ode}, w.p. at least 0.75, we have,\n\\begin{equation}\n {\\bm{Y}}(t)^\\top {\\bm{Y}}(t)\\le c_1\\epsilon^2 e^{-\\ln{\\frac{1}{\\epsilon^{0.5}}}}+c_2 \\epsilon^{2.5}=(c_1+c_2)\\epsilon^{2.5}\n\\end{equation}\nand\n\\begin{equation}\n {\\bm{W}}(t)^\\top {\\bm{W}}(t)\\ge c_3\\epsilon^2 e^{\\ln{\\frac{1}{\\epsilon^{0.5}}}}-c_4 \\epsilon^{2.5}=c_3\\epsilon^{1.5} - c_4\\epsilon^{2.5}\n\\end{equation}\n\nWe can calculate the limit\n\\begin{align}\n \\lim_{\\epsilon\\rightarrow 0} \\frac{\\|{\\bm{Y}}(t)\\|^2}{\\|{\\bm{W}}(t)\\|^2}\\le\\lim_{\\epsilon\\rightarrow 0}\\frac{(c_1+c_2)\\epsilon^{2.5}}{c_3\\epsilon^{1.5}-c_4\\epsilon^{2.5}}=0\n\\end{align}\nFrom Lemma \\ref{lemma:upper_bound_in_time}, $\\|{\\bm{A}}(t)\\|_F\\le M_2\\epsilon$, so we can calculate the limit,\n\n\\begin{equation}\n \\lim_{\\epsilon\\rightarrow 0} \\frac{\\|{\\bm{A}}(t)\\|_F^2}{\\|{\\bm{W}}(t)\\|^2} \\le\\lim_{\\epsilon\\rightarrow 0}\\frac{M_2\\epsilon^2}{c_3\\epsilon^{1.5}-c_4\\epsilon^{2.5}}=0\n\\end{equation}\nwhich concludes the proof.\n\\end{proof}\n\n\n\\begin{proof}[\\textbf{Proof of Lemma~\\ref{lemma:upper_bound_in_time}}]\nApplying Lemma~\\ref{lemma:bound_init} with $d>20$ results with the bounds holding at initialization with probabilities $\\ge 1-2exp(-9\\cdot 20^2\/64)$ for ${\\bm{A}}$, and $\\ge 1-2exp(-9\\cdot 20\/64)$ for ${\\bm{B}}$ and ${\\bm{C}}$. The probability for ${\\bm{A}},{\\bm{B}},{\\bm{C}}$ satisfying the inequalities simultaneously $\\ge \\left(1-2exp(-9\\cdot 20 \/64)\\right)^3\\approx 0.83>0.75$.\n\nSuppose that the norm bounds of \\Eqref{eq:init_bound} are satisfied at $t=0$. In particular, $\\exists M_1,M_2$ such that \n\\begin{equation}\n \\|{\\bm{B}}(0)\\|,\\|{\\bm{C}}(0)\\|<2\\epsilon4\\epsilon^{0.25}$.\n\nDenote by $t_A$ the minimal time for which $\\|{\\bm{A}}(t_A)\\|=M_2\\epsilon$. Similarly, $t_B,t_C$ are the times for which $\\|{\\bm{B}}(t_B)\\|=\\|{\\bm{C}}(t_C)\\|=M_1\\epsilon^{0.75}$.\nDenote $\\tilde{t}=\\min\\lbrace t_A,t_B,t_C\\rbrace$, the minimal time for which the inequalities above are violated. Lastly, denote $t'=\\min\\left\\lbrace \\tilde{t}, \\frac{1}{2w_0}\\ln{\\left(\\frac{1}{\\epsilon^{0.5}}\\right)}\\right\\rbrace$\n\n\n\n\n\nRecall the derivative of ${\\bm{B}}$ with respect to time (see \\Secref{sec:grad_derivation}),\n\\begin{equation}\n \\dot{{\\bm{B}}} = -\\sum_{i=0}^{k-1} \\nabla \\ell_i ({\\bm{A}}^\\top)^i {\\bm{C}}^\\top.\n\\end{equation}\nUsing Cauchy-Schwartz inequality, we have that for all $t\\in[0,t']$, the norm of ${\\bm{B}}$ is upper bounded by\n\\begin{equation}\n\\label{eq:B_dot_CS_inequality1}\n \\left\\|\\dot{{\\bm{B}}}\\right\\| = \\left\\|-\\sum_{i=0}^{k-1} \\nabla \\ell_i ({\\bm{A}}^\\top)^i {\\bm{C}}^\\top\\right\\| \\leq \\sum_{i=0}^{k-1} \\left|\\nabla \\ell_i\\right| \\left\\|{\\bm{A}}^\\top\\right\\|_F^i \\big\\|{\\bm{C}}^\\top\\big\\|.\n\\end{equation}\n\nWe now bound the norms of $\\nabla\\ell_i, {\\bm{A}}$ and ${\\bm{C}}$ in order to transfer the inequality to a differential one. \n\nDenote $M = \\underset{i}{\\max}(|w_i|)+M_1^2\\epsilon^{1.5}$, then we have\n\\begin{align}\n M &= \\underset{i}{\\max}(|w_i|)+M_1^2\\epsilon^{1.5} \\geq \\underset{i}{\\max}(|w_i|)+\\|{\\bm{C}}\\| \\|{\\bm{B}}\\| \\geq \\underset{i}{\\max}(|w_i|)+|{\\bm{C}}{\\bm{B}}|, \\\\ &\\geq \\underset{i}{\\max}(|w_i|)+\\underset{i}{\\max}(|{\\bm{C}}{\\bm{A}}^i {\\bm{B}}|) \\geq \\underset{i}{\\max}(|w_i|+|{\\bm{C}}{\\bm{A}}^i {\\bm{B}}|) \\geq \\underset{i}{\\max}(|\\nabla \\ell_i|).\n\\end{align}\n\nFor the norm of ${\\bm{C}}$, recall the conservation law from Lemma~\\ref{lemma:preserve_norm_diff} for the norms of ${\\bm{B}}$ and ${\\bm{C}}$:\n\\begin{equation}\n \\forall t,\\; \\frac{d}{dt}\\left(\\left\\|{\\bm{B}}(t)\\right\\|-\\left\\|{\\bm{C}}(t)\\right\\|\\right) = 0.\n\\end{equation}\nFrom the assumption that the initial conditions are met,\n\\begin{equation}\n \\|{\\bm{B}}(0)\\|, \\|{\\bm{C}}(0)\\| < 2 \\epsilon \\Rightarrow (|\\|{\\bm{B}}(0)\\|-\\|{\\bm{C}}(0)\\||) < 4 \\epsilon.\n\\end{equation}\nTherefore, we get\n\\begin{equation}\n \\forall t,\\; \\|{\\bm{C}}(t)\\| < \\|{\\bm{B}}(t)\\|+4 \\epsilon.\n\\end{equation}\nNote also that by assuming $M_2 <\\frac{1}{\\epsilon}$, we have $M_2\\epsilon<1$ and\n\\begin{equation}\n \\sum_{i=0}^{k-1}(M_2\\epsilon)^i< k.\n\\end{equation}\nPlugging the above steps into \\eqref{eq:B_dot_CS_inequality1}, we have:\n\\begin{align}\n\\label{eq:B_dot_inequality_bound}\n \\|\\dot{{\\bm{B}}}\\| \\leq \\sum_{i=0}^{k-1} |\\nabla \\ell_i| \\|({\\bm{A}}^\\top)\\|_F^i \\|{\\bm{C}}^\\top\\| &< M (\\|{\\bm{B}}\\|+4 \\epsilon) \\sum_{i=0}^{k-1} \\|({\\bm{A}}^\\top)\\|_F^i \\\\\n &< Mk (\\|{\\bm{B}}\\|+4 \\epsilon).\n\\end{align}\nDenoting $\\gamma = \\|{\\bm{B}}\\|^2 = {\\bm{B}}^\\top {\\bm{B}}$, then\n\\begin{equation}\n \\dot{\\gamma} = \\dot{{\\bm{B}}^\\top}{\\bm{B}} + {\\bm{B}}^\\top\\dot{{\\bm{B}}} = 2{\\bm{B}}^\\top\\dot{{\\bm{B}}}\n\\end{equation}\nTaking absolute value and then plugging \\eqref{eq:B_dot_inequality_bound} results with\n\\begin{equation}\n |\\dot{\\gamma}| = |2{\\bm{B}}^\\top\\dot{{\\bm{B}}}| \\leq 2 \\|{\\bm{B}}^\\top\\| \\|\\dot{{\\bm{B}}}\\| < 2k \\|{\\bm{B}}\\| M (\\|{\\bm{B}}\\|+4 \\epsilon).\n\\end{equation}\nUsing the definition of $\\gamma$, we get that\n\\begin{equation}\n |\\dot{\\gamma}| < 2k M \\left(|\\gamma|+4 \\epsilon \\sqrt{|\\gamma|}\\right)\n\\end{equation}\n\nNotice that for all $t$ such that $\\|{\\bm{B}}(t)\\| = \\sqrt{\\gamma(t)}<4 \\epsilon4\\epsilon^{0.25}$.} Suppose there exists $t_2\\le t'$ such that$\\sqrt{\\gamma(t_2)}>4 \\epsilon$, from continuity and the intermediate value theorem, there exist a maximal $t_1$ such that $0 < t_14 \\epsilon$, therefore the following bound holds,\n\\begin{equation}\n |\\dot{\\gamma}| < 4k M |\\gamma|\n\\end{equation}\nfor all $t\\in[t_1,t_2]$.\nIntegrating by $t$ for $t_{int}\\in [t_1,t_2]$,\n\n\\begin{equation}\n \\int_{t_1}^{t_{int}} \\frac{1}{|\\gamma|}|\\dot{\\gamma}| dt < \\int_{t_1}^{t_{int}}4k M dt,\n\\end{equation}\nsubstituting integration variables and using $0\\le t_1\\le t_{int}\\le t_2\\le \\frac{1}{2w_0}\\ln{\\left(\\frac{1}{\\epsilon^{0.5}}\\right)}$,\n\\begin{equation}\n \\int_{\\gamma(t_1)}^{\\gamma(t_{int})} \\frac{1}{|\\gamma|}d\\gamma \\leq 4k M \\left(t_{int}-t_1\\right) < 4kM \\left(\\frac{1}{2w_0}\\ln{\\left(\\frac{1}{\\epsilon^{0.5}}\\right)}-0\\right).\n\\end{equation}\nThe above evaluates to,\n\\begin{equation}\n \\ln\\left( \\frac{|\\gamma(t_{int})|}{|\\gamma(t_1)|} \\right) < 4k M \\frac{1}{2w_0}\\ln{\\left(\\frac{1}{\\epsilon^{0.5}}\\right)}\n\\end{equation}\nwhich may be further manipulated to reach,\n\\begin{equation}\n |\\gamma(t_{int})| < e^{4k M \\frac{1}{2w_0}\\ln{\\left(\\frac{1}{\\epsilon^{0.5}}\\right)}} |\\gamma(t_1)| = (4 \\epsilon)^2 e^{4k M \\frac{1}{2w_0}\\ln{\\left(\\frac{1}{\\epsilon^{0.5}}\\right)}}.\n\\end{equation}\nThe final bound on the norm of $\\gamma(t_{int})$ is therefore,\n\\begin{equation}\n |\\gamma(t_{int})| < \\frac{16 \\epsilon^2}{\\epsilon^{0.5}} e^{\\frac{4k M}{2w_0}}=16\\epsilon^{1.5}e^{\\frac{4k M}{2w_0}}.\n\\end{equation}\n\nDenoting $M_1^2 = 16 \\cdot e^{\\frac{4k M}{2w_0}}$, and taking the square root of the above,\n\\begin{equation}\n \\|{\\bm{B}}(t_{int})\\| < M_1 \\epsilon^{0.75}.\n\\end{equation}\nWe have shown that for all $0\\le t\\le t'\\le\\frac{1}{w_0}\\ln{\\left(\\frac{1}{\\epsilon^{0.5}}\\right)}$, there exists $M_1$ s.t $\\|{\\bm{B}}(t)\\| |\\nabla \\ell_i|$ and the bounds found for $\\|{\\bm{C}}\\|, \\|{\\bm{B}}\\|0$, $ln(x)|\\nabla \\ell_i|$. We can bound the terms in \\eqref{eq:first_bound_on_Y_TY_dot} by \n\\begin{equation}\n |\\nabla\\ell_i|\\left(|{\\bm{Y}}^\\top({\\bm{A}}^i)_S{\\bm{Y}}| + |{\\bm{Y}}^\\top({\\bm{A}}^i)_{\\bar{S}}{\\bm{W}}|\\right)\\le 8\\cdot M \\cdot M_3^3\\epsilon^{2.5}\n\\end{equation}\n\nPlugging back into \\eqref{eq:first_bound_on_Y_TY_dot}:\n\\begin{equation}\n{\\bm{Y}}^\\top\\dot{{\\bm{Y}}} \\le \\nabla\\ell_0 {\\bm{Y}}^\\top {\\bm{Y}} +\\sum_{i=1}^{k-1} 8\\cdot M \\cdot M_3^3\\epsilon^{2.5}\n\\end{equation}\nWe can also bound $\\nabla\\ell_0=({\\bm{C}}{\\bm{B}}-w_0)\\le -w_0 +|{\\bm{C}}{\\bm{B}}|\\le -w_0+\\|{\\bm{C}}\\|\\|{\\bm{B}}\\|\\le -w_0+M_1^2\\epsilon^{1.5}$. Note also that we multiply by ${\\bm{Y}}^\\top {\\bm{Y}}$ so we can bound\n\\begin{equation}\n \\nabla\\ell_0{\\bm{Y}}^\\top {\\bm{Y}}\\le -w_0 {\\bm{Y}}^\\top {\\bm{Y}} + \\underbrace{M_1^2\\epsilon^{1.5}(M_21\\epsilon^{0.75})^2}_{=4M_1^4\\epsilon^3}\n\\end{equation}\nputting back together, we get\n\\begin{equation}\n{\\bm{Y}}^\\top\\dot{{\\bm{Y}}} \\le -w_0 {\\bm{Y}}^\\top {\\bm{Y}} + 4M_1^4\\epsilon^3 + (k-1)(M_1+1)8\\cdot M \\cdot M_3^3\\epsilon^{2.5}\n\\end{equation}\nIn particular, there exists $M_4$ such that \n\\begin{equation}\n{\\bm{Y}}^\\top\\dot{{\\bm{Y}}} \\le -w_0 {\\bm{Y}}^\\top {\\bm{Y}} + M_4\\epsilon^{2.5}\n\\end{equation}\nRecall that we were interested in bounding $\\frac{d}{dt}({\\bm{Y}}^\\top {\\bm{Y}})=\\dot{{\\bm{Y}}}^\\top {\\bm{Y}} + {\\bm{Y}}^\\top \\dot{{\\bm{Y}}}=2{\\bm{Y}}^\\top\\dot{{\\bm{Y}}}$,\n\\begin{equation}\n \\frac{d}{dt}({\\bm{Y}}^\\top {\\bm{Y}})\\le -2w_0{\\bm{Y}}^\\top {\\bm{Y}}+M_4\\epsilon^{2.5}\n\\end{equation}\nDenoting $z(t)\\equiv {\\bm{Y}}(t)^\\top {\\bm{Y}}(t)$ and $x(t)\\equiv {\\bm{W}}(t)^\\top {\\bm{W}}(t)$, and using Lemma \\ref{lemma:integral_bound}, we have the desired bounds\n\\begin{equation}\n z(t) \\le \\frac{1}{2w_0}\\left[(33w_0 d \\epsilon^2)e^{-2w_0 t} + M_4 \\epsilon^{2.5} \\right]\n\\end{equation}\nIn particular, we can write\n\\begin{equation}\n z(t) \\le c_1\\epsilon^2e^{-2w_0 t} + c_2 \\epsilon^{2.5}\n\\end{equation}\n\nand\n\\begin{equation}\n x(t) \\ge c_3\\epsilon^2e^{2w_0 t} - c_4 \\epsilon^{2.5}\n\\end{equation}\nwhere $c_i$'s are positive constants.\n\nNote that the derivation of $x(t)$ is exactly the same as $z(t)$ with opposite signs and bounding from below instead.\n\n\\end{proof}\n\n\\subsubsection{Integral bound of differential equations}\n\n\n\\begin{lemma}\\label{lemma:integral_bound}\nAssume $\\dot{z}< -2w_0 z +M_4 \\epsilon^{2.5} <0, \\dot{x}> 2w_0 z -M_4 \\epsilon^{2.5} >0$ , where $w_0>0$. \nThen, under the assumptions of Lemma \\ref{lemma:bound_init}:\n\\begin{equation}\n z(t_1) < \\frac{1}{2w_0}\\left(\\exp(-2w_0t_1) \\cdot (75 w_0 d \\epsilon^2) + M_4 \\epsilon^{2.5} \\right)\n\\end{equation}\n\\begin{equation}\n x(t_2) > \\frac{1}{2w_0}\\left(\\exp(2w_0t_2) \\cdot (\\frac{w_0\\epsilon^2}{25d}) + M_4 \\epsilon^{2.5} \\right)\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}[\\textbf{Proof of Lemma~\\ref{lemma:integral_bound}}]\nAssume $\\dot{z}< -2w_0 z +M_4 \\epsilon^{2.5}$ , where $w_0>0, 2w_0z>M_4 \\epsilon^{2.5}$.\nSimilarly, assume $\\dot{x}> 2w_0 x -M_4 \\epsilon^{2.5}$.\n\nThen:\n\\begin{equation}\n \\frac{\\dot{z}}{2w_0 z -M_4 \\epsilon^{2.5}}< -1\n\\end{equation}\n\\begin{equation}\n \\frac{\\dot{x}}{2w_0 x -M_4 \\epsilon^{2.5}}>1\n\\end{equation}\n\nIntegrating both sides by dt, and using integration by substitution, we get:\n\\begin{equation}\n -t_1 = \\int_0^{t_1} -1 > \\int_0^{t_1} \\frac{1}{2w_0 z -M_4 \\epsilon^{2.5}} \\frac{dz}{dt} dt = \\int_{z(0)}^{z(t_1)} \\frac{1}{2w_0 z -M_4 \\epsilon^{2.5}} dz\n\\end{equation}\n\\begin{equation}\n t_2 = \\int_0^{t_2} 1 < \\int_0^{t_2} \\frac{1}{2w_0 x -M_4 \\epsilon^{2.5}} \\frac{dx}{dt} dt = \\int_{x(0)}^{x(t_2)} \\frac{1}{2w_0 x -M_4 \\epsilon^{2.5}} dx\n\\end{equation}\n\nWe note:\n\\begin{align}\n \\int_{z(0)}^{z(t_1)} \\frac{1}{2w_0 z -M_4 \\epsilon^{2.5}} dz &= \\frac{1}{2w_0} [\\ln(2w_0 z(t_1) -M_4 \\epsilon^{2.5}) -\\ln(2w_0 z(0) -M_4 \\epsilon^{2.5} )] \\\\\n \\Rightarrow\\int_{z(0)}^{z(t_1)} \\frac{1}{2w_0 z -M_4 \\epsilon^{2.5}} dz &= \\frac{1}{2w_0} \\left[ \\ln \\left( \\frac{2w_0 z(t_1) -M_4 \\epsilon^{2.5} }{2w_0 z(0) -M_4 \\epsilon^{2.5} }\\right) \\right] \\\\\n \\int_{x(0)}^{x(t_2)} \\frac{1}{2w_0 x -M_4 \\epsilon^{2.5}} dx &= \\frac{1}{2w_0} \\left[ \\ln \\left( \\frac{2w_0 x(t_2) -M_4 \\epsilon^{2.5} }{2w_0 x(0) -M_4 \\epsilon^{2.5} }\\right) \\right] \n\\end{align}\nCombining equations, we have:\n\\begin{equation}\n \\frac{1}{2w_0} \\left[ \\ln \\left( \\frac{2w_0 z(t_1) -M_4 \\epsilon^{2.5} }{2w_0 z(0) -M_4 \\epsilon^{2.5} } \\right) \\right] < -t_1\n\\end{equation}\n\\begin{equation}\n \\Rightarrow \\ln \\left( \\frac{2w_0 z(t_1) -M_4 \\epsilon^{2.5} }{2w_0 z(0) -M_4 \\epsilon^{2.5} } \\right) < -2w_0t_1\n\\end{equation}\n\\begin{equation}\n \\Rightarrow \\frac{2w_0 z(t_1) -M_4 \\epsilon^{2.5} }{2w_0 z(0) -M_4 \\epsilon^{2.5} } < \\exp(-2w_0t_1)\n\\end{equation}\n\\begin{equation}\n \\Rightarrow 2w_0 z(t_1) -M_4 \\epsilon^{2.5} < \\exp(-2w_0t_1) \\cdot \\left(2w_0 z(0) -M_4 \\epsilon^{2.5} \\right)\n\\end{equation}\n\\begin{equation}\n \\Rightarrow z(t_1) < \\frac{1}{2w_0}\\left[\\exp(-2w_0t_1) \\cdot (2w_0 z(0) -M_4 \\epsilon^{2.5}) + M_4 \\epsilon^{2.5} \\right]\n\\end{equation}\n\n\\begin{equation}\n x(t_2) > \\frac{1}{2w_0}\\left[\\exp(2w_0t_2) \\cdot (2w_0 x(0) -M_4 \\epsilon^{2.5}) + M_4 \\epsilon^{2.5} \\right]\n\\end{equation}\n\nNote that $z(0) = {\\bm{Y}}^\\top(0){\\bm{Y}}(0)$. \nBy linearity of sum of variances, ${\\bm{Y}}(0)$'s entries are distributed according to $\\mathcal{N}(0,\\sqrt{2}\\cdot \\epsilon)$, by Lemma~\\ref{lemma:bound_init}:\n\\begin{equation}\n \\frac{\\sqrt{2}}{2}\\epsilon<\\|{\\bm{Y}}(0)\\|<\\frac{3\\sqrt{2}}{2}\\epsilon\n\\end{equation}\nFrom Cauchy-Schwartz, $z(0)<3\\epsilon^2$ with high probability.\n${\\bm{W}}$ is distributed as ${\\bm{Y}}$, therefore, $x(0)> \\frac{1}{2}\\epsilon^2$.\nAssuming $M_4 \\epsilon^{2.5} < \\frac{w_0 \\epsilon^2}{2}$, we have:\n\n\\begin{equation}\n z(t_1) < \\frac{1}{2w_0}\\left(\\exp(-2w_0t_1) \\cdot (6 w_0 \\epsilon^2) + M_4 \\epsilon^{2.5} \\right)\n\\end{equation}\n\\begin{equation}\n x(t_2) > \\frac{1}{2w_0}\\left(\\exp(2w_0t_2) \\cdot (w_0\\epsilon^2) + M_4 \\epsilon^{2.5} \\right)\n\\end{equation}\n\nConcluding the proof.\n\\end{proof}\n\\section{Conclusion}\\label{sec:conclusions}\n\nThis paper studies the question of extrapolation in RNNs, and more specifically, of whether a student RNN trained on data generated by a teacher RNN can capture the behavior of the teacher over sequences longer than those seen in training. \nWe focus on overparameterized students that can perfectly fit training sequences while producing a wide range of behaviors over longer sequences. \nSuch a student will fail to extrapolate, unless the teacher possesses a certain structure, and the learning algorithm is biased towards solutions adhering to that structure.\nWe show~---~theoretically for linear RNNs and empirically for both linear and non-linear RNNs~---~that such implicit extrapolation takes place when the teacher has a low dimensional state space and the learning algorithm is GD\n\nExisting studies of implicit extrapolation in (linear) RNNs \\citep{emami2021implicit,cohen2022extrapolation} suggest that GD is biased towards solutions with short-term memory.\nWhile low dimensional state space and short-term memory may coincide in some cases, in general they do not, and a solution with low dimensional state space may entail long-term memory.\nOur theory and experiments show that in settings where low dimensional state space and short-term memory contradict each other, the implicit extrapolation chooses the former over the latter\n\nWe note that linear RNNs fundamentally differ from the commonly studied model of linear (feed-forward) NNs (see, e.g., \\cite{arora2018optimization,ji2018gradient,arora2019convergence,arora2019implicit,razin2020implicit}). \nOne of the key differences is that a linear RNN entails different powers of a parameter (transition) matrix, leading to a loss function which roughly corresponds to a sum of losses for multiple linear NNs having different architectures and shared weights.\nThis makes the analysis of linear RNNs technically challenging, warranting introduction of new theoretical approaches such as the moment matching view employed in \\Secref{sec:exact_extrapolation}. \n\nAn important direction for future work is extending our theory to non-linear RNNs. \nWe believe it is possible, in the same way that theories for linear (feed-forward) NNs were extended to account for non-linear NNs (see, e.g., \\citet{razin2021implicit,razin2022implicit, lyu2019gradient}).\nAn additional direction to explore is the applicability of our results to the recently introduced S4 model \\citep{gu2022efficiently}.\n\\section{Experiments}\n\\label{sec:experiments}\n\nIn this section we present experiments corroborating our theoretical analysis (\\Secref{sec:analysis}).\nThe latter establishes that, under certain conditions, a linear RNN with state space dimension~$d$ extrapolates when learning from a teacher network with state space dimension~\\smash{$\\hat{d}$} via training sequences of length~$k$, irrespective of how large~$d$ is compared to \\smash{$\\hat{d}$} and~$k$.\nA key condition underlying the result is that $k$ is larger than $2 \\hat{d}$.\n\\Secref{sec:exp:sym_teacher} below considers the theoretically analyzed setting, and empirically evaluates extrapolation as~$k$ varies.\nIts results demonstrate a phase transition, in the sense that extrapolation takes place when \\smash{$k > 2 \\hat{d}$}, in compliance with theory, but on the other hand fails when $k$ falls below~\\smash{$2 \\hat{d}$}, in which case the theory indeed does not guarantee extrapolation.\n\\Secref{sec:exp:step_func} displays the same phenomenon with linear RNNs that do not adhere to some of the assumptions made by the theory (in particular the assumption of symmetric transition matrices, and those concerning balancedness).\nFinally, \\Secref{sec:exp:non_linear_teacher} considers non-linear RNNs (specifically, Gated Recurrent Unit networks \\cite{chung2014empirical}), and shows that they too exhibit a phase transition in extrapolation as the training sequence length varies.\nFor brevity, we defer some of the details of our implementation, as well as additional experiments, to \\Appref{sec:apdx:additional_experiments}.\n\n\\subsection{Theoretically Analyzed Setting} \\label{sec:exp:sym_teacher}\n\nOur first experiment considers the setting described in \\Secref{sec:setup} and theoretically analyzed in \\Secref{sec:analysis}.\nAs representative values for the state space dimensions of the teacher and (overparameterized) student, we choose \\smash{$\\hat{d} = 5$} and $d = 40$ respectively (higher state space dimensions for the student, namely $d = 100$ and $d = 200$, yielded qualitatively identical results).\nFor a given training sequence length~$k$, the student is learned via GD applied directly to the population loss defined in \\Eqref{eq:population_loss} (applying GD to the empirical loss defined in \\Eqref{eq:empirical_loss}, with $N = 10,000$ training examples, led to similar results).\n\\Figref{fig:phase_transition_v2}(a) reports the extrapolation error (quantified by the $\\ell_\\infty$ distance between the impulse response of the learned student and that of the teacher) as a function of~$k$.\nAs can be seen, extrapolation exhibits a phase transition that accords with our theory: when \\smash{$k > 2 \\hat{d}$} extrapolation error is low, whereas when $k$ falls below~\\smash{$2 \\hat{d}$} extrapolation error is high.\n\n\\subsection{Other Settings With Linear Recurrent Neural Networks} \\label{sec:exp:step_func}\n\nTo assess the generality of our findings, we experiment with linear RNNs in settings that do not adhere to some of the assumptions made by our theory.\nSpecifically, we evaluate settings in which:\n\\emph{(i)}~the teacher is unbalanced, meaning \\smash{$\\hat{{\\bm{B}}} \\neq \\hat{{\\bm{C}}}^\\top$}, and its transition matrix~\\smash{$\\hat{{\\bm{A}}}$} is non-symmetric;\n\\emph{(ii)}~the student's transition matrix~${\\bm{A}}$ is not restricted to be symmetric;\n\\emph{(iii)}~learning is implemented by optimizing the empirical loss defined in \\Eqref{eq:empirical_loss} (rather than the population loss defined in \\Eqref{eq:population_loss});\nand\n\\emph{(iv)}~optimization is based on Adam \\cite{kingma2014adam} (rather than GD), emanating from standard near-zero initialization which is generally unbalanced (namely, ${\\bm{B}}\\neq {\\bm{C}}^\\top$).\n\\Figref{fig:phase_transition_v2}(b) reports the results of an experiment where the state space dimensions of the teacher and (overparameterized) student are \\smash{$\\hat{d} = 10$} and $d = 50$ respectively (higher state space dimensions for the student, namely $d = 100$ and $d = 200$, yielded qualitatively identical results), and where the teacher implements a delay line of \\smash{$\\hat{d}$} time steps (for details see \\Appref{sec:apdx:unbalanced_teacher_generation}).\nSimilar results obtained with randomly generated teachers are reported in \\Appref{sec:apdx:additional_experiments}.\nAs can be seen, despite the fact that our theory does not apply to the evaluated settings, its conclusions still hold~---~extrapolation error is low when the training sequence length~$k$ is greater than~\\smash{$2 \\hat{d}$}, and high when $k$ falls below~\\smash{$2 \\hat{d}$}.\n\n\\begin{figure}[H]\n \\centering\n \\subfigure[Theoretically analyzed setting]{\\includegraphics[width=0.45\\textwidth,height=3.6cm]{figures\/extrapolation_as_func_of_k_theoretical_setup.png}}\n \\subfigure[Other setting with linear RNN]{\\includegraphics[width=0.45\\textwidth,height=3.6cm]{figures\/extrapolation_as_func_of_k_step_func_init_1e-06_v2.png}} \n \\caption{\n Demonstration of implicit extrapolation with linear RNNs.\n Plots show extrapolation error (average over three random seeds, with shaded region marking standard deviation) as a function of training sequence length~$k$, for a student with state space dimension~$d$ learning from a teacher with state space dimension~\\smash{$\\hat{d}$}, where \\smash{$d \\gg \\hat{d}$}.\n (a) Models adhere to the setting described in \\Secref{sec:setup} and theoretically analyzed in \\Secref{sec:analysis}, with \\smash{$\\hat{d} = 5$}, $d = 40$.\n (b) Models do not adhere to some of the assumptions made by the theory, and \\smash{$\\hat{d} = 10$}, $d = 50$.\n Notice that extrapolation exhibits a phase transition that accords with theory~---~when \\smash{$k > 2 \\hat{d}$} extrapolation error is low, and when $k$ falls below~\\smash{$2 \\hat{d}$} extrapolation error is high.\n For further details see Sections \\ref{sec:exp:sym_teacher} and~\\ref{sec:exp:step_func} and \\Appref{sec:apdx:additional_experiments}.\n }\n \\label{fig:phase_transition_v2}\n\\end{figure}\n\n\\subsection{Non-Linear Recurrent Neural Networks} \\label{sec:exp:non_linear_teacher}\n\nAs a final experiment, we explore implicit extrapolation with non-linear RNNs, namely GRU networks.\nSpecifically, we evaluate the extent to which a student GRU with state space dimension $d_g = 100$ extrapolates when learning from a teacher GRU with state space dimension \\smash{$\\hat{d}_g = 10$} (higher state space dimensions for the student, namely $d_g = 200$ and $d_g = 500$, yielded qualitatively identical results).\nThe student is learned by optimizing an empirical loss comprising training sequences of length~$k_g$, where $k_g$ is predetermined.\nOptimization is based on Adam emanating from standard near-zero initialization.\n\\figref{fig:gru_extrapolation}(a) reports the extrapolation error (quantified by the $\\ell_\\infty$ distance between the response of the learned student and that of the teacher, averaged across randomly generated input sequences) for different choices of~$k_g$.\nAs can be seen, similarly to the case with linear RNNs (see Sections \\ref{sec:exp:sym_teacher} and~\\ref{sec:exp:step_func}), there exists a critical threshold for the training sequence length~$k_g$, above which extrapolation error is low and below which extrapolation error is high.\\footnote{%\nNote that this critical threshold is around four times the teacher's state space dimension, whereas with linear RNNs the critical threshold was around two times the teacher's state space dimension.\nTheoretically explaining this difference is viewed as an interesting direction for future work.\n}\n\\figref{fig:gru_extrapolation}(b) displays the average output response over different inputs of the teacher alongside those of two students~---~one trained with sequences of length $k_g = 30$, and the other with sequences of length $k_g = 60$.\\footnote{%\nNote that with GRU networks, in contrast to linear RNNs, the impulse response does not identify the input-output mapping realized by a network.\nIt is presented in \\figref{fig:gru_extrapolation}(b) for demonstrative purposes.\n}\nAs expected, the impulse response of each student tracks that of the teacher for the first $k_g$ time steps (where $k_g$ is student-dependent).\nHowever, while the student for which $k_g = 30$ fails to track the teacher beyond $k_g$ time steps, the student for which $k_g = 60$ succeeds, thereby exemplifying implicit extrapolation.\n\n\\begin{figure}\n\\centering\n\\subfigure[Extrapolation vs.~training sequence length]\n{\\includegraphics[width=0.45\\textwidth,height=3.6cm]{figures\/extrapolation_as_func_of_k_gru.png}}\n\\subfigure[Average output over several inputs]\n{\\includegraphics[width=0.52\\textwidth,height=3.6cm]{figures\/gru_impulse_responses.png}}\n\\caption{\nDemonstration of implicit extrapolation with non-linear RNNs, namely GRU networks.\nPlots show results for a student with state space dimension $d_g = 100$ learning from a teacher with state space dimension \\smash{$\\hat{d}_g = 10$} using training sequences of length~$k_g$, where $k_g$ varies.\n(a)~Extrapolation error (average over ten random seeds, with shaded region marking standard deviation) as a function of~$k_g$.\n(b)~Average output over several inputs of teacher and student for different choices of~$k_g$.\nNotice that, similarly to the case with linear RNNs, there exists a critical threshold for~$k_g$ above which extrapolation error is low and below which extrapolation error is high.\nSee more details in \\Appref{sec:apdx:additional_experiments}.\n}\n\\label{fig:gru_extrapolation}\n\\end{figure}\n\\section{Introduction}\n\nNeural Networks (NNs) are often \\emph{overparameterized}, in the sense that their representational capacity far exceeds what is necessary for fitting training data.\nSurprisingly, training overparameterized NNs via (variants of) Gradient Descent (GD) tends to produce solutions that generalize well, despite existence of many solutions that do not.\nThis \\emph{implicit generalization} phenomenon attracted considerable scientific interest, resulting in various theoretical explanations (see, e.g., \\cite{woodworth2020kernel,yun2020unifying,ZhangBHRV17,li2020towards,ji2018gradient,lyu2019gradient}).\n\nRecent studies have surfaced a new form of implicit bias which arises in Recurrent Neural Networks (RNNs) and their variants (e.g., Long Short-Term Memory \\cite{lstm} and Gated Recurrent Units \\cite{chung2014empirical}).\nFor such models, the length of sequences in training is often shorter than in testing, and it is not clear to what extent a learned solution will be able to \\emph{extrapolate} beyond the sequence lengths seen in training.\nIn the overparameterized regime, where the representational capacity of the learned model exceeds what is necessary for fitting short sequences, there may exist solutions which generalize but do not extrapolate, meaning that their accuracy is high over short sequences but arbitrarily poor over long ones (see \\cite{cohen2022extrapolation}). \nIn practice however, when training RNNs using GD, accurate extrapolation is often observed. \nWe refer to this phenomenon as the \\emph{implicit extrapolation} of~GD. \n\nAs opposed to the implicit generalization of GD, little is formally known about its implicit extrapolation. \nExisting theoretical analyses of the latter focus on linear RNNs~---~also known as \\emph{Linear Dynamical Systems} (LDS)~---~and either treat infinitely wide models \\citep{emami2021implicit}, or models of finite width that learn from a memoryless teacher \\citep{cohen2022extrapolation}.\nIn these regimes, GD has been argued to exhibit an implicit bias towards short-term memory.\nWhile such results are informative, their generality remains in question, particularly since infinitely wide NNs are known to substantially differ from their finite-width counterparts, and since a memoryless teacher essentially neglects the main characteristic of RNNs (memory).\n\nIn this paper, we theoretically investigate the implicit extrapolation of GD when applied to overparameterized finite-width linear RNNs learning from a teacher with memory. \nWe consider models with symmetric transition matrices, in the case where a student (learned model) with state space dimension~$d$ is trained on sequences of length~$k$ generated by a teacher with state space dimension~\\smash{$\\hat{d}$}.\nOur interest lies in the overparameterized regime, where $d$ is greater than both $k$ and~\\smash{$\\hat{d}$}, meaning that the student has state space dimension large enough to fully agree with the teacher on sequences of length~$k$, while potentially disagreeing with it on longer sequences. As a necessary assumption on initialization, we follow prior work and focus on a certain balancedness condition, which is known (see experiments in \\cite{cohen2022extrapolation}, as well as our theoretical analysis) to capture near-zero initialization as commonly employed in practice.\n\nOur main theoretical result states that GD originating from a balanced initialization leads the student to extrapolate, \\emph{irrespective of how large its state space dimension is}. \nKey to the result is a surprising connection to a moment matching theorem from \\cite{cohen2011use}, whose proof relies on ideas from compressed sensing \\citep{Elad2010Sparse,Eldar2021CompressedSensing} and neighborly polytopes \\citep{gale1963neighborly}. \nThis connection may be of independent interest, and in particular may prove useful in deriving other results concerning implicit properties of GD.\nWe corroborate our theory with experiments, which demonstrate extrapolation via learning low dimensional state spaces in both the analyzed setting and ones involving non-linear RNNs.\n\nThe implicit extrapolation of GD is a new and exciting area of inquiry.\nOur results suggest that short-term memory is not enough for explaining it as previously believed.\nWe hope the techniques developed in this paper will contribute to a further understanding of this phenomenon.\n\\section{Linear Recurrent Neural Networks \\label{sec:setup}}\n\nOur theoretical analysis applies to single-input single-output (SISO) linear RNNs with symmetric transition matrices.\nGiven a state space dimension $d \\in \\mathbb{N}$, this model is defined by the update rule:\n\\begin{equation}\n \\label{eq:lin_rnn}\n {\\bm{s}}_{t+1}={\\bm{A}} {\\bm{s}}_t+{\\bm{B}} x_t,\n \\quad\n y_t={\\bm{C}} {\\bm{s}}_t,\n \\quad\n t = 0 , 1 , 2 , \\ldots\n \\text{\\,,}\n\\end{equation}\nwhere\n${\\bm{A}} \\in\\mathbb{R}^{d\\times d}$, ${\\bm{B}} \\in \\mathbb{R}^{d\\times 1}$ and ${\\bm{C}} \\in \\mathbb{R}^{1\\times d}$ are configurable parameters, with the \\emph{transition matrix}~${\\bm{A}}$ satisfying ${\\bm{A}} = {\\bm{A}}^\\top$;\n$x_0 , x_1 , \\ldots \\in \\mathbb{R}$ form an input sequence;\n$y_0 , y_1 , \\ldots \\in \\mathbb{R}$ form the corresponding output sequence; and \n${\\bm{s}}_t\\in\\mathbb{R}^{d\\times 1}$ represents the internal state at time~$t$, assumed to be equal to zero at the outset (i.e.~it is assumed that ${\\bm{s}}_0=\\mathbf{0}$).\nAs with any linear time-invariant system \\citep{porat1996course}, the input-output mapping realized by the RNN is determined by its impulse response.\n\\begin{definition}\\label{def:impulse_response}\nThe \\textbf{impulse response} of the RNN is the output sequence corresponding to the input sequence $( x_0 , x_1 , x_2 , \\ldots ) = ( 1, 0 , 0 , \\ldots )$.\nNamely, it is the sequence $({\\bm{C}}{\\bm{B}},{\\bm{C}}{\\bm{A}}{\\bm{B}},{\\bm{C}}{\\bm{A}}^2{\\bm{B}},\\dots)$.\n\\end{definition}\nFor brevity, we employ the shorthand $\\Theta := ({\\bm{A}},{\\bm{B}},{\\bm{C}})$.\nThe $d \\times d$ symmetric transition matrix~${\\bm{A}}$ is parameterized through a $d ( d + 1 ) \/2$-dimensional vector holding its upper triangular elements, and with a slight overloading of notation, the symbol~${\\bm{A}}$ is also used to refer to this parameterization.\n\nWe note that our theory readily extends to multiple-input multiple-output (MIMO) networks, and the focus on the SISO case is merely for simplicity of presentation.\nNote also that the restriction to symmetric transition matrices is customary in both theory \\citep{hazan2018spectral} and practice \\citep{gupta2022diagonal}, and represents a generalization of the \\emph{canonical modal form}, which under mild non-degeneracy conditions does not limit generality \\citep{boyd2006ee263}.\n\nGiven a length~$k$ input sequence, $\\mathbf{x}=(x_0,\\dots,x_{k-1}) \\in \\mathbb{R}^k$, consider the output at the last time step, i.e.~$y:=y_k \\in \\mathbb{R}$, and denote it by~$RNN ( {\\bm{x}} )$.\nUsing this output as a label, we define an empirical loss induced by a training set $S=\\left\\lbrace \\left({\\bm{x}}^{(1)},y^{(1)}\\right),\\dots, \\left({\\bm{x}}^{(N)},y^{(N)}\\right) \\right\\rbrace \\subset \\mathbb{R}^k \\times \\mathbb{R}$:\n\\begin{equation}\n \\mathcal{L}_S({\\bm{A}},{\\bm{B}},{\\bm{C}})=\\frac{1}{N}\\sum_{i=1}^N \\ell\\left( RNN\\left({\\bm{x}}^{(i)}\\right), y^{(i)}\\right),\n\\end{equation}\nwhere $\\ell(y,\\hat{y})=(y-\\hat{y})^2$ is the square loss. \nBy the update rule of the RNN (\\Eqref{eq:lin_rnn}), we have:\n\\begin{equation}\\label{eq:empirical_loss}\n \\mathcal{L}_S({\\bm{A}},{\\bm{B}},{\\bm{C}}) = \\frac{1}{N}\\sum_{i=1}^N \\left( \\sum_{j=0}^{k-1}{\\bm{C}}{\\bm{A}}^{k-1-j}{\\bm{B}} x_j^{(i)} - y^{(i)} \\right)^2.\n\\end{equation}\nSuppose that ground truth labels are generated by an RNN as defined in \\Eqref{eq:lin_rnn}, and denote the state space dimension and parameters of this \\emph{teacher} network by \\smash{$\\hat{d}$} and \\smash{$\\hat{\\Theta} = (\\hat{{\\bm{A}}},\\hat{{\\bm{B}}},\\hat{{\\bm{C}}})$} respectively.\nWe employ the common assumption (e.g., see \\cite{hardt2016gradient}) by which input sequences are drawn from a whitened distribution, i.e.~a distribution where $\\mathbb{E}\\left[ x_j x_{j'} \\right]$ equals~$1$ if $j = j'$ and~$0$ otherwise.\nThe population loss over length~$k$ sequences can then be written as (see Lemma~\\ref{lemma:expected_loss}):\n\\begin{equation}\\label{eq:population_loss}\n \\mathcal{L}({\\bm{A}},{\\bm{B}},{\\bm{C}})=\\sum_{j=0}^{k-1} \\left({\\bm{C}}{\\bm{A}}^j {\\bm{B}}-\\hat{{\\bm{C}}}\\hat{{\\bm{A}}}^j\\hat{{\\bm{B}}}\\right)^2\n \\text{\\,.}\n\\end{equation}\n\\Eqref{eq:population_loss} implies that a solution $\\Theta = ( {\\bm{A}} , {\\bm{B}} , {\\bm{C}} )$ achieves zero population loss over length~$k$ sequences if and only if \\smash{${\\bm{C}} {\\bm{A}}^j {\\bm{B}}=\\hat{{\\bm{C}}} \\hat{{\\bm{A}}}^j \\hat{{\\bm{B}}}$} for $j=0,\\dots,k-1$.\nTo what extent does such a solution imply that the \\emph{student} (i.e., the learned RNN) extrapolates to longer sequences?\nThis depends on how close ${\\bm{C}} {\\bm{A}}^j {\\bm{B}}$ is to \\smash{$\\hat{{\\bm{C}}} \\hat{{\\bm{A}}}^j \\hat{{\\bm{B}}}$} for $j \\geq k$.\n\\begin{definition}\\label{def:extrapolation}\nFor $\\epsilon \\geq 0$ and $q \\in \\mathbb{N}$, we say that the student \\textbf{$\\epsilon$-extrapolates with horizon $q$} with respect to (w.r.t) the teacher if: \n\\begin{equation}\n |{\\bm{C}}{\\bm{A}}^j {\\bm{B}}-\\hat{{\\bm{C}}}\\hat{{\\bm{A}}}^j\\hat{{\\bm{B}}}| \\leq \\epsilon,\n \\quad\n \\forall j \\in \\{ 0 , 1 , \\ldots , q - 1 \\}\n \\text{\\,.}\n\\end{equation}\nIf the above holds for all $q \\in \\mathbb{N}$ then the student is said to \\textbf{$\\epsilon$-extrapolate} w.r.t the teacher, and if it holds for all $q \\in \\mathbb{N}$ with $\\epsilon = 0$ then the student is simply said to \\textbf{extrapolate} w.r.t the teacher.\n\\end{definition}\nPer Definition~\\ref{def:extrapolation}, $\\epsilon$-extrapolation with horizon~$q$ is equivalent to the first $q$~elements of the student's impulse response being $\\epsilon$-close to those of the teacher's, whereas extrapolation means that the student's impulse response fully coincides with the teacher's.\nThe latter condition implies that the student realizes the same input-output mapping as the teacher, for any sequence length (this corresponds to the notion of system identification; see \\Secref{sec:related_work}).\nWhen the student is \\emph{overparameterized}, in the sense that $d$ is greater than $k$ and~\\smash{$\\hat{d}$}, it may perfectly generalize, i.e.~lead the population loss over length~$k$ sequences (\\Eqref{eq:population_loss}) to equal zero, and yet fail to extrapolate as stated in the following proposition.\n\\begin{proposition}\\label{prop:symmetric_lds_expressivity}\nAssume $d > k$, and let $\\epsilon \\geq 0$ and $q \\in \\{ k + 1 , k + 2 , \\ldots\\}$.\nThen, for any teacher parameters \\smash{$\\hat{\\Theta}$}, there exist student parameters $\\Theta$ with which the population loss in \\Eqref{eq:population_loss} equals zero, and yet the student does \\emph{\\textbf{not}} $\\epsilon$-extrapolate with horizon $q$.\n\\end{proposition}\n\\begin{sproof}[Proof sketch (for complete proof see \\Appref{sec:apdx:perfect_generalization_failed_extrapolation}).]\nThe result follows from the fact that the first~$d$ elements of the student's impulse response can be assigned freely via a proper choice of~$\\Theta$.\n\\end{sproof}\nWe are interested in the extent to which student parameters learned by GD extrapolate in the overparameterized regime.\nProposition~\\ref{prop:symmetric_lds_expressivity} implies that, regardless of how many (length~$k$) sequences are used in training, if GD leads to any form of extrapolation, it must be a result of some implicit bias induced by the algorithm.\nNote that in our setting, extrapolation cannot be explained via classic tools from statistical learning theory, as evaluation over sequences longer than those seen in training violates the standard assumption of train and test data originating from the same distribution.\n\nTo decouple the question of extrapolation from that of generalization, we consider the case where the training set~$S$ is large, or more formally, where the empirical loss~$\\mathcal{L}_S ( \\cdot )$ (\\Eqref{eq:empirical_loss}) is well represented by the population loss~$\\mathcal{L} ( \\cdot )$ (\\Eqref{eq:population_loss}). \nWe model GD with small step size via Gradient Flow (GF), as customary in the theory of NNs~---~see \\citet{saxe2013exact, gunasekar2017implicit, arora2018optimization, arora2019implicit, lyu2019gradient, li2020towards, azulay2021implicit} for examples and~\\cite{elkabetz2021continuous} for theoretical justification.\nThat is, we analyze the following dynamics:\n\\begin{equation}\n \\dot{\\alpha} ( \\tau ) := \\frac{d}{d \\tau} \\alpha ( \\tau ) = - \\frac{\\partial}{\\partial \\alpha} \\mathcal{L} \\big( {\\bm{A}} ( \\tau ) , {\\bm{B}} ( \\tau ) , {\\bm{C}} ( \\tau ) \\big)\n ~ , ~\n \\tau \\geq 0\n \\text{\\,,}\n\\end{equation}\nwhere $\\alpha\\in\\lbrace {\\bm{A}},{\\bm{B}},{\\bm{C}} \\rbrace$. \nIf no assumption on initialization is made, no form of extrapolation can be established (indeed, the initial point may be a global minimizer of~$\\mathcal{L} ( \\cdot )$ that fails to extrapolate, and GF will stay there).\nFollowing prior work (see \\cite{cohen2022extrapolation}), we assume that initialization adheres to the following balancedness condition:\n\\begin{definition}\\label{def:balanced}\nAn RNN with parameters $\\Theta = ({\\bm{A}},{\\bm{B}},{\\bm{C}})$ is said to be \\textbf{balanced} if ${\\bm{B}}={\\bm{C}}^\\top$.\n\\end{definition}\nIt was shown empirically in \\cite{cohen2022extrapolation} that the balancedness condition captures near-zero initialization as commonly employed in practice.\nWe support this finding theoretically in Section~\\ref{sec:relaxed_balanced_assumption}.\nAside from the initialization of the student, we will also assume that the teacher adheres to the balancedness condition.\n\n\n\n\n\n\n\\section{Related Work}\\label{sec:related_work}\nThe study of linear RNNs, or LDS, has a rich history dating back to at least the early works of Kalman \\citep{kalman1960general, kalman1963mathematical}. \nAn extensively studied question relevant to extrapolation is that of \\textit{system identification}, which explores when the parameters of a teacher LDS can be recovered (see \\cite{ljung1999system}).\nAnother related topic concerns finding compact realizations of systems, i.e.~realizations of the same input-output mapping as a given LDS, with a state space dimension that is lower (see \\cite{antoulas2005approximation}).\nDespite the relation, our focus is fundamentally different from the above~---~we ask what happens when one learns an LDS using GD. \nSince GD is not explicitly designed to find a low dimensional state space, it is not clear that the application of GD to an overparameterized student allows system identification through a compact realization. \nThe fact that it does relates to the implicit properties of GD, and to our knowledge has not been investigated in the classic LDS literature.\n\nThe implicit generalization of GD in training RNNs has been a subject of theoretical study for at least several years (see, e.g.,~\\cite{hardt2016gradient, allen2019convergence, lim2021noisy}).\nIn contrast, works analyzing the implicit extrapolation of GD have surfaced only recently, specifically in~\\cite{emami2021implicit} and~\\cite{cohen2022extrapolation}.\\footnote{%\nWe note that there have been works studying extrapolation in the context of non-recurrent NNs, e.g. \\cite{xu2020neural}. This type of extrapolation deals with the behavior of learned functions outside the support of the training distribution, and is therefore fundamentally different from the type of extrapolation we consider (which deals with the behavior of learned functions over sequences longer than those seen in training).\n}\n\\cite{emami2021implicit}~analyzes linear RNNs in the infinite width regime, suggesting that in this case GD is implicitly biased towards impulse responses corresponding to short-term memory. \n\\cite{cohen2022extrapolation}~studies finite-width linear RNNs (as we do), showing that when the teacher is memoryless (has state space dimension zero), GD emanating from a balanced initialization successfully extrapolates.\nOur work tackles an arguably more realistic and challenging setting~---~we analyze the regime in which the teacher has memory.\nOur results suggest that the implicit extrapolation of GD does not originate from a bias towards short-term memory, but rather a tendency to learn low dimensional state spaces.\n\nOn the empirical side, extrapolation of NNs to sequence lengths beyond those seen in training has been experimentally demonstrated in numerous recent works, covering both modern language and attention models \\citep{press2022train, anil2022exploring, zhang2022unveiling}, and RNNs with transition matrices of particular forms \\citep{gu2022efficiently, gu2021combining, gu2020hippo, gupta2022diagonal}. \nThe current paper is motivated by these findings, and takes a step towards theoretically explaining them.\n\\section{TODOs}\n\\begin{itemize}\n \\item Results of \\Secref{sec:exp:sym_teacher} - this experiment should be\n based on GD, not Adam\n \\item Would be better if state space dimensions for student and teacher in all experiments would be identical.\n \\item Improve figures, e.g.:\n \\begin{itemize}\n \\item Reduce height (i.e.~make figures shorter)\n \\item Increase axis label font size\n \\item Change \"Sequence Length\" to \"Training Sequence Length\"\n \\item Add axis labels to Figure~2(b)\n \\item Tighten axis limits (e.g.~no need for negative y values in Figure~1 and negative x values in Figure~2(b))\n \\item In Figure~2(b), replace~$k$ by~$k_g$, or better yet, by \"training sequence length\"\n \\end{itemize}\n \\item Would be nicer if number of seeds was the same as across all experiments.\n \\item \\st{Capitalize references to equations (i.e.~have \"Equation X\" instead of \"equation X\")}\n \\item \\st{Use \"Appendix\" (rather than \"Section\") in references to supplementary material}\n \\item Add Wasserstein distance to \\Secref{sec:setup}.\n \\item Add $\\odot$ to denote Hadamard product.\n\\end{itemize}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Background \\& Summary}\n\nDue to their demonstrated uniqueness and persistence \\cite{bargary2017individual}, human eye movements are a desirable modality for biometric applications \\cite{jain2015guidelines}. Since their original consideration in the early 2000s \\cite{kasprowski2004eye}, eye movement biometrics have received substantial attention within the security literature \\cite{katsini2020role}. Recent interest in this domain is accelerating, due to the proliferation of gaze tracking sensors throughout modern consumer products, including automotive interfaces, traditional computing platforms, and head-mounted devices for virtual and augmented reality applications. Beyond this increase in sensor ubiquity, eye movements are an advantageous modality for emerging biometric systems due to their ability to support continuous authentication \\cite{eberz2016looks} and liveliness detection \\cite{komogortsev2015attack}, along with their ease of fusion with other appearance-based traits in both the eye \\cite{winston2019comprehensive} and periocular region \\cite{woodard2010periocular}.\n\nDespite considerable research progress in eye movement biometrics over the past two decades, several open research areas remain. Namely, ensuring performance robustness with respect to data quality, and further investigating both task dependency and requisite recording duration is necessary to transition this technology to widespread commercial adoption. Moreover, the exploration of emerging deep learning techniques, which have proven successful for more traditional biometric modalities \\cite{sundararajan2018deep}, has been limited for eye movement biometrics. This investigation is impeded by the challenges associated with the large-scale collection of eye movement data, along with the lack of task-diverse, publicly-available, large-scale data repositories. \n\nTo promote further development in eye movement biometrics research, this manuscript describes a newly-released dataset consisting of 12,334 monocular eye movement recordings captured from 322 individuals while performing seven discrete tasks. The considered task battery includes guided stimuli intended to induce specific eye movements of interest, along with multiple objective-oriented and free-viewing tasks, such as reading, movie viewing, and game playing. Hereby denoted as GazeBase, this data were captured over a 37 month period during nine rounds of recording, with two contiguous sessions completed during each recording period. The data collection workflow is summarized in \\figurerefname~\\ref{fig:SummaryFig}.\n\n\n\n\\begin{figure}[ht!] \n\\centering\n\\includegraphics[width=\\linewidth]{DraftMainFigwBorder.jpg} \n\\caption{Summary of the GazeBase dataset collection. \\textbf{ \\textit{Top:}} Experimental set-up. \\textbf{ \\textit{Middle:}} Screenshots of the stimuli from four of the seven tasks. a) is a screenshot of the gaze-driven gaming task, b) is a screenshot of the reading task, c-d) is a single screenshot from one of the two video viewing tasks. e-g) shows the stationary bull's-eye target utilized in the calibration and validation process, along with the fixation and two saccade tasks. The screenshot is obtained during the random saccade task. \\textbf{ \\textit{Bottom:}} A timeline of the multiple recording rounds (round identifiers are labeled on top of each rectangle).} \n\n\n\n\\label{fig:SummaryFig}\n\\end{figure}\n\nAlthough subsets of this data have been utilized in prior work\\cite{abdulin2017method, friedman2018novel, rigas2018study, friedman2019assessment, lohr2019evaluating, friedman2020temporal, friedman2017method, griffith2018towards,griffith2020prediction,griffith2018towards1,griffith2020shift,griffith2020texture}, this recent dissemination is the first release of the entire set of gaze recordings and corresponding target locations for applicable stimuli. As the experimental parameters of the data collection were chosen to maximize the utility of the resulting data for biometric applications, GazeBase is well suited for supporting further investigation of emerging machine learning biometric techniques to the eye movement domain, such as metric learning \\cite{abdelwahab2019deep}. Beyond this target application, the resulting dataset is also useful for exploring numerous additional research hypotheses in various areas of interest, including eye movement classification and prediction. Applications employing machine learning techniques will benefit both from the scale of available data, along with the diversity in tasks considered and subjects recorded. Moreover, this dissemination will help improve quality in subsequent research by providing a diverse set of recordings for benchmarking across the community \\cite{jain2015guidelines}.\n\n\\section*{Methods}\n\\subsection*{Subjects}\nSubjects were initially recruited from the undergraduate student population at Texas State University through email and targeted in-class announcements. A total of 322 subjects (151 self-identifying as female, 171 self-identifying as male) were enrolled in the study and completed the Round 1 collection in its entirety. Subjects for Rounds 2 - 9 were recruited exclusively from the prior round's subject pool. All subjects had normal or corrected to normal visual acuity. Aggregate subject demographic information is presented in Table~\\ref{tab:DemographicInfo}. The distribution of subjects' age at the time of the Round 1 collection is shown in \\figurerefname~\\ref{fig:AgeDist}. The number of subjects completing the entire task battery in each round is summarized in Table~\\ref{tab:RoundInfo}, along with the recording dates for each round of collection. \n\n\\begin{table}[ht]\n\\centering\n\\begin{tabular}{llllll}\n\\toprule\nEthnicity: & Asian & Black & Caucasian & Hispanic & Mixed \\\\\n\\midrule\nNum. of Subjects: &10 & 32 & 178 & 76 & 27 \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Self-reported ethnicity of subjects}\n\\label{tab:DemographicInfo}\n\\end{table}\n\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.7\\linewidth]{AgeDist.png}\n\\caption{Distribution of subjects' ages at the time of enrollment}\n\\label{fig:AgeDist}\n\\end{figure}\n\n\\begin{table}[ht]\n\\centering\n\\begin{tabular}{lll}\n\\toprule\nRound ID & Num. of Subjects & Date Range \\\\\n\\midrule\n1 & 322 & 09\/13 - 02\/14\\\\\n2 & 136 & 02\/14 - 03\/14\\\\\n3 & 105 & 03\/14 - 04\/14\\\\\n4 & 101 & 04\/14 - 04\/14\\\\\n5 & 78 & 09\/14 - 11\/14\\\\\n6 & 59 & 03\/15 - 05\/15\\\\\n7 & 35 & 10\/15 - 11\/15\\\\\n8 & 31 & 03\/16 - 05\/16\\\\\n9 & 14 & 10\/16 - 11\/16\\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Total number of subjects and recording date range of each round}\n\\label{tab:RoundInfo}\n\\end{table}\n\n\n All subjects provided informed consent under a protocol approved by the Institutional Research Board at Texas State University prior to each round of recording. As part of the consent process, subjects acknowledged that the resulting data may be disseminated in a de-identified form. \n\n\\subsection*{Data Acquisition Overview}\nData was captured under the supervision of a trained experimental proctor. Before initiating the recording process, the proctor provided a general overview of the experiment to the subject, along with a summary of best practices for maximizing the quality of the captured data. Namely, subjects were instructed to maintain a stable head and body position, and to attempt to avoid excessive blinking. Based upon initial recording experiences, it was ultimately suggested that subjects avoid wearing mascara to the recording session as part of the appointment confirmation email. This suggestion was initiated during the first round of recording and maintained throughout the remainder of the collection. \n\nSubjects wearing eyeglasses were asked to attempt the experiment with glasses removed. This protocol was chosen due to the known challenges associated with recording individuals wearing eyeglasses using the target capture modality. If subjects were unable to complete the experimental protocol with glasses removed, an attempt was made to complete the experiment while wearing eyeglasses. Subjects that could not be successfully calibrated or recorded after multiple attempts were withdrawn from the study. Subjects could also self-withdrawal at any point during the recording process. A total of 13 subjects were withdrawn from the initial round of the study. GazeBase contains only data from subjects completing the entire recording protocol for a given round. \n\nMonocular (left) eye movements were captured at a 1,000 Hz sampling rate using an EyeLink 1000 eye tracker (SR Research, Ottawa, Ontario, Canada) in a desktop mount configuration. The EyeLink 1000 is a video oculography device which operates using the pupil-corneal reflection principle, where gaze locations are estimated from pupil-corneal reflection vectors using a polynomial mapping developed during calibration \\cite{sr2010eyelink}. Stimuli were presented to the subject on a 1680 x 1050 pixel (474 x 297 mm) ViewSonic (ViewSonic Corporation, Brea, California, USA) monitor. Instrumentation control and recording monitoring were performed by the proctor using a dedicated host computer as shown in \\figurerefname~\\ref{fig:SummaryFig}. Recordings were performed in a quiet laboratory environment without windows. Consistent ambient lighting was provided by ceiling-mounted fluorescent light fixtures. \n\nSubjects were seated 550 mm in front of the display monitor. Subjects' heads were stabilized using a chin and forehead rest. Once the participant was initially seated, the chin rest was adjusted to level the subjects' left eye at the primary gaze position, located 36 mm above the center of the monitor. This vertical offset from the monitor center was chosen to ensure the comfort of taller participants given restrictions on adjusting the chair and monitor height. Chair height was initially adjusted as necessary to ensure the comfort of the subject, followed by additional fine tuning of the chin rest as required to align the left eye with the primary gaze position. The lens focus was manually tuned as necessary in order to ensure the sharpness of the eye image as viewed by the proctor on the host display. \n\nSubjects completed two sessions of recording for each round of collection. While the proctor suggested that subjects take a five-minute break between sessions if needed, subjects were free to decline the break if desired. Subjects could also request breaks at any time during the recording process as noted during the consent process. During each recording, the gaze location was monitored by the proctor to ensure compliance with the individual task protocols. \n\nThe gaze position and corresponding stimuli were innately expressed in terms of pixel display coordinates. These values were converted to degrees of the visual angle (dva) according to the geometry of the recording setup. Although iris images were also captured as part of this collection before the initiation of eye movement recordings, they are not distributed as part of GazeBase. Prior collections including both gaze traces and matching iris images may be found at the following link - \n\\href{http:\/\/userweb.cs.txstate.edu\/~ok11\/etpad_v2.html}{http:\/\/userweb.cs.txstate.edu\/\\textasciitilde ok11\/etpad\\_v2.html}\n\n\n\\subsection*{Calibration and Validation}\nA calibration and validation procedure were performed before the recording of each task to ensure data quality. To initiate the calibration process, pupil and corneal reflection thresholds were established. While manual tuning was exclusively used for some initial recordings, the automatic thresholding function of the instrumentation software was ultimately employed to develop initial estimates, with manual fine-tuning performed as required. \n\nOnce threshold parameter values were tuned to ensure successful tracking of the pupil and corneal reflection, a nine-point rectangular grid calibration was performed. During this process, subjects were instructed to fixate at the center of bulls-eye calibration target positioned on a black background. The bulls-eye target consisted of a larger white circle with an approximate diameter of one dva enclosing a small black dot as shown in \\figurerefname~\\ref{fig:SummaryFig}. The stability of target fixations was monitored by the proctor on the host monitor using a vendor-provided software interface. If necessary, the proctor provided additional instructions to improve image capture quality (i.e.: increase eye opening, etc.). Once the software determined that the subject had successfully fixated on a target, the calibration process advanced to the next target. Calibration was terminated when a stable fixation had been captured for all nine points in the grid, thereby producing the aforementioned mapping for estimating gaze location. \n\nA nine-point validation process was subsequently performed to ensure calibration accuracy. Validation points outside of the primary position were disjoint from those utilized in the calibration grid. Validation for each target was manually terminated by the proctor (contrasting from the calibration procedure, which used automatic termination) upon the determination of a stable target fixation. The spatial accuracy of each fixation on the corresponding validation target, hereby referred to as the validation error, was computed after completion of the validation process. Validation error was determined by computing the Euclidean distance between each target and the estimated gaze location. A maximum and average validation error of less than 1.5 and 1.0 dva., respectively, was established as a guideline accuracy criteria for accepting the calibration. However, acceptance of the calibration was ultimately determined by the proctor based upon visual inspection of the discrepancy between the estimated and true target location for each validation point, with additional discretion applied to calibrations failing to meet this quantitative accuracy goal. The calibration protocol was repeated before the recording of each task. \n\n\\subsection*{Task Battery Overview}\nA battery of seven tasks were performed during each session of the recording. Tasks were performed in the numbered order described in the following subsections. Acronyms utilized to describe each task within the distributed dataset are defined within each subsection title. \n\\subsubsection*{Task 1: Horizontal Saccade Task (HSS) }\nThe HSS task was designed to elicit visually-guided horizontal saccades of constant amplitude through the periodic displacement of a peripheral target. Subjects were instructed to fixate on the center of the bull's-eye target utilized during calibration. The target was displayed on a black background and was initially placed at the primary gaze position. The target was regularly displaced between two positions located $\\pm 15$ dva horizontally from the center of the screen, thereby ideally eliciting a 30 degree horizontal saccade upon each jump displacement. The target's position was maintained for one second between displacements, with 100 transitions occurring during each recording. The proctor notified the subjects of the approximate time remaining within the 100-second recording session at 20 second intervals. An identical stimulus was used for the HSS task across subjects, sessions, and rounds. \n\n\n\n\\subsubsection*{Task 2: Video Viewing Task 1 (VD1)}\nThe VD1 task was designed to elicit natural eye movements occurring during the free-viewing of a cinematic video. Subjects were instructed to watch the first 60 seconds of a trailer for the movie ``The Hobbit: The Desolation of Smaug''. No audio was played during the video clip. The same video segment was used for the VD1 task across subjects and rounds. Due to variability in instrumentation settings, the video stimulus was only displayed for the initial 57 seconds during the second session of each recording. \n\n\n\\subsubsection*{Task 3: Fixation Task (FXS)}\nThe FXS task was designed to elicit fixational eye movements through the static presentation of a central fixation target located at the primary gaze position. Subjects were instructed to fixate on the previously described bull's-eye target which was maintained at the center of the display for 15 seconds. The proctor asked that subjects avoid blinking if possible during the duration of the task before initiating the recording. An identical stimulus was used for the FXS task across subjects, sessions, and rounds. \n\n\n\n\n\\subsubsection*{Task 4: Random Saccade Task (RAN)}\nThe RAN task was designed to elicit visually-guided oblique saccades of variable amplitude through the periodic displacement of a peripheral target. Similar to the HSS task, subjects were instructed to follow the bull's-eye target by fixating at its center. The target was displaced at random locations across the display monitor, ranging from $\\pm 15$ and $\\pm 9$ dva in the horizontal and vertical directions, respectively. The minimum amplitude between adjacent target displacements was two degrees of the visual angle. Similar to the HSS task, the target was displayed on a black background, with each position maintained for one second. As the trajectory of the target was randomized for each recording iteration, the stimulus varied across subjects, sessions, and rounds. The distribution of target locations was chosen to ensure uniform coverage across the display. \n\n\n\n\n\\subsubsection*{Task 5: Reading Task (TEX)}\nThe TEX task was designed to capture subjects' eye movements during reading. Subjects were instructed to silently read a passage from the poem ``The Hunting of the Snark'' by Lewis Carroll. The task was automatically terminated after 60 seconds irrespective of the subjects' reading progress. Subjects did not receive explicit instructions of what to do if they finished reading before the end of the 60 second period. Instead, several possible actions were suggested, including rereading the passage. Because of this ambiguity in instructions, the gaze position towards the end of the recording may vary from the expected per-line pattern typically encountered during reading. Another passage of the poem was displayed as the stimulus for the second session within a given round, with the same pair of sections utilized for all subjects and rounds.\n\n\n\\subsubsection*{Task 6: Balura Game (BLG)}\nThe BLG task was designed to capture subjects' eye movements while interacting with a gaze-driven gaming environment. During the game, blue and red balls moving at a slow fixed speed were presented on a black background. Subjects were instructed to attempt to remove all red balls from the display area as quickly as possible. Red balls were eliminated when the subject fixated on them, while blue balls could not be eliminated. Visual feedback was provided to subjects by placing a highlighted border around each ball upon the detected onset of a fixation on the ball. The game was terminated when no additional red balls were remaining. Further details regarding the game may be found at the following link - \\href{https:\/\/digital.library.txstate.edu\/handle\/10877\/4158}{https:\/\/digital.library.txstate.edu\/handle\/10877\/4158}. \n\nIn some instances, steady fixations on red balls did not produce the desired elimination behavior. Based upon this limitation, the proctor instructed subjects to not maintain elongated fixations if the red balls were not eliminating as intended. Instead, subjects were instructed to move their gaze away from the ball, and subsequently re-fixate on the non-eliminating ball. As the initial position and trajectory of each ball was set randomly for each recording, the stimulus varied across subjects, sessions, and rounds. \n\n\n\n\\subsubsection*{Task 7: Video Viewing Task 2 (VD2)}\nSubjects were instructed to watch the subsequent 60 seconds of the trailer used in the VD1 task. Similar to the VD1 task, no audio was presented to the subjects. The same video was used for the VD2 task across subjects and rounds. Similar to the VD1 task, the duration of the VD2 stimulus in Session 2 was truncated to 57 seconds due to variability in instrumentation settings. \n\n\\section*{Data Records}\n\nGazeBase is available for download on figshare~\\cite{griffith_gazebase_2020}. GazeBase is distributed under a \\href{https:\/\/creativecommons.org\/licenses\/by\/4.0\/}{Creative Commons Attribution 4.0 International (CC BY 4.0) license}. Gazebase may be used without restriction for non-commercial applications, with all resulting publications providing citation to this manuscript. All data have been de-identified in accordance with the informed consent provided by subjects. \n\nGazeBase is organized in a hierarchical directory structure by round, subject, session, and task, respectively. Data records are compressed at the subject folder level. Each task folder contains a single csv file with the following naming convention - `S\\_rxxx\\_Sy\\_tsk', with the relevant parameters for each field summarized in Table \\ref{tab:FileConvention}. The first line of each file contains the variable identifiers for each column, which are summarized in Table \\ref{tab:FileVariables}. \n\n\\begin{table}[ht]\n\\centering\n\\begin{tabular}{lll}\n\\toprule\nNaming Parameter & Definition & Valid Values \\\\\n\\midrule\nr & Round Number & 1 - 9 \\\\\nxxx & Subject ID & 1 - 335 (excluding incomplete subjects) \\\\\ny & Session Number & 1 - 2 \\\\\ntsk & Task Description & {`HSS', `VD1', `FXS', `RAN', `TEX', `BLG', `VD2'} \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Description of file naming convention}\n\\label{tab:FileConvention}\n\\end{table}\n\n\n\\begin{table}[ht]\n\\centering\n\\begin{tabular}{lll}\n\\toprule\nVariable Identifier & Definition \\\\\n\\midrule\nn & Timestamp (ms) \\\\\nx & Horizontal Gaze Position (dva) \\\\\ny & Vertical Gaze Position (dva) \\\\\nval & Sample Validity (0 implies valid sample) \\\\\nxT & Horizontal Target Position (dva) (where applicable) \\\\\nyT & Vertical Target Position (dva) (where applicable) \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Description of file variables}\n\\label{tab:FileVariables}\n\\end{table}\n\nMissing samples within each file are denoted by a non-zero value in the val field, with the corresponding gaze position specified as NaNs. Missing samples result from the failure to extract either the pupil or corneal reflection from the captured image, which may occur under scenarios of blinks or partial occlusions of the eye. For tasks not employing a target (i.e.: VD1, TEX, BLG, VD2), target entries (i.e.: columns xT and yT) are populated with NaNs at each sample. \n\n\n\\section*{Technical Validation} \\label{sec:Val}\nSubstantial efforts were undertaken to maintain data quality throughout the experimental design and collection process, initiating with the selection of the recording instrument. The EyeLink 1000 was selected for data collection due to its high spatial accuracy and precision characteristics, which have resulted in its widespread adoption across the research community \\cite{nystrom2020tobii}. The EyeLink 1000 is routinely employed as a quality benchmark in the research literature when evaluating emerging camera-based eye tracking sensors (i.e.: \\cite{ehinger2019new, lohr2019evaluating,raynowska2018validity}). To ensure adherence to best practices throughout the data collection, all experimental proctors were trained by personnel with considerable prior experience using the device. This expertise was developed during the lab's prior data collections using the EyeLink 1000 (e.g. \\href{https:\/\/userweb.cs.txstate.edu\/~ok11\/software.html}{https:\/\/userweb.cs.txstate.edu\/\\textasciitilde ok11\/software.html}). \n\nThe experimental protocol was also designed to maximize the quality of the captured data. Namely, subjects were instructed to maintain a stable head position and sufficient opening of the eyelids due to the known relationship between these factors and associated raw eye positional data quality \\cite{nystrom2013influence}. A dedicated calibration and validation process was also employed for each task to avoid calibration decay across the collection. Box plots of the distributions of mean and maximum validation errors for each round of recording are presented in \\figuresrefname~\\ref{fig:MeanVal} and \\ref{fig:MaxVal}, respectively. As shown, the median values of the mean validation errors in each round are less than the upper bound of the specified typical spatial accuracy (0.5 dva) of the instrument. Moreover, the significant dispersion of the two metrics, indicating considerable variability in calibration quality across individuals, is consistent with prior observations in the literature\\cite{hornof2002cleaning}. \n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.7\\linewidth]{MeanValData.png}\n\\caption{Distribution of mean validation error across recordings versus round. The central mark in each box corresponds to the median value, with the lower and upper edges of the box corresponding to the 25th and 75th percentiles of the distribution, respectively. The whiskers extend to the outlier boundaries for each round, which are set at 1.5 times the interquartile range of the distribution above and below the box boundaries. Outliers are marked using the + symbol.}\n\\label{fig:MeanVal}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.7\\linewidth]{MaxValData.png}\n\\caption{Distribution of the maximum validation error across recordings versus round. See Fig.~\\ref{fig:MeanVal} for an explanation of box plot parameters.}\n\\label{fig:MaxVal}\n\\end{figure}\n\n\n\n\n\\section*{Code availability}\n\nThe distributed csv files were generated by first converting the edf output files produced by the Eyelink 1000 to a text-based asc file format. These files were subsequently converted to csv files of the specified format using a customized MATLAB script. Data may be extracted from the repository into the target computing environment using traditional csv import functions. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\n\\chapter{How to install \\LaTeX} \n\n\\section*{Windows OS}\n\n\\subsection*{TeXLive package - full version}\n\\begin{enumerate}\n\\item\tDownload the TeXLive ISO (2.2GB) from\\\\\n\\href{https:\/\/www.tug.org\/texlive\/}{https:\/\/www.tug.org\/texlive\/}\n\\item\tDownload WinCDEmu (if you don't have a virtual drive) from \\\\\n\\href{http:\/\/wincdemu.sysprogs.org\/download\/}\n{http:\/\/wincdemu.sysprogs.org\/download\/}\n\\item\tTo install Windows CD Emulator follow the instructions at\\\\\n\\href{http:\/\/wincdemu.sysprogs.org\/tutorials\/install\/}\n{http:\/\/wincdemu.sysprogs.org\/tutorials\/install\/}\n\\item\tRight click the iso and mount it using the WinCDEmu as shown in \\\\\n\\href{http:\/\/wincdemu.sysprogs.org\/tutorials\/mount\/}{\nhttp:\/\/wincdemu.sysprogs.org\/tutorials\/mount\/}\n\\item\tOpen your virtual drive and run setup.pl\n\\end{enumerate}\n\nor\n\n\\subsection*{Basic MikTeX - \\TeX~ distribution}\n\\begin{enumerate}\n\\item\tDownload Basic-MiK\\TeX (32bit or 64bit) from\\\\\n\\href{http:\/\/miktex.org\/download}{http:\/\/miktex.org\/download}\n\\item\tRun the installer \n\\item\tTo add a new package go to Start >> All Programs >> MikTex >> Maintenance (Admin) and choose Package Manager\n\\item\tSelect or search for packages to install\n\\end{enumerate}\n\n\\subsection*{TexStudio - \\TeX~ editor}\n\\begin{enumerate}\n\\item\tDownload TexStudio from\\\\\n\\href{http:\/\/texstudio.sourceforge.net\/\\#downloads}\n{http:\/\/texstudio.sourceforge.net\/\\#downloads} \n\\item\tRun the installer\n\\end{enumerate}\n\n\\section*{Mac OS X}\n\\subsection*{MacTeX - \\TeX~ distribution}\n\\begin{enumerate}\n\\item\tDownload the file from\\\\\n\\href{https:\/\/www.tug.org\/mactex\/}{https:\/\/www.tug.org\/mactex\/}\n\\item\tExtract and double click to run the installer. It does the entire configuration, sit back and relax.\n\\end{enumerate}\n\n\\subsection*{TexStudio - \\TeX~ editor}\n\\begin{enumerate}\n\\item\tDownload TexStudio from\\\\\n\\href{http:\/\/texstudio.sourceforge.net\/\\#downloads}\n{http:\/\/texstudio.sourceforge.net\/\\#downloads} \n\\item\tExtract and Start\n\\end{enumerate}\n\n\n\\section*{Unix\/Linux}\n\\subsection*{TeXLive - \\TeX~ distribution}\n\\subsubsection*{Getting the distribution:}\n\\begin{enumerate}\n\\item\tTexLive can be downloaded from\\\\\n\\href{http:\/\/www.tug.org\/texlive\/acquire-netinstall.html}\n{http:\/\/www.tug.org\/texlive\/acquire-netinstall.html}.\n\\item\tTexLive is provided by most operating system you can use (rpm,apt-get or yum) to get TexLive distributions\n\\end{enumerate}\n\n\\subsubsection*{Installation}\n\\begin{enumerate}\n\\item\tMount the ISO file in the mnt directory\n\\begin{verbatim}\nmount -t iso9660 -o ro,loop,noauto \/your\/texlive####.iso \/mnt\n\\end{verbatim}\n\n\\item\tInstall wget on your OS (use rpm, apt-get or yum install)\n\\item\tRun the installer script install-tl.\n\\begin{verbatim}\n\tcd \/your\/download\/directory\n\t.\/install-tl\n\\end{verbatim}\n\\item\tEnter command `i' for installation\n\n\\item\tPost-Installation configuration:\\\\\n\\href{http:\/\/www.tug.org\/texlive\/doc\/texlive-en\/texlive-en.html\\#x1-320003.4.1}\n{http:\/\/www.tug.org\/texlive\/doc\/texlive-en\/texlive-en.html\\#x1-320003.4.1} \n\\item\tSet the path for the directory of TexLive binaries in your .bashrc file\n\\end{enumerate}\n\n\\subsubsection*{For 32bit OS}\nFor Bourne-compatible shells such as bash, and using Intel x86 GNU\/Linux and a default directory setup as an example, the file to edit might be \\begin{verbatim}\nedit $~\/.bashrc file and add following lines\nPATH=\/usr\/local\/texlive\/2011\/bin\/i386-linux:$PATH; \nexport PATH \nMANPATH=\/usr\/local\/texlive\/2011\/texmf\/doc\/man:$MANPATH;\nexport MANPATH \nINFOPATH=\/usr\/local\/texlive\/2011\/texmf\/doc\/info:$INFOPATH;\nexport INFOPATH\n\\end{verbatim}\n\\subsubsection*{For 64bit OS}\n\\begin{verbatim}\nedit $~\/.bashrc file and add following lines\nPATH=\/usr\/local\/texlive\/2011\/bin\/x86_64-linux:$PATH;\nexport PATH \nMANPATH=\/usr\/local\/texlive\/2011\/texmf\/doc\/man:$MANPATH;\nexport MANPATH \nINFOPATH=\/usr\/local\/texlive\/2011\/texmf\/doc\/info:$INFOPATH;\nexport INFOPATH\n\n\\end{verbatim}\n\n\n\n\\subsubsection*{Fedora\/RedHat\/CentOS:}\n\\begin{verbatim} \nsudo yum install texlive \nsudo yum install psutils \n\\end{verbatim}\n\n\n\\subsubsection*{SUSE:}\n\\begin{verbatim}\nsudo zypper install texlive\n\\end{verbatim}\n\n\n\\subsubsection*{Debian\/Ubuntu:}\n\\begin{verbatim} \nsudo apt-get install texlive texlive-latex-extra \nsudo apt-get install psutils\n\\end{verbatim}\n\n\\chapter{Installing the CUED class file}\n\n\\LaTeX.cls files can be accessed system-wide when they are placed in the\n\/tex\/latex directory, where is the root directory of the user's \\TeX installation. On systems that have a local texmf tree (), which\nmay be named ``texmf-local'' or ``localtexmf'', it may be advisable to install packages in , rather than as the contents of the former, unlike that of the latter, are preserved after the \\LaTeX system is reinstalled and\/or upgraded.\n\nIt is recommended that the user create a subdirectory \/tex\/latex\/CUED for all CUED related \\LaTeX class and package files. On some \\LaTeX systems, the directory look-up tables will need to be refreshed after making additions or deletions to the system files. For \\TeX Live systems this is accomplished via executing ``texhash'' as root. MIK\\TeX users can run ``initexmf -u'' to accomplish the same thing.\n\nUsers not willing or able to install the files system-wide can install them in their personal directories, but will then have to provide the path (full or relative) in addition to the filename when referring to them in \\LaTeX.\n\n\n\\section{Matroid Axioms}\n\n\n\n\\chapter{Getting started} \n\n\\ifpdf\n \\graphicspath{{Chapter1\/Figs\/Raster\/}{Chapter1\/Figs\/PDF\/}{Chapter1\/Figs\/}}\n\\else\n \\graphicspath{{Chapter1\/Figs\/Vector\/}{Chapter1\/Figs\/}}\n\\fi\n\n\n\\section{What is loren ipsum? Title with math \\texorpdfstring{$\\sigma$}{[sigma]}}\n\nLorem Ipsum is simply dummy text of the printing and typesetting industry (see \nSection~\\ref{section1.3}). Lorem Ipsum~\\citep{Aup91} has been the industry's \nstandard dummy text ever since the 1500s, when an unknown printer took a galley \nof type and scrambled it to make a type specimen book. It has survived not only \nfive centuries, but also the leap into electronic typesetting, remaining \nessentially unchanged. It was popularised in the 1960s with the release of \nLetraset sheets containing Lorem Ipsum passages, and more recently with desktop \npublishing software like Aldus PageMaker including versions of Lorem \nIpsum~\\citep{AAB95,Con90,LM65}.\n\nThe most famous equation in the world: $E^2 = (m_0c^2)^2 + (pc)^2$, which is \nknown as the \\textbf{energy-mass-momentum} relation as an in-line equation.\n\nA {\\em \\LaTeX{} class file}\\index{\\LaTeX{} class file@LaTeX class file} is a file, which holds style information for a particular \\LaTeX{}.\n\n\n\\begin{align}\nCIF: \\hspace*{5mm}F_0^j(a) = \\frac{1}{2\\pi \\iota} \\oint_{\\gamma} \\frac{F_0^j(z)}{z - a} dz\n\\end{align}\n\n\\nomenclature[z-cif]{$CIF$}{Cauchy's Integral Formula} \n\\nomenclature[a-F]{$F$}{complex function} \n\\nomenclature[g-p]{$\\pi$}{ $\\simeq 3.14\\ldots$} \n\\nomenclature[g-i]{$\\iota$}{unit imaginary number $\\sqrt{-1}$} \n\\nomenclature[g-g]{$\\gamma$}{a simply closed curve on a complex plane} \n\\nomenclature[x-i]{$\\oint_\\gamma$}{integration around a curve $\\gamma$}\n\\nomenclature[r-j]{$j$}{superscript index} \n\\nomenclature[s-0]{$0$}{subscript index} \n\n\n\\section{Why do we use loren ipsum?}\n\n\nIt is a long established fact that a reader will be distracted by the readable content of a page when looking at its layout. The point of using Lorem Ipsum is that it has a more-or-less normal distribution of letters, as opposed to using `Content here, content here', making it look like readable English. Many desktop publishing packages and web page editors now use Lorem Ipsum as their default model text, and a search for `lorem ipsum' will uncover many web sites still in their infancy. Various versions have evolved over the years, sometimes by accident, sometimes on purpose (injected humour and the like).\n\n\\section{Where does it come from?} \n\\label{section1.3}\n\nContrary to popular belief, Lorem Ipsum is not simply random text. It has roots in a piece of classical Latin literature from 45 BC, making it over 2000 years old. Richard McClintock, a Latin professor at Hampden-Sydney College in Virginia, looked up one of the more obscure Latin words, consectetur, from a Lorem Ipsum passage, and going through the cites of the word in classical literature, discovered the undoubtable source. Lorem Ipsum comes from sections 1.10.32 and 1.10.33 of \"de Finibus Bonorum et Malorum\" (The Extremes of Good and Evil) by Cicero, written in 45 BC. This book is a treatise on the theory of ethics, very popular during the Renaissance. The first line of Lorem Ipsum, \"Lorem ipsum dolor sit amet..\", comes from a line in section 1.10.32.\n\nThe standard chunk of Lorem Ipsum used since the 1500s is reproduced below for those interested. Sections 1.10.32 and 1.10.33 from ``de Finibus Bonorum et Malorum\" by Cicero are also reproduced in their exact original form, accompanied by English versions from the 1914 translation by H. Rackham\n\n``Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.\"\n\nSection 1.10.32 of ``de Finibus Bonorum et Malorum\", written by Cicero in 45 BC: ``Sed ut perspiciatis unde omnis iste natus error sit voluptatem accusantium doloremque laudantium, totam rem aperiam, eaque ipsa quae ab illo inventore veritatis et quasi architecto beatae vitae dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit aut fugit, sed quia consequuntur magni dolores eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci velit, sed quia non numquam eius modi tempora incidunt ut labore et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure reprehenderit qui in ea voluptate velit esse quam nihil molestiae consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla pariatur?\"\n\n1914 translation by H. Rackham: ``But I must explain to you how all this mistaken idea of denouncing pleasure and praising pain was born and I will give you a complete account of the system, and expound the actual teachings of the great explorer of the truth, the master-builder of human happiness. No one rejects, dislikes, or avoids pleasure itself, because it is pleasure, but because those who do not know how to pursue pleasure rationally encounter consequences that are extremely painful. Nor again is there anyone who loves or pursues or desires to obtain pain of itself, because it is pain, but because occasionally circumstances occur in which toil and pain can procure him some great pleasure. To take a trivial example, which of us ever undertakes laborious physical exercise, except to obtain some advantage from it? But who has any right to find fault with a man who chooses to enjoy a pleasure that has no annoying consequences, or one who avoids a pain that produces no resultant pleasure?\"\n\nSection 1.10.33 of ``de Finibus Bonorum et Malorum\", written by Cicero in 45 BC: ``At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Temporibus autem quibusdam et aut officiis debitis aut rerum necessitatibus saepe eveniet ut et voluptates repudiandae sint et molestiae non recusandae. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.\"\n\n1914 translation by H. Rackham: ``On the other hand, we denounce with righteous indignation and dislike men who are so beguiled and demoralized by the charms of pleasure of the moment, so blinded by desire, that they cannot foresee the pain and trouble that are bound to ensue; and equal blame belongs to those who fail in their duty through weakness of will, which is the same as saying through shrinking from toil and pain. These cases are perfectly simple and easy to distinguish. In a free hour, when our power of choice is untrammelled and when nothing prevents our being able to do what we like best, every pleasure is to be welcomed and every pain avoided. But in certain circumstances and owing to the claims of duty or the obligations of business it will frequently occur that pleasures have to be repudiated and annoyances accepted. The wise man therefore always holds in these matters to this principle of selection: he rejects pleasures to secure other greater pleasures, or else he endures pains to avoid worse pains.\"\n\n\\nomenclature[z-DEM]{DEM}{Discrete Element Method}\n\\nomenclature[z-FEM]{FEM}{Finite Element Method}\n\\nomenclature[z-PFEM]{PFEM}{Particle Finite Element Method}\n\\nomenclature[z-FVM]{FVM}{Finite Volume Method}\n\\nomenclature[z-BEM]{BEM}{Boundary Element Method}\n\\nomenclature[z-MPM]{MPM}{Material Point Method}\n\\nomenclature[z-LBM]{LBM}{Lattice Boltzmann Method}\n\\nomenclature[z-MRT]{MRT}{Multi-Relaxation \nTime}\n\\nomenclature[z-RVE]{RVE}{Representative Elemental Volume}\n\\nomenclature[z-GPU]{GPU}{Graphics Processing Unit}\n\\nomenclature[z-SH]{SH}{Savage Hutter}\n\\nomenclature[z-CFD]{CFD}{Computational Fluid Dynamics}\n\\nomenclature[z-LES]{LES}{Large Eddy Simulation}\n\\nomenclature[z-FLOP]{FLOP}{Floating Point Operations}\n\\nomenclature[z-ALU]{ALU}{Arithmetic Logic Unit}\n\\nomenclature[z-FPU]{FPU}{Floating Point Unit}\n\\nomenclature[z-SM]{SM}{Streaming Multiprocessors}\n\\nomenclature[z-PCI]{PCI}{Peripheral Component Interconnect}\n\\nomenclature[z-CK]{CK}{Carman - Kozeny}\n\\nomenclature[z-CD]{CD}{Contact Dynamics}\n\\nomenclature[z-DNS]{DNS}{Direct Numerical Simulation}\n\\nomenclature[z-EFG]{EFG}{Element-Free Galerkin}\n\\nomenclature[z-PIC]{PIC}{Particle-in-cell}\n\\nomenclature[z-USF]{USF}{Update Stress First}\n\\nomenclature[z-USL]{USL}{Update Stress Last}\n\\nomenclature[s-crit]{crit}{Critical state}\n\\nomenclature[z-DKT]{DKT}{Draft Kiss Tumble}\n\\nomenclature[z-PPC]{PPC}{Particles per cell}\n\n\\chapter{My second chapter}\n\n\\ifpdf\n \\graphicspath{{Chapter2\/Figs\/Raster\/}{Chapter2\/Figs\/PDF\/}{Chapter2\/Figs\/}}\n\\else\n \\graphicspath{{Chapter2\/Figs\/Vector\/}{Chapter2\/Figs\/}}\n\\fi\n\n\n\\section[Short title]{Reasonably long section title}\n\nI'm going to randomly include a picture Figure~\\ref{fig:minion}.\n\n\nIf you have trouble viewing this document contact Krishna at: \\href{mailto:kks32@cam.ac.uk}{kks32@cam.ac.uk} or raise an issue at \\url{https:\/\/github.com\/kks32\/phd-thesis-template\/}\n\n\n\\begin{figure}[htbp!] \n\\centering \n\\includegraphics[width=1.0\\textwidth]{minion}\n\\caption[Minion]{This is just a long figure caption for the minion in Despicable Me from Pixar}\n\\label{fig:minion}\n\\end{figure}\n\n\n\\section*{Enumeration}\nLorem ipsum dolor sit amet, consectetur adipiscing elit. Sed vitae laoreet lectus. Donec lacus quam, malesuada ut erat vel, consectetur eleifend tellus. Aliquam non feugiat lacus. Interdum et malesuada fames ac ante ipsum primis in faucibus. Quisque a dolor sit amet dui malesuada malesuada id ac metus. Phasellus posuere egestas mauris, sed porta arcu vulputate ut. Donec arcu erat, ultrices et nisl ut, ultricies facilisis urna. Quisque iaculis, lorem non maximus pretium, dui eros auctor quam, sed sodales libero felis vel orci. Aliquam neque nunc, elementum id accumsan eu, varius eu enim. Aliquam blandit ante et ligula tempor pharetra. Donec molestie porttitor commodo. Integer rutrum turpis ac erat tristique cursus. Sed venenatis urna vel tempus venenatis. Nam eu rhoncus eros, et condimentum elit. Quisque risus turpis, aliquam eget euismod id, gravida in odio. Nunc elementum nibh risus, ut faucibus mauris molestie eu.\n Vivamus quis nunc nec nisl vulputate fringilla. Duis tempus libero ac justo laoreet tincidunt. Fusce sagittis gravida magna, pharetra venenatis mauris semper at. Nullam eleifend felis a elementum sagittis. In vel turpis eu metus euismod tempus eget sit amet tortor. Donec eu rhoncus libero, quis iaculis lectus. Aliquam erat volutpat. Proin id ullamcorper tortor. Fusce vestibulum a enim non volutpat. Nam ut interdum nulla. Proin lacinia felis malesuada arcu aliquet fringilla. Aliquam condimentum, tellus eget maximus porttitor, quam sem luctus massa, eu fermentum arcu diam ac massa. Praesent ut quam id leo molestie rhoncus. Praesent nec odio eget turpis bibendum eleifend non sit amet mi. Curabitur placerat finibus velit, eu ultricies risus imperdiet ut. Suspendisse lorem orci, luctus porta eros a, commodo maximus nisi.\n\nNunc et dolor diam. Phasellus eu justo vitae diam vehicula tristique. Vestibulum vulputate cursus turpis nec commodo. Etiam elementum sit amet erat et pellentesque. In eu augue sed tortor mollis tincidunt. Mauris eros dui, sagittis vestibulum vestibulum vitae, molestie a velit. Donec non felis ut velit aliquam convallis sit amet sit amet velit. Aliquam vulputate, elit in lacinia lacinia, odio lacus consectetur quam, sit amet facilisis mi justo id magna. Curabitur aliquet pulvinar eros. Cras metus enim, tristique ut magna a, interdum egestas nibh. Aenean lorem odio, varius a sollicitudin non, cursus a odio. Vestibulum ante ipsum primis in faucibus orci luctus et ultrices posuere cubilia Curae; \n\\begin{enumerate}\n\\item The first topic is dull\n\\item The second topic is duller\n\\begin{enumerate}\n\\item The first subtopic is silly\n\\item The second subtopic is stupid\n\\end{enumerate}\n\\item The third topic is the dullest\n\\end{enumerate}\nMorbi bibendum est aliquam, hendrerit dolor ac, pretium sem. Nunc molestie, dui in euismod finibus, nunc enim viverra enim, eu mattis mi metus id libero. Cras sed accumsan justo, ut volutpat ipsum. Nam faucibus auctor molestie. Morbi sit amet eros a justo pretium aliquet. Maecenas tempor risus sit amet tincidunt tincidunt. Curabitur dapibus gravida gravida. Vivamus porta ullamcorper nisi eu molestie. Ut pretium nisl eu facilisis tempor. Nulla rutrum tincidunt justo, id placerat lacus laoreet et. Sed cursus lobortis vehicula. Donec sed tortor et est cursus pellentesque sit amet sed velit. Proin efficitur posuere felis, porta auctor nunc. Etiam non porta risus. Pellentesque lacinia eros at ante iaculis, sed aliquet ipsum volutpat. Suspendisse potenti.\n\nUt ultrices lectus sed sagittis varius. Nulla facilisi. Nullam tortor sem, placerat nec condimentum eu, tristique eget ex. Nullam pretium tellus ut nibh accumsan elementum. Aliquam posuere gravida tellus, id imperdiet nulla rutrum imperdiet. Nulla pretium ullamcorper quam, non iaculis orci consectetur eget. Curabitur non laoreet nisl. Maecenas lacinia, lorem vel tincidunt cursus, odio lorem aliquet est, gravida auctor arcu urna id enim. Morbi accumsan bibendum ipsum, ut maximus dui placerat vitae. Nullam pretium ac tortor nec venenatis. Nunc non aliquet neque. \n\n\\section*{Itemize}\n\\begin{itemize}\n\\item The first topic is dull\n\\item The second topic is duller\n\\begin{itemize}\n\\item The first subtopic is silly\n\\item The second subtopic is stupid\n\\end{itemize}\n\\item The third topic is the dullest\n\\end{itemize}\n\n\\section*{Description}\n\\begin{description}\n\\item[The first topic] is dull\n\\item[The second topic] is duller\n\\begin{description}\n\\item[The first subtopic] is silly\n\\item[The second subtopic] is stupid\n\\end{description}\n\\item[The third topic] is the dullest\n\\end{description}\n\n\n\\clearpage\n\n\\tochide\\section{Hidden section}\n\\textbf{Lorem ipsum dolor sit amet}, \\textit{consectetur adipiscing elit}. In magna nisi, aliquam id blandit id, congue ac est. Fusce porta consequat leo. Proin feugiat at felis vel consectetur. Ut tempus ipsum sit amet congue posuere. Nulla varius rutrum quam. Donec sed purus luctus, faucibus velit id, ultrices sapien. Cras diam purus, tincidunt eget tristique ut, egestas quis nulla. Curabitur vel iaculis lectus. Nunc nulla urna, ultrices et eleifend in, accumsan ut erat. In ut ante leo. Aenean a lacinia nisl, sit amet ullamcorper dolor. Maecenas blandit, tortor ut scelerisque congue, velit diam volutpat metus, sed vestibulum eros justo ut nulla. Etiam nec ipsum non enim luctus porta in in massa. Cras arcu urna, malesuada ut tellus ut, pellentesque mollis risus.Morbi vel tortor imperdiet arcu auctor mattis sit amet eu nisi. Nulla gravida urna vel nisl egestas varius. Aliquam posuere ante quis malesuada dignissim. Mauris ultrices tristique eros, a dignissim nisl iaculis nec. Praesent dapibus tincidunt mauris nec tempor. Curabitur et consequat nisi. Quisque viverra egestas risus, ut sodales enim blandit at. Mauris quis odio nulla. Cras euismod turpis magna, in facilisis diam congue non. Mauris faucibus nisl a orci dictum, et tempus mi cursus.\n\nEtiam elementum tristique lacus, sit amet eleifend nibh eleifend sed \\footnote{My footnote goes blah blah blah! \\dots}. Maecenas dapibu augue ut urna malesuada, non tempor nibh mollis. Donec sed sem sollicitudin, convallis velit aliquam, tincidunt diam. In eu venenatis lorem. Aliquam non augue porttitor tellus faucibus porta et nec ante. Proin sodales, libero vitae commodo sodales, dolor nisi cursus magna, non tincidunt ipsum nibh eget purus. Nam rutrum tincidunt arcu, tincidunt vulputate mi sagittis id. Proin et nisi nec orci tincidunt auctor et porta elit. Praesent eu dolor ac magna cursus euismod. Integer non dictum nunc.\n\n\n\\begin{landscape}\n\n\\section*{Subplots}\nI can cite Wall-E (see Fig.~\\ref{fig:WallE}) and Minions in despicable me (Fig.~\\ref{fig:Minnion}) or I can cite the whole figure as Fig.~\\ref{fig:animations}\n\n\n\\begin{figure}\n \\centering\n \\begin{subfigure}[b]{0.3\\textwidth}\n \\includegraphics[width=\\textwidth]{TomandJerry}\n \\caption{Tom and Jerry}\n \\label{fig:TomJerry} \n \\end{subfigure} \n \\begin{subfigure}[b]{0.3\\textwidth}\n \\includegraphics[width=\\textwidth]{WallE}\n \\caption{Wall-E}\n \\label{fig:WallE}\n \\end{subfigure} \n \\begin{subfigure}[b]{0.3\\textwidth}\n \\includegraphics[width=\\textwidth]{minion}\n \\caption{Minions}\n \\label{fig:Minnion}\n \\end{subfigure}\n \\caption{Best Animations}\n \\label{fig:animations}\n\\end{figure}\n\n\n\\end{landscape}\n\n\n\\chapter{My third chapter}\n\n\\ifpdf\n \\graphicspath{{Chapter3\/Figs\/Raster\/}{Chapter3\/Figs\/PDF\/}{Chapter3\/Figs\/}}\n\\else\n \\graphicspath{{Chapter3\/Figs\/Vector\/}{Chapter3\/Figs\/}}\n\\fi\n\n\\section{First section of the third chapter}\nAnd now I begin my third chapter here \\dots\n\nAnd now to cite some more people~\\citet{Rea85,Ancey1996}\n\n\\subsection{First subsection in the first section}\n\\dots and some more \n\n\\subsection{Second subsection in the first section}\n\\dots and some more \\dots\n\n\\subsubsection{First subsub section in the second subsection}\n\\dots and some more in the first subsub section otherwise it all looks the same\ndoesn't it? well we can add some text to it \\dots\n\n\\subsection{Third subsection in the first section}\n\\dots and some more \\dots\n\n\\subsubsection{First subsub section in the third subsection}\n\\dots and some more in the first subsub section otherwise it all looks the same\ndoesn't it? well we can add some text to it and some more and some more and\nsome more and some more and some more and some more and some more \\dots\n\n\\subsubsection{Second subsub section in the third subsection}\n\\dots and some more in the first subsub section otherwise it all looks the same\ndoesn't it? well we can add some text to it \\dots\n\n\\section{Second section of the third chapter}\nand here I write more \\dots\n\n\\section{The layout of formal tables}\nThis section has been modified from ``Publication quality tables in \\LaTeX*''\n by Simon Fear.\n\nThe layout of a table has been established over centuries of experience and \nshould only be altered in extraordinary circumstances. \n\nWhen formatting a table, remember two simple guidelines at all times:\n\n\\begin{enumerate}\n \\item Never, ever use vertical rules (lines).\n \\item Never use double rules.\n\\end{enumerate}\n\nThese guidelines may seem extreme but I have\nnever found a good argument in favour of breaking them. For\nexample, if you feel that the information in the left half of\na table is so different from that on the right that it needs\nto be separated by a vertical line, then you should use two\ntables instead. Not everyone follows the second guideline:\n\nThere are three further guidelines worth mentioning here as they\nare generally not known outside the circle of professional\ntypesetters and subeditors:\n\n\\begin{enumerate}\\setcounter{enumi}{2}\n \\item Put the units in the column heading (not in the body of\n the table).\n \\item Always precede a decimal point by a digit; thus 0.1\n {\\em not} just .1.\n \\item Do not use `ditto' signs or any other such convention to\n repeat a previous value. In many circumstances a blank\n will serve just as well. If it won't, then repeat the value.\n\\end{enumerate}\n\nA frequently seen mistake is to use `\\textbackslash begin\\{center\\}' \\dots `\\textbackslash end\\{center\\}' inside a figure or table environment. This center environment can cause additional vertical space. If you want to avoid that just use `\\textbackslash centering'\n\n\n\\begin{table}\n\\caption{A badly formatted table}\n\\centering\n\\label{table:bad_table}\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline \n& \\multicolumn{2}{c}{Species I} & \\multicolumn{2}{c|}{Species II} \\\\ \n\\hline\nDental measurement & mean & SD & mean & SD \\\\ \\hline \n\\hline\nI1MD & 6.23 & 0.91 & 5.2 & 0.7 \\\\\n\\hline \nI1LL & 7.48 & 0.56 & 8.7 & 0.71 \\\\\n\\hline \nI2MD & 3.99 & 0.63 & 4.22 & 0.54 \\\\\n\\hline \nI2LL & 6.81 & 0.02 & 6.66 & 0.01 \\\\\n\\hline \nCMD & 13.47 & 0.09 & 10.55 & 0.05 \\\\\n\\hline \nCBL & 11.88 & 0.05 & 13.11 & 0.04\\\\ \n\\hline \n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\caption{A nice looking table}\n\\centering\n\\label{table:nice_table}\n\\begin{tabular}{l c c c c}\n\\hline \n\\multirow{2}{*}{Dental measurement} & \\multicolumn{2}{c}{Species I} & \\multicolumn{2}{c}{Species II} \\\\ \n\\cline{2-5}\n & mean & SD & mean & SD \\\\ \n\\hline\nI1MD & 6.23 & 0.91 & 5.2 & 0.7 \\\\\n\nI1LL & 7.48 & 0.56 & 8.7 & 0.71 \\\\\n\nI2MD & 3.99 & 0.63 & 4.22 & 0.54 \\\\\n\nI2LL & 6.81 & 0.02 & 6.66 & 0.01 \\\\\n\nCMD & 13.47 & 0.09 & 10.55 & 0.05 \\\\\n\nCBL & 11.88 & 0.05 & 13.11 & 0.04\\\\ \n\\hline \n\\end{tabular}\n\\end{table}\n\n\n\\begin{table}\n\\caption{Even better looking table using booktabs}\n\\centering\n\\label{table:good_table}\n\\begin{tabular}{l c c c c}\n\\toprule\n\\multirow{2}{*}{Dental measurement} & \\multicolumn{2}{c}{Species I} & \\multicolumn{2}{c}{Species II} \\\\ \n\\cmidrule{2-5}\n & mean & SD & mean & SD \\\\ \n\\midrule\nI1MD & 6.23 & 0.91 & 5.2 & 0.7 \\\\\n\nI1LL & 7.48 & 0.56 & 8.7 & 0.71 \\\\\n\nI2MD & 3.99 & 0.63 & 4.22 & 0.54 \\\\\n\nI2LL & 6.81 & 0.02 & 6.66 & 0.01 \\\\\n\nCMD & 13.47 & 0.09 & 10.55 & 0.05 \\\\\n\nCBL & 11.88 & 0.05 & 13.11 & 0.04\\\\ \n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\n\n\n\n\n\n\\section*{Acknowledgments}\n\nI would like to offer my sincere gratitude to my advisor Prof. Dr. Winfried Hochst\u00e4ttler for\nalways providing me with the necessary input and feedback during all stages of my studies and research. \nHis profound knowledge and open-hearted support have been a great resource throughout the whole process of preparing this work.\n\nFurthermore, I would like to thank all my coworkers and colleagues both at the chair for Discrete Mathematics and Optimization\nand at the Department of Mathematics and Computer Science of the FernUniversit\u00e4t in Hagen, as well as those doing related research\nwho are scattered around the world, for the fruitful discussions with them and for sharing their knowledge with me. \n\nI am indebted to the FernUniversit\u00e4t that granted me a scholarship for the last six months, which allowed me to focus \nentirely on my thesis.\n\nFinally, I would like to thank my wife, my parents, my family, and my friends for their moral support,\nespecially during the more intensive days of preparation.\n\\chapter{Preliminaries}\n\n\\marginpar{Jan 3rd}\nIn this chapter, we introduce those aspects of matroid theory that\nare most important to the comprehension of the later chapters. For a thorough\nintroduction to matroid theory, we would like to redirect the reader to the\nfollowing books, in no particular order.\n\n\\begin{itemize}\n\\item {\\em Matroid Theory} \n\t\tby J.G.~Oxley \\cite{Ox11} is a comprehensive resource on matroid theory covering\n\t\tmost of the current state of the art. Matroids are introduced using a variety\n\t\tof cryptomorphic axiom systems starting from independence axioms and base\n\t\taxioms. This book is the authoritative standard reference for matroid\n\t\ttheory and we guarantee that all definitions made in this work are compatible\n\t\twith those found in J.G.~Oxley's book.\n\\item {\\em Matroid Theory}\n\t\tby D.J.A.~Welsh \\cite{We76} is an introduction to matroid theory that also covers\n\t\tthe greedy algorithm, transversal theory, Menger's Theorem and gammoids,\n\t\tpolymatroids, and infinite generalizations of matroids. Although this book is not the\n\t\tmost recent one on this topic, it is the book that we would like to recommend \n\t\tto anyone\n\t\twho wants to read {\\em only one} book on matroid theory, as it presents the theory in\n\t\tremarkable clarity.\n\\item {\\em On the Foundations of Combinatorial Theory: Combinatorial Geometries} \n\t\tby H.H.~Crapo and G.-C.~Rota \\cite{CR70} is a remarkably well structured\n\t\tintroduction to matroid theory with lattice theory as a starting point.\n\t\tUnfortunately, a regular edition never followed the preliminary edition.\n\\end{itemize}\n\n\\needspace{4\\baselineskip}\n\\section*{Notation}\n\n\\marginpar{Jan 3rd}\nAll notation used in this work is either standard mathematical notation, or\ndeclared in the corresponding definitions. We would like to point out one less\ncommon notational detail: If we denote a set $X=\\SET{a,b,c}$ we are stating\nthat the set $X$ consists of the elements $a$, $b$, and $c$; but we do not\nrequire any two or all three of $a$,$b$,$c$ to be distinct elements. Thus\n$|X|=1$, $|X|=2$, and $|X|=3$ are possibly true assertions with this\nnotation. But if we denote a set $Y=\\dSET{a,b,c}$,\\label{n:dset} then we are stating that\n$Y$ consists of the elements $a$, $b$, and $c$; and that no two of these\nelements are equal, therefore $|Y|=3$ is the only possibility here.\n\nWe will denote the set of non-negative integers by $\\mathbb{N}=\\SET{0,1,2,\\ldots}$,\nthe set of integers by $\\mathbb{Z} = \\SET{0,1,-1,2,-2,\\ldots}$, the field of the rational numbers by ${\\mathbb{Q}}$, and\nthe field of the real numbers by $\\mathbb{R}$.\n The cardinality of a set $X$ is denoted by $\\left| X \\right|$, the power set of $X$ is denoted by $2^X$.\n The set of subsets of $X$ with cardinality $n$ is denoted by $\\binom{X}{n}$. The set of all maps $f\\colon X \\longrightarrow Y$ is denoted by $Y^X$.\n\n\nIf $f\\colon X\\longrightarrow Y$ is a map and $X'\\subseteq X$, then we denote the set of images of $x'\\in X'$ under $f$ by\n$f[X'] = \\SET{f(x')\\mid x'\\in X'}$.\\label{n:fsquareX} We denote the restriction of $f$ to $X'$ by $f|_{X'}$.\\label{n:frestrictX'}\n\nWhenever ${\\mathcal{A}} \\subseteq 2^X$ is a family of sets, we denote\nthe union of all those sets by \\( \\bigcup {\\mathcal{A}} = \\bigcup_{A\\in{\\mathcal{A}}} A \\).\\label{n:bigcup}\nIf ${\\mathcal{A}}\\not= \\emptyset$, we denote the intersection of all sets in ${\\mathcal{A}}$ by\n\\(\\bigcap {\\mathcal{A}} = \\bigcap_{A\\in{\\mathcal{A}}} A\\). For ${\\mathcal{A}}=\\emptyset$, we set $\\bigcap {\\mathcal{A}} = \\bigcap_X \\emptyset = X$.\\label{n:bigcap}\n\n\\PRFR{Mar 7th}\nWe use the $O$-notation in the usual way: If $f,g,h \\colon \\mathbb{N} \\longrightarrow \\mathbb{R}$ are maps,\nwe write $f = O(g)$ in order to denote that $\\limsup_{x\\rightarrow \\infty} \\left| \\frac{f(x)}{g(x)} \\right| < \\infty$.\nWe write $O(g) = O(h)$ if the implication $f = O(g) \\Rightarrow f = O(h)$ holds for all $f\\in \\mathbb{R}^\\mathbb{N}$. Please keep in mind that\n$O(g) = O(h)$ is not\nequivalent to $O(h) = O(g)$. (!) Instead, the $O$-notation is asymmetric and has to be read from left-to-right.\nWe also use the straight-forward generalization of the $O$-notation to several non-negative integer variables in\nan informal way, for instance we would write $O(x^2 y^3) = O(2^x y^4)$. Similarly, we write $f = \\Omega(g)$ in order to\ndenote that $\\limsup_{x\\rightarrow \\infty} \\left| \\frac{f(x)}{g(x)} \\right| > 0$.\n\n\n\n\n\\section{Matroid Basics}\n\\marginpar{Jan 3rd}\n\nIn this section, we give a quick and incomplete review of some axiomatizations\nof matroids. A more complete picture as well as some proofs\\footnote{Some\naxiomatizations can be found in the exercise sections, where, of cause, the\nproofs are left for the reader.} of cryptomorphy can be obtained from J.G.~Oxley's\nbook \\cite{Ox11}. \n\n\n\\subsection{Independence Axioms}\n\\marginpar{Jan 3rd}\n\nAll definitions, lemmas, theorems, and proofs in this subsection are canonical and \ncan be found in \\cite{Ox11}. \nReaders familiar with matroid theory may safely skip this section.\n\n\\begin{definition}\\label{def:indepAxioms}\\label{n:Is}\\marginpar{Jan 3rd}\n\tLet $E$ be a finite set, ${\\mathcal{I}} \\subseteq 2^{E}$. Then the pair $(E,{\\mathcal{I}})$\\label{n:EI} is\n\tan \\deftext{independence matroid}, or shorter \\deftext{matroid}, if\n\tthe following properties hold:\n\t\\begin{enumerate}\n\t\t\\item[(I1)] $\\emptyset \\in {\\mathcal{I}}$,\n\t\t\\item[(I2)] for $I\\in{\\mathcal{I}}$ and every $J\\subseteq I$, we have $J\\in{\\mathcal{I}}$.\n\t\t\\item[(I3)] If $J,I\\in{\\mathcal{I}}$ and $|J|<|I|$, then there is some $i\\in I\\backslash J$, such that $J\\cup\\SET{i}\\in {\\mathcal{I}}$.\n\t\\end{enumerate}\n\tLet $X\\subseteq E$, we say that $X$ is \\deftext{independent} in the matroid $M=(E,{\\mathcal{I}})$,\n\t if $X\\in{\\mathcal{I}}$. Otherwise, we say that $X$ is \\deftext{dependent} in $M$.\n\\end{definition}\n\n\\begin{example}\\label{ex:freematroid}\\marginpar{Jan 3rd}\n\tLet $E$ be any finite set, then the \\deftext{free matroid} on the ground set $E$\n\tshall be the matroid $M=(E,{\\mathcal{I}})$ where all subsets of $E$ are independent, i.e. where ${\\mathcal{I}} = 2^{E}$.\n\\end{example}\n\n\\noindent\nMatroids have the natural concept of isomorphy.\n\n\\begin{definition}\\PRFR{Feb 15th}\n\tLet $M=(E,{\\mathcal{I}})$ and $N=(E',{\\mathcal{I}}')$ be matroids. A bijective map\n\t$$\\phi\\colon E\\longrightarrow E'$$ is called \\deftext[matroid isomorphism]{matroid isomorphism between $\\bm M$ and $\\bm N$}, if\n\tfor all $X\\subseteq E$\n\t\\[ X\\in {\\mathcal{I}} \\quad\\Longleftrightarrow\\quad \\phi[X] \\in {\\mathcal{I}}' \\]\n\tholds. As usual, an \\deftext[M-automorphism@$M$-automorphism]{$\\bm M$-automorphism} is a matroid isomorphism between $M$ and itself.\n\\end{definition}\n\n\n\\noindent For now, we will stick to the independence axioms of matroids and define the typical matroid\n concepts in terms of their independence systems.\n\n\\begin{definition}\\label{def:directSum}\\PRFR{Mar 7th}\n\tLet $M=(E,{\\mathcal{I}})$ and $N=(E',{\\mathcal{I}}')$ be matroids such that $E\\cap E' = \\emptyset$.\n\tThen the \\deftext[direct sum of matroids]{direct sum of $\\bm M$ and $\\bm N$} is the matroid $M \\oplus N = (E\\cup E', {\\mathcal{I}}_\\oplus)$ \n\twhere\n\t\\[ {\\mathcal{I}}_\\oplus = \\SET{X\\cup X' ~\\middle|~ X\\in {\\mathcal{I}},\\,X'\\in {\\mathcal{I}}'}. \\qedhere\\]\n\\end{definition}\n\n\\begin{lemma}\\PRFR{Mar 7th}\n\tLet $M=(E,{\\mathcal{I}})$ and $N=(E',{\\mathcal{I}}')$ be matroids such that $E\\cap E' = \\emptyset$. Then $M\\oplus N$ is indeed a matroid.\n\\end{lemma}\n\\begin{proof}\\PRFR{Mar 7th}\n\tEach matroid axiom may be easily deduced from the fact that every summand satisfies that axiom:\n\t$\\emptyset\\in {\\mathcal{I}}_\\oplus$ since $\\emptyset \\in {\\mathcal{I}}$ and $\\emptyset \\in {\\mathcal{I}}'$, {\\em(I1)} holds.\n\t Let $X\\cup X' \\in {\\mathcal{I}}_\\oplus$ for\n\tsome $X\\in {\\mathcal{I}}$ and $X'\\in {\\mathcal{I}}'$. Let $Y\\subseteq X\\cup X'$, then $Y = (Y\\cap X) \\cup (Y\\cap X')$, and since $(Y\\cap X) \\subseteq X$\n\tand $(Y\\cap X')\\subseteq X'$, we have $(Y\\cap X) \\in {\\mathcal{I}}$ and $(Y\\cap X')\\in {\\mathcal{I}}'$, therefore $Y\\in {\\mathcal{I}}_\\oplus$, \n\t{\\em(I2)} holds. Let $X\\cup X'\\in {\\mathcal{I}}_\\oplus$ and $Y\\cup Y'\\in {\\mathcal{I}}_\\oplus$ with\n\t$\\left| X\\cup X' \\right| < \\left| Y\\cup Y' \\right|$, i.e. $X,Y\\in {\\mathcal{I}}$ and $X',Y'\\in {\\mathcal{I}}'$, and\n\t$\\left| X \\right| + \\left| X' \\right| < \\left| Y \\right| + \\left| Y' \\right|$. By symmetry we may assume without loss of generality that $\\left| X \\right| < \t\\left| Y \\right|$. Then there is some $y\\in Y\\backslash X$ such that $X\\cup\\SET{y} \\in {\\mathcal{I}}$,\n\ttherefore $X\\cup\\SET{y}\\cup X' \\in {\\mathcal{I}}_\\oplus$, thus {\\em(I3)} holds.\n\\end{proof}\n\n\n\\begin{definition} \\marginpar{Jan 3rd}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid. Every maximal element of ${\\mathcal{I}}$ is called \n\ta \\deftext{base} of $M$. For $F\\subseteq E$, every maximal element of\n\t$\\SET{I\\in{\\mathcal{I}} \\mid I\\subseteq F}$ is called a \\deftext[base of F@base of $F$]{base of $\\bm F$} in $M$.\n\tThe family of all bases of $M$ shall be denoted by \\label{n:BcalM} ${\\mathcal{B}}(M)$, and the family of all bases of $F$ in $M$\n\tshall be denoted by\\label{n:BcalMF} ${\\mathcal{B}}_M(F)$.\n\\end{definition}\n\n\\noindent It is an important property of matroids, that for every\n$F\\subseteq E$, the bases of $F$ have the same cardinality; and that every\nindependent subset of $F$ can be augmented to a base of $F$. Likewise, any\nset independent in a matroid $M$ can be augmented to a base of $M$.\n\n\\needspace{7\\baselineskip}\n\\begin{lemma}\\label{lem:augmentation}\\marginpar{Jan 3rd}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, and let $F\\subseteq H\\subseteq E$ with $F\\in{\\mathcal{I}}$.\n\tThen there is a subset $G\\in {\\mathcal{I}}$ with $F\\subseteq G\\subseteq H$, such that\n\t\\( \\left| G\\right| = \\max \\SET{\\vphantom{A^A} \\left| I\\right| ~\\middle|~ I\\in {\\mathcal{I}},\\,I\\subseteq H} \\).\n\\end{lemma}\n\\begin{proof}\\marginpar{Jan 3rd}\n\tLet ${\\mathcal{I}}' = \\SET{I\\in {\\mathcal{I}}\\mid F\\subseteq I \\subseteq H}$. Clearly, $F\\in {\\mathcal{I}}'$ and ${\\mathcal{I}}'$ is finite,\n\t therefore there is an element $G\\in {\\mathcal{I}}'$ which is maximal with respect to set-inclusion $\\subseteq$. Now assume that $\\left| G \\right| < \\left| I \\right|$ for some $I\\in {\\mathcal{I}}$ with $I\\subseteq H$. By {\\em (I3)} there is an element $i\\in I\\backslash G$\n\tsuch that $G\\cup\\SET{i}\\in {\\mathcal{I}}$. But $i\\in I\\subseteq H$, therefore $G\\cup\\SET{i} \\in {\\mathcal{I}}'$, which contradicts the choice of $G$ as $\\subseteq$-maximal element of ${\\mathcal{I}}'$. Thus\n\t$\\left|G\\right| = \\max \\SET{\\vphantom{A^A} \\left| I\\right| ~\\middle|~ I\\in {\\mathcal{I}},\\,I\\subseteq H}$.\n\\end{proof}\n\n\\needspace{4\\baselineskip}\n\\begin{corollary}\\label{cor:equicardinality}\\marginpar{Jan 3rd}\nLet $M=(E,{\\mathcal{I}})$ be a matroid, $H\\subseteq E$. Let $F,G$ be maximal elements in $\\SET{X\\in{\\mathcal{I}}\\mid X\\subseteq H}$ with respect to set-inclusion. Then $\\left| F \\right|=\\left| G\\right|$.\n\\end{corollary}\n\\begin{proof}\\marginpar{Jan 3rd}\n\tIf, without loss of generality, $\\left| F\\right| < \\left| G\\right|$, then $F$ cannot\n\tbe maximal with respect to set-inclusion, because then Lemma~\\ref{lem:augmentation} gives\n\ta proper independent superset of $F$ in $H$.\n\\end{proof}\n\n\\begin{corollary}\\label{cor:basisexchange}\\marginpar{Jan 3rd}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, $F\\subseteq E$ and $B_{1},B_{2}\\subseteq F$ be bases of $F$ in $M$. Then the following property is satisfied:\n\t\\begin{enumerate}[align=parleft,leftmargin=2cm,labelsep=1.5cm]\\label{n:B3p}\n\t\\item[(B3')] For every element $x\\in B_{1}\\backslash B_{2}$ there is an element $y\\in B_{2}\\backslash B_{1}$, such that\n\t$\\left(B_{1}\\BSET{x}\\right)\\cup\\SET{y}$ is a base of $F$ in $M$.\n\\end{enumerate}\n\\end{corollary}\n\\begin{proof}\\marginpar{Jan 3rd}\n\n\tSince $\\left| B_1 \\right| = \\left| \\left(B_{1}\\BSET{x}\\right)\\cup\\SET{y}\n\t\\right|$ for any $x\\in B_{1}\\backslash B_{2}$ and $y\\in B_{2}\\backslash B_{1}$, it\n\tsuffices to show, that for each such $x$, there is a corresponding $y$ with\n\t$\\left(B_{1}\\BSET{x}\\right)\\cup\\SET{y}\\in {\\mathcal{I}}$. We give an indirect\n\targument. Assume that for $x\\in B_{1}\\backslash B_{2}$, there is no $y\\in\n\tB_{2}\\backslash B_{1}$ with $\\left(B_{1}\\BSET{x}\\right)\\cup\\SET{y}$ independent\n\tin $M$. Then $B_{1}\\BSET{x}$ is a base of $B' =\n\t\\left(B_1\\BSET{x}\\right) \\cup \\left(B_2\\backslash B_1 \\right)$. Clearly, $B' =\n\t\\left(B_{1} \\cup B_{2}\\right)\\BSET{x}$, but $x\\notin B_{2}$, therefore\n\t$B_{2} \\subseteq B'$. Now $B_{2}\\in {\\mathcal{I}}$ together with $\\left| B_2 \\right| > \\left| B_1\\BSET{x} \\right|$ contradicts that\n\t$B_{1}\\BSET{x}$ is a base of $B'$. Therefore, there is some $y\\in\n\tB_{2}\\backslash B_{1}$ such that $\\left(B_{1}\\BSET{x}\\right)\\cup\\SET{y}$ is a base of\n\t$F$ in $M$.\n\\end{proof}\n\n\n\\begin{lemma}\\label{lem:basisexchangesymmetric}\\marginpar{Jan 3rd}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, $F\\subseteq E$ and $B_{1},B_{2}\\subseteq F$ be bases of $F$ in $M$. For every element $y\\in B_{2}\\backslash B_{1}$ there is an element $x\\in B_{1}\\backslash B_{2}$, such that\n\t$\\left(B_{1}\\BSET{x}\\right)\\cup\\SET{y}$ is a base of $F$ in $M$.\n\\end{lemma}\n\n\\noindent D.J.A.~Welsh gives the following nice and short proof of\n this lemma in \\cite{We76}.\n\n\\begin{proof}\\marginpar{Jan 3rd}\n Let $y\\in B_{2}\\backslash B_{1}$, thus $\\SET{y}\\in{\\mathcal{I}}$. From Lemma~\\ref{lem:augmentation} we\n obtain that there is a basis $B'$ of $F' = B_1\\cup\\SET{y}$ with $\\SET{y}\\subseteq B'$.\n Since $B_1$ is a base of $F$ and a proper subset of $F'\\subseteq F$, $F'$ is dependent. Thus $B'$ is a proper subset of $F'$ and therefore there is an element\n $x\\in B_{1}\\backslash B'$. Since $B_{1}$ and $B'$ are bases of $F'=B_{1}\\cup\\SET{y}=B'\\cup\\SET{x}$ in $M$,\n and $B_{1}$ and $B_{2}$ are bases of $F$ in $M$,\n we have $\\left| B' \\right| = \\left| B_{1}\\right| = \\left| B_{2}\\right|$,\n so $B'=(B_{1}\\BSET{x})\\cup\\SET{y}$ is a base of $F$ in $M$, too.\n\\end{proof}\n\n\n\\begin{definition} \\marginpar{Jan 3rd}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid. A set $C\\subseteq E$ is called \\deftext{circuit} of $M$, if $C$ is dependent, yet any proper subset of $C$ is independent in $M$. The set of circuits of $M$ is denoted by\\label{n:CM}\n\t\\[ {\\mathcal{C}}(M) = \\SET{\\vphantom{A^A}C\\subseteq E ~\\middle|~C\\notin {\\mathcal{I}},\\,\\forall c\\in C\\colon \\,C\\BSET{c}\\in {\\mathcal{I}}}. \\qedhere\\]\n\\end{definition}\n\n\\noindent Obviously, we may restore ${\\mathcal{I}}$ from ${\\mathcal{C}}(M)$ since the independent sets of $M$ are those subsets of $E$,\n which do not contain a circuit. The following property of ${\\mathcal{C}}(M)$\n is called \\deftext{strong circuit elimination} and also plays a role\n in axiomatizing matroids using axioms governing its family of circuits.\n\n\\begin{lemma}[\\cite{Ox11}, Proposition~1.4.12]\\label{lem:strongCircuitElimination}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, and let $C_1, C_2\\in{\\mathcal{C}}(M)$ be circuits of $M$.\n\tFurthermore, let $e\\in C_1\\cap C_2$ and $f\\in C_1\\backslash C_2$.\n\tThen there is a circuit $C'\\in {\\mathcal{C}}(M)$ such that\n\t$f\\in C'$ and $C' \\subseteq \\left( C_1\\cup C_2 \\right)\\BSET{e}$.\n\\end{lemma}\n\n\\noindent For a proof, see \\cite{Ox11}, p.29.\n\n\\begin{definition}\\label{def:matroidLoop}\\label{def:parallelEdgesMatroid}\\marginpar{Jan 3rd}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, $l\\in E$. Then $l$ is called a \\deftext[loop (matroid)]{loop in $\\bm M$}, if\n\tthe singleton\n\t$\\SET{l}$ is a circuit of $M$.\n\tLet $p_1,p_2\\in E$ such that $p_1\\not= p_2$. Then $p_1$ and $p_2$ are called \\deftext[parallel edges (matroid)]{parallel edges in $\\bm M$},\n\tif $\\SET{p_1,p_2}$ is a circuit of $M$.\n\tLet $c\\in E$ such that for all bases $B$ of $M$, $c\\in B$. Then $c$ is called a \\deftext[coloop]{coloop in $\\bm M$}.\n\\end{definition}\n\n\n\\begin{definition}\\label{def:rank}\\marginpar{Jan 3rd}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid. The \\deftext{rank function} of $M$ shall be\n\t the map\\label{n:rkM}\n\t\\[ \\mathrm{rk}_M \\colon 2^{E} \\longrightarrow \\mathbb{N},\\,X\\mapsto \\max \\SET{\\vphantom{A^A}\\left|Y\\right| ~\\middle|~ Y\\subseteq X,\\,Y\\in {\\mathcal{I}}} .\\]\n\tIf the matroid $M$ is clear from the context, we denote $\\mathrm{rk}_{M}$ by $\\mathrm{rk}$.\n\\end{definition}\n\n\\noindent Again, ${\\mathcal{I}}$ may be retrieved from $\\mathrm{rk}_{M}$ since the independent sets are precisely those elements of the \ndomain $2^E$ of $\\mathrm{rk}_M$, for which the cardinality and the image under the rank function coincide.\n\n\\begin{lemma}\\label{lem:rankMonotone}\\marginpar{Jan 3rd}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, and $X\\subseteq Y\\subseteq E$. Then $\\mathrm{rk}(X) \\leq \\mathrm{rk}(Y)$.\n\\end{lemma}\n\\begin{proof}\\marginpar{Jan 3rd}\n\tSince $\\SET{I\\in {\\mathcal{I}}\\mid I\\subseteq X} \\subseteq \\SET{I\\in {\\mathcal{I}}\\mid I\\subseteq Y}$\n\tthe maximum expression for $\\mathrm{rk}(Y)$ ranges over a superset of the expression for $\\mathrm{rk}(X)$\n\tand therefore cannot be smaller.\n\\end{proof}\n\n\\begin{definition}\\label{def:clM}\\label{def:FcalM}\\marginpar{Jan 3rd}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid. A set $F\\subseteq E$ is called \\deftext{flat} of $M$,\n\tif for all $x\\in E\\backslash F$, the equality $\\mathrm{rk}(F\\cup\\SET{x}) = \\mathrm{rk}(F) + 1$\n\tholds. The family of all flats of $M$ is denoted by\\label{n:FM}\n\t\\[ {\\mathcal{F}}(M) = \\SET{ \\vphantom{A^A}X\\subseteq E ~\\middle|~ \\forall y\\in E\\backslash X\\colon\\, \\mathrm{rk}_M(X) < \\mathrm{rk}_M(X\\cup\\SET{y})}.\\]\n\tThe \\deftext[closure operator of M@closure operator of $M$]{closure operator of $\\bm M$} is defined to be the map\\label{n:clM}\n\t\\[ \\mathrm{cl}_{M}\\colon 2^{E} \\longrightarrow 2^{E},\\, X\\mapsto \\bigcap\\SET{F\\in {\\mathcal{F}}(M)\\mid X\\subseteq F}.\\]\n\tIf the matroid $M$ is clear from the context, we denote $\\mathrm{cl}_{M}$ by $\\mathrm{cl}$.\n\\end{definition}\n\n\\noindent\n\tClearly, for every matroid $M=(E,{\\mathcal{I}})$, the ground set $E\\in{\\mathcal{F}}(M)$ is a flat, and therefore the defining expression of $\\mathrm{cl}(X)$ is well-defined, as it is never an intersection of an empty family.\n The following properties are easy consequences from the definition of\nthe closure operator.\n\n\n\\begin{lemma}\\label{lem:clFlips}\\marginpar{Jan 3rd}\n Let $M=(E,{\\mathcal{I}})$ be a matroid, $X\\subseteq Y\\subseteq E$.\n Then $X\\subseteq \\mathrm{cl}(X) \\subseteq \\mathrm{cl}(Y)$.\n\\end{lemma}\n\n\\begin{proof}\n Since $\\emptyset \\not= \\SET{F\\in {\\mathcal{F}}(M)\\mid Y\\subseteq F} \\subseteq \\SET{F\\in {\\mathcal{F}}(M)\\mid X\\subseteq F}$, we have \n \\[ X\\subseteq \\mathrm{cl}(X) = \\bigcap\\SET{F\\in {\\mathcal{F}}(M)\\mid X\\subseteq F} \\subseteq \\bigcap\\SET{F\\in {\\mathcal{F}}(M)\\mid Y\\subseteq F} = \\mathrm{cl}(Y).\n \\qedhere \\]\n\\end{proof}\n\n\\begin{lemma}\\label{lem:clKeepsRank}\\marginpar{Jan 3rd}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, $X\\subseteq E$. Then $\\mathrm{rk}(X) = \\mathrm{rk}(\\mathrm{cl}(X))$.\n\\end{lemma}\n\\begin{proof}\n\tBy Lemma~\\ref{lem:clFlips} we have $X\\subseteq \\mathrm{cl}(X)$ and by Lemma~\\ref{lem:rankMonotone} we obtain that $\\mathrm{rk}(X) \\leq \\mathrm{rk}(\\mathrm{cl}(X))$. Now consider the\n\tfamily ${\\mathcal{E}} = \\SET{Y\\subseteq E\\mid X\\subseteq Y {\\mathop{\\text{~and~}}} \\mathrm{rk}(X) = \\mathrm{rk}(Y)}.$\n\tSince $X\\in {\\mathcal{E}}$ and $E$ is finite, there is a maximal element $F\\in {\\mathcal{E}}$ with respect to set-inclusion. Since $F$ is maximal, we have that $F\\in{\\mathcal{F}}(M)$.\n\tThus $\\mathrm{cl}(X) \\subseteq F$ and so $\\mathrm{rk}(\\mathrm{cl}(X)) \\leq \\mathrm{rk}(F) = \\mathrm{rk}(X)$ holds, and\n\tconsequently $\\mathrm{rk}(X) = \\mathrm{rk}(\\mathrm{cl}(X))$.\n\\end{proof}\n\n\n\\begin{lemma}\\label{lem:flatsintersectinflat}\\marginpar{Jan 3rd}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, $X\\subseteq E$. Then for every ${\\mathcal{F}}' \\subseteq {\\mathcal{F}}(M)$, $\\bigcap_E {\\mathcal{F}}' \\in {\\mathcal{F}}(M)$. Furthermore, for all $X\\subseteq E$\n\t $$\\mathrm{cl}(X) \\in {\\mathcal{F}}(M) \\text{ ~~and~~ } \\mathrm{cl}(\\mathrm{cl}(X)) = \\mathrm{cl}(X).$$\n\\end{lemma}\n\n\\begin{proof}\\marginpar{Jan 3rd}\t\n Let ${\\mathcal{F}}' \\subseteq {\\mathcal{F}}(M)$, and let $F' = \\bigcap_E\n{\\mathcal{F}}' = \\SET{x\\in E\\mid \\forall F\\in {\\mathcal{F}}'\\colon\\, x\\in F}$. Let $e\\in\nE\\backslash F'$, then there is some $F\\in {\\mathcal{F}}'$ with $e\\notin F$. Since\n$\\mathrm{rk}(F\\cup\\SET{e}) > \\mathrm{rk}(F)$ holds, for every base $B$ of $F$, we must have\n$B\\cup\\SET{e} \\in {\\mathcal{I}}$. Now let $B'\\subseteq F'$ be a base of $F'$, then by\nLemma~\\ref{lem:augmentation}, there is a base $B$ of $F$ with $B'\\subseteq B$.\nSince $B'\\cup\\SET{e}\\subseteq B\\cup\\SET{e}$, we obtain that\n$\\mathrm{rk}(F'\\cup\\SET{e}) \\geq \\left| B'\\cup\\SET{e} \\right| > \\left| B' \\right| =\n\\mathrm{rk}(F')$. Thus $F'\\in {\\mathcal{F}}(M)$.\n\n\\noindent Let $X\\subseteq E$,\nsince the closure operator $\\mathrm{cl}$ is defined to be the intersection of a family of flats of $M$,\nwe have\n$\\mathrm{cl}(X)\\in {\\mathcal{F}}(M)$. Therefore $\\mathrm{cl}(X)$ is the unique minimal element of\n $\\SET{F\\in{\\mathcal{F}}(M)\\mid X\\subseteq F}$ with respect to set-inclusion $\\subseteq$. Thus we have the following equality between subfamilies of ${\\mathcal{F}}(M)$ $$\\SET{F\\in{\\mathcal{F}}(M)\\mid X\\subseteq F}=\\SET{F\\in{\\mathcal{F}}(M)\\mid \\mathrm{cl}(X)\\subseteq F},$$\nwhich yields $\\mathrm{cl}(\\mathrm{cl}(X))=\\mathrm{cl}(X)$.\n\\end{proof}\n\n\\needspace{4\\baselineskip}\n\n\\begin{lemma}\\label{lem:clofbase}\\marginpar{Jan 3rd}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, $X \\subseteq Y \\subseteq E$.\n\tThen $\\mathrm{cl}(X) = \\mathrm{cl}(Y)$ if and only if there is a base $B$ of $Y$ with\n\t$B\\subseteq X$.\n\\end{lemma}\n\\begin{proof}\\marginpar{Jan 3rd}\n\tAssume that $\\mathrm{cl}(X) = \\mathrm{cl}(Y)$, then $\\mathrm{rk}(X) = \\mathrm{rk}(\\mathrm{cl}(X)) = \\mathrm{rk}(\\mathrm{cl}(Y)) = \\mathrm{rk}(Y)$ by Lemma~\\ref{lem:clKeepsRank}. Let $B$ be a base of $X$, then $\\mathrm{rk}(B) = \\mathrm{rk}(Y)$, so $B\\subseteq X\\subseteq Y$ is also a base of $Y$. Now assume that $\\mathrm{cl}(X)\\not=\\mathrm{cl}(Y)$, thus there is some $y\\in \\mathrm{cl}(Y) \\backslash \\mathrm{cl}(X)$ such that for some base $B$ of $\\mathrm{cl}(X)$ in $M$, $B\\cup\\SET{y}\\in{\\mathcal{I}}$ is independent. Thus $\\mathrm{rk}(Y) = \\mathrm{rk}(\\mathrm{cl}(Y)) > \\mathrm{rk}(X)$ and therefore no base $B'$ of $Y$ is a subset of $X$.\n\\end{proof}\n\n\\studyremark{\n\\remred{I AM REMOVING THIS}\n\n\\begin{definitionX}\n Let $M=(E,{\\mathcal{I}})$ be a matroid, $Z\\subseteq E$ is called \\deftext{cyclic flat} of $M$,\n if $Z\\in{\\mathcal{F}}(M)$ is a flat and if it is a union of circuits, i.e. if\n \\( F = \\displaystyle \\bigcup_{C\\in{\\mathcal{C}}(M),\\,C\\subseteq F} C\\).\n The family of all cyclic flats of $M$ is denoted by\\label{n:ZM}\n \\({\\mathcal{Z}} (M)\\).\n\\end{definitionX}\n\n\\begin{lemmaX}\n Let $M=(E,{\\mathcal{I}})$ be a matroid, $F\\in{\\mathcal{F}}(M)$.\n Then $F$ is a cyclic flat if and only if for all $f\\in F$, $\\mathrm{rk}(F) = \\mathrm{rk}(F\\BSET{f})$.\n\\end{lemmaX}\n\n\\begin{proof}\n\tLet $F$ be a cyclic flat, and let $f\\in F$. Then there is a circuit\n\t$C\\subseteq F$ with $f\\in C$. Therefore, $C\\BSET{f}\\in{\\mathcal{I}}$ and by\n\tLemma~\\ref{lem:augmentation} there is a subset $X\\subseteq F\\BSET{f}$ with\n\t$C\\BSET{f}\\subseteq X$ and $\\left| X \\right| = \\mathrm{rk}(F\\BSET{f})$. Assume\n\tthat $\\mathrm{rk}(F\\BSET{f}) < \\mathrm{rk}(F)$, then again by Lemma~\\ref{lem:augmentation}\n\tthere is a set $X'\\subseteq F\\cup\\SET{f}$ with $X\\subseteq X'$ and $\\left|\n\tX' \\right| = \\mathrm{rk}(F) > \\left| X \\right|$. But $X$ is maximally independent\n\tin $F\\BSET{f}$, thus the only possibility is $X'=X\\cup\\SET{f}$, but then\n\t$C\\subseteq X'$ which contradicts that $X'\\in{\\mathcal{I}}$. Thus $\\mathrm{rk}(F) = \\left|\n\tX \\right| = \\mathrm{rk}(F\\BSET{f})$. Conversely, let $F\\in{\\mathcal{F}}(M)$ such that for\n\tall $f\\in F$, $\\mathrm{rk}(F) = \\mathrm{rk}(F\\BSET{f})$. Assume that for some $f\\in F$,\n\tthere is no circuit $C\\subseteq F$ with $f\\in C$. Let $X\\subseteq\n\tF\\BSET{f}$ be of maximal cardinality, such that $X\\in{\\mathcal{I}}$, then $\\left|\n\tX \\right| = \\mathrm{rk}(F\\BSET{f})$. Then $X\\cup\\SET{f}\\in {\\mathcal{I}}$ because\n\t$X\\cup\\SET{f}\\subseteq F$ which has no circuit containing $f$. But then\n\t$\\mathrm{rk}(F) \\geq \\left|X \\right| + 1 > \\left| X \\right| = \\mathrm{rk}(F\\BSET{f})$\n\tyields a contradiction. Therefore, for every $f\\in F$ there is a circuit\n\t$C\\subseteq F$ with $f\\in C$.\n\\end{proof}\n}\n\n\\subsection{Rank Axioms}\n\nThere are at least two natural ways to axiomatize matroids through their corresponding rank functions.\n\n\\begin{theorem}\\label{thm:rankAxioms}\\marginpar{Jan 3rd}\n Let $E$ be a finite set, $\\rho\\colon 2^{E}\\longrightarrow \\mathbb{N}$ a map. The following are\n equivalent:\n \\begin{enumerate}\\renewcommand{\\theenumi}{(\\roman{enumi})}\\renewcommand{\\labelenumi}{\\theenumi}\n \\item There is a matroid $M=(E,{\\mathcal{I}})$ with $\\mathrm{rk}_{M} = \\rho$,\n \\item $\\rho$ satisfies the properties {\\em (R1') -- (R3')}, and\n \\item $\\rho$ satisfies the properties {\\em (R1) -- (R3)};\n \\end{enumerate}\n where\n \\begin{itemize}\\label{n:Rxp}\n \t\\item[(R1')] $\\rho(\\emptyset) = 0$,\n \\item[(R2')] $\\rho(X) \\leq \\rho(X\\cup\\SET{y}) \\leq \\rho(X) + 1$ for all $X\\subseteq E$ and all $y\\in E$,\n \\item[(R3')] if $\\rho(X) = \\rho(X\\cup\\SET{y})=\\rho(X\\cup\\SET{z})$, then $\\rho(X) = \\rho(X\\cup\\SET{y,z})$, for all $X\\subseteq E$ and all $y,z\\in E$;\n \\end{itemize}\n \\begin{itemize}\\label{n:Rx}\n \t\\item[(R1)] $0 \\leq \\rho(X) \\leq \\left|X\\right|$ for all $X\\subseteq E$,\n \\item[(R2)] if $X\\subseteq Y$, then $\\rho(X) \\leq \\rho(Y)$ for all $X,Y\\subseteq E$,\n \\item[(R3)] $\\rho(X\\cup Y) + \\rho(X\\cap Y) \\leq \\rho(X) + \\rho(Y)$ for all $X,Y\\subseteq E$.\n \\end{itemize}\n \\end{theorem}\n\\noindent We named the rank axioms coherent with J.G.~Oxley's book \\cite{Ox11}; D.J.A.~Welsh's {\\em Matroid Theory} \\cite{We76} denotes {\\em (R1)--(R3)} with {\\em (R1')--(R3')}, and vice-versa, yet the proof is more along the lines of section~1.6 in D.J.A.~Welsh's book \\cite{We76}.\n\n\\begin{proof}\n\\underline{The implication {\\em (i) $\\Rightarrow$ (ii)}.}\\marginpar{Jan 3rd}\n\\goldstar{Marc, Oct 27, whole proof}\n\\par \\vspace*{.6\\baselineskip}\\noindent \\mbox{---\\hspace*{2.5mm}} By {\\em (I1)} we\nobtain $\\mathrm{rk}(\\emptyset) = \\left|\\emptyset\\right| = 0$, thus {\\em (R1')} holds\nfor $\\mathrm{rk}$. \n\n\\par \\vspace*{.6\\baselineskip}\\noindent \\mbox{---\\hspace*{2.5mm}} \\marginpar{Jan 3rd}\nLet $X'\\in {\\mathcal{I}}$ with $X'\\subseteq X\\cup\\SET{y}$ such that\n$\\mathrm{rk}(X\\cup\\SET{y}) = \\left|X'\\right|$. By {\\em (I2)} $X'\\BSET{y}\\in{\\mathcal{I}}$,\ntherefore $\\mathrm{rk}(X\\cup\\SET{y}) \\leq \\mathrm{rk}(X) + 1$. Since every subset of $X$ is a\nsubset of $X\\cup\\SET{y}$, too, we obtain {\\em (R2')} for $\\mathrm{rk}$: $\\mathrm{rk}(X)\\leq\n\\mathrm{rk}(X\\cup\\SET{y}) \\leq \\mathrm{rk}(X) + 1$. \n\n\n\\par \\vspace*{.6\\baselineskip}\\noindent \\mbox{---\\hspace*{2.5mm}} \\marginpar{Jan 3rd}\nWe prove {\\em (R3')} via contraposition and show that $\\rho(X) \\not= \\rho(X\\cup\\SET{x,y})$ implies that $\\rho(X)\\not=\\rho(X\\cup\\SET{x})$\nor $\\rho(X)\\not=\\rho(X\\cup\\SET{y})$.\nWe may assume the non-trivial case $y,z\\notin X$.\nIf $\\mathrm{rk}(X\\cup\\SET{y,z}) > \\mathrm{rk}(X)$, then every $X'\\subseteq X\\cup\\SET{y,z}$,\nwhich has maximal cardinality such that $X'\\in{\\mathcal{I}}$, must have a non-empty\nintersection $X'\\cap \\SET{y,z}\\not= \\emptyset$, because $X'\\not\\subseteq X$.\nWithout loss of generality we may assume that $y\\in X'$. If $y=z$ or $z\\notin X'$ or $\\mathrm{rk}(X) =\n\\mathrm{rk}(X\\cup\\SET{y,z})-2$, we obtain that $\\mathrm{rk}(X\\cup\\SET{y})=\\mathrm{rk}(X)+1$. The\nremaining case is that $\\dSET{y,z} \\subseteq X'$ and $\\mathrm{rk}(X) = |X'|-1$. Let\n$\\tilde{X}\\subseteq X$ be a subset with maximal cardinality such that it is still independent, i.e. $\\tilde{X}\\in{\\mathcal{I}}$.\nSince $X'\\BSET{y,z}\\in{\\mathcal{I}}$, {\\em (I3)} yields that there is an $x\\in\\tilde{X}\\backslash X'$\nsuch that $\\left( X'\\BSET{y,z}\\right) \\cup \\SET{x}\\in {\\mathcal{I}}$. Applying {\\em\n(I3)} again yields that either $\\left(X'\\BSET{y}\\right) \\cup \\SET{x}\\in {\\mathcal{I}}$\nor $\\left( X'\\BSET{z}\\right) \\cup \\SET{x} \\in {\\mathcal{I}}$, therefore either $\\mathrm{rk}(X)\n< \\mathrm{rk}(X\\cup\\SET{y})$ or $\\mathrm{rk}(X) < \\mathrm{rk}(X\\cup\\SET{z})$. This establishes {\\em (R3')}.\n\n\\bigskip\n\\noindent \\underline{The implication {\\em (ii) $\\Rightarrow$ (iii)}:} \\marginpar{Jan 3rd}\n\n\\par \\vspace*{.6\\baselineskip}\\noindent \\mbox{---\\hspace*{2.5mm}} We show {\\em (R1)} by\ninduction on $|X|$. From {\\em (R1')} we obtain $0 \\leq \\rho(\\emptyset) = 0\n\\leq \\left|\\emptyset\\right|$. Now, let $X\\subseteq E$ and $x\\in X$. By\ninduction hypothesis, we have $0\\leq \\rho(X\\BSET{x}) \\leq\n\\left|X\\BSET{x}\\right|=\\left|X\\right| - 1$. {\\em (R2')} yields\n$\\rho(X\\BSET{x}) \\leq \\rho(X) \\leq \\rho(X\\BSET{x}) + 1$, which combines with\nthe previous inequality to the desired $ 0 \\leq \\rho(X\\BSET{x}) \\leq \\rho(X) \\leq \\left(\n\\left|X\\right| - 1\\right) + 1 = \\left| X \\right|$.\n\n\\par \\vspace*{.6\\baselineskip}\\noindent \\mbox{---\\hspace*{2.5mm}} \\marginpar{Jan 3rd}\nIn order to show {\\em (R2)}\nit suffices to consider $X\\subseteq Y\\subseteq E$. We prove $\\rho(X) \\leq\n\\rho(Y)$ by induction on $\\left| Y\\backslash X \\right|$. The base case implies $X=Y$\nthus $\\rho(X)\\leq \\rho(Y)$ holds trivially. Now let $y\\in Y\\backslash X$. By\ninduction hypothesis, $\\rho(X) \\leq \\rho(Y\\BSET{y})$ holds. From {\\em (R2')}\nwe obtain $\\rho(Y\\BSET{y}) \\leq \\rho(Y)$, and thus $\\rho(X) \\leq\n\\rho(Y\\BSET{y}) \\leq \\rho(Y)$ holds.\n\n\\marginpar{Jan 3rd}\n\\par \\vspace*{.6\\baselineskip}\\noindent \\mbox{---\\hspace*{2.5mm}} We prove that the following auxiliary property ...\n\\begin{enumerate}[align=parleft,leftmargin=2cm,labelsep=1.5cm]\\label{n:R2pp}\n \\item[\\em (R2'')] If $\\rho(X\\cup\\SET{y})=\\rho(X)+ 1$ and $X'\\subseteq X$, then $\\rho(X'\\cup\\SET{y})=\\rho(X')+ 1$; \\\\ for all $X\\subseteq E$, $y\\in E$.\n\\end{enumerate}\n... follows from {\\em (ii)} by induction on\n $\\left| X\\backslash X'\\right|$. The base case $X=X'$ is trivial. For the induction step, let $x\\in X\\backslash X'$, and\n assume that the implication is not vacuously true. By induction hypothesis\n $\\rho(X'\\cup\\SET{x,y}) = \\rho(X'\\cup\\SET{x}) + 1$. Using {\\em (R2')} we obtain the inequalities $\\rho(X') \\leq \\rho(X'\\cup\\SET{x}) \\leq \\rho(X') + 1$, similarly\n $\\rho(X') \\leq \\rho(X'\\cup\\SET{y}) \\leq \\rho(X') + 1$, and furthermore\n $\\rho(X'\\cup\\SET{y})\\leq \\rho(X'\\cup\\SET{x,y})\\leq \\rho(X'\\cup\\SET{y}) + 1$. \n We establish {\\em (R2'')} by the following case analysis:\n \\begin{enumerate}\n \\item[{\\em (a)}] $\\rho(X'\\cup\\SET{x}) = \\rho(X') + 1$, by induction hypothesis $\\rho(X'\\cup\\SET{x,y})=\\rho(X')+2$ and as a consequence of the last inequality $\\rho(X'\\cup \\SET{y}) = \\rho(X') + 1$.\n \\item[{\\em (b)}] $\\rho(X'\\cup\\SET{x}) = \\rho(X')$. If we assume that $\\rho(X'\\cup\\SET{y})=\\rho(X')$, we could use {\\em (R3')} in order to deduce\n $\\rho(X'\\cup\\SET{x,y})=\\rho(X')$, which would contradict the induction hypothesis. Therefore,\n $\\rho(X'\\cup\\SET{y})=\\rho(X')+1$.\n\\end{enumerate} \n\n\\marginpar{Jan 3rd}\n\\par \\vspace*{.6\\baselineskip}\\noindent \\mbox{---\\hspace*{2.5mm}} \nIn order to show that {\\em (R3)} holds for all $X,Y\\subseteq E$, we may use an\ninductive argument over $(\\left|X\\backslash Y\\right|,\\left|Y\\backslash X\\right|)$ with\nrespect to the well-founded natural coordinate-wise partial order. The base case\n$\\left|X\\backslash Y\\right| = 0 = \\left|Y\\backslash X\\right|$ implies that $X=Y$ and\ntherefore $\\rho(X\\cap Y)+\\rho(X\\cup Y)= 2\\rho(X) = \\rho(X) + \\rho(Y)$ holds.\nDue to the commutativity of the operations $\\cap$, $\\cup$, and $+$, it suffices to proof the\ninduction step from $(X\\BSET{x},Y)$ to $(X,Y)$ for $x\\in X\\backslash Y$, as the step\nfrom $(X,Y\\BSET{y})$ to $(X,Y)$ for $y\\in Y\\backslash X$ follows symmetrically. By\ninduction hypothesis, we may assume that $\\rho\\left(\\left(X\\BSET{x}\\right)\\cup\nY\\right) + \\rho\\left(\\left(X\\BSET{x}\\right)\\cap Y\\right) \\leq \\rho(X\\BSET{x})\n+ \\rho(Y)$ holds. Since $x\\in X\\backslash Y$, we see that $x\\notin Y$ and thus\n$\\left(X\\BSET{x}\\right)\\cap Y = X\\cap Y$ as well as\n$\\left(X\\BSET{x}\\right)\\cup Y = \\left(X\\cup Y\\right)\\BSET{x}$, so we may\nwrite the induction hypothesis as $\\rho\\left(\\left(X\\cup Y\\right)\\BSET{x}\n\\right) + \\rho(X\\cap Y) \\leq \\rho\\left(X\\BSET{x}\\right) + \\rho(Y)$. Property\n{\\em (R2')} implies that \\linebreak $\\rho(X\\cup Y) = \\rho\\left(\\left(X\\cup\nY\\right)\\BSET{x} \\right) + \\alpha$ and $\\rho(X) = \\rho(X\\BSET{x})+ \\beta$ for\nsome $\\alpha,\\beta\\in\\SET{0,1}$. The desired inequality $\\rho(X\\cup\nY)+\\rho(X\\cap Y)\\leq \\rho(X)+ \\rho(Y)$ follows from the fact that $\\alpha \\leq\n\\beta$, which is a consequence of {\\em (R2'')} where $X\\BSET{x}$ takes the\nrole of $X'$, $\\left(X\\BSET{x}\\right)\\cup Y$ takes the role of $X$ and $x$\ntakes the role of $y$.\n\n\\bigskip\n\\noindent \\underline{The implication {\\em (iii) $\\Rightarrow$ (i)}:} \n\n\\marginpar{Jan 3rd}\n\\par \\vspace*{.6\\baselineskip}\\noindent \\mbox{---\\hspace*{2.5mm}} First, we prove that {\\em (iii)} implies property {\\em (R2')} that\n$\\rho$ is unit-increasing, let $X\\subseteq E$ and $y\\in E$. If $y\\in X$ the\nproperty holds trivially, let $y\\notin X$. The first inequality $\\rho(X)\\leq\n\\rho(X\\cup\\SET{y})$ holds due to {\\em (R2)}. With {\\em (R3)} we obtain\n$\\rho(X\\cup\\SET{y}) + \\rho(X\\cap \\SET{y}) \\leq \\rho(X) + \\rho(\\SET{y})$, and\nsince $X\\cap\\SET{y}=\\emptyset$ we may use {\\em (R1)} twice to obtain\n$\\rho(\\SET{y}) \\leq 1$ and $\\rho(\\emptyset) = 0$, from which we may infer the\nsecond inequality of {\\em (R2')}, namely $\\rho(X\\cup\\SET{y}) \\leq \\rho(X) +\n1$.\n\n\\marginpar{Jan 3rd}\n\\par \\vspace*{.6\\baselineskip}\\noindent \\mbox{---\\hspace*{2.5mm}} We prove that {\\em (iii)} implies property\n\\begin{enumerate}[align=parleft,leftmargin=2cm,labelsep=1.5cm]\\label{n:R4}\n \\item[\\em (R4)] $\\left(\\forall y\\in Y\\colon\\,\\rho(X\\cup\\SET{y}) = \\rho(X)\\right) \\Rightarrow \\rho(X\\cup Y)= \\rho(X)$ for all $X,Y\\subseteq E$.\n\\end{enumerate}\nBy induction on $|Y\\backslash X|$. The base cases $|Y\\backslash X|\\in \\SET{0,1}$ are trivial. Now\nlet $v,w\\in Y\\backslash X$. By induction hypothesis, $\\rho(X) = \\rho(X\\cup Y \\BSET{v}) = \\rho(X\\cup Y \\BSET{w}) = \\rho(X\\cup Y\\BSET{v,w})$. Using {\\em (R3)} we obtain\n$\\rho(X\\cup Y\\BSET{v,w}) + \\rho(X\\cup Y) \\leq \\rho(X\\cup Y\\BSET{v}) + \\rho(X\\cup Y\\BSET{w})$. Together with the induction hypothesis we get $\\rho(X\\cup Y) \\leq \\rho(X)$ and the property {\\em (R2)} that $\\rho$ is isotone yields $\\rho(X\\cup Y)=\\rho(X)$.\n\n\\marginpar{Jan 3rd}\n\\par \\vspace*{.6\\baselineskip}\\noindent \\mbox{---\\hspace*{2.5mm}} \nNext, we prove that {\\em (iii)} also implies the following property:\n\\begin{enumerate}[align=parleft,leftmargin=2cm,labelsep=1.5cm]\\label{n:R5}\n\\item[\\em (R5)] For every $X\\subseteq E$ there is a subset $X'\\subseteq X$, such that\n $\\left|X'\\right| = \\rho(X') = \\rho(X)$.\n \n \n\\end{enumerate}\nBy induction on $|X|$. The base case $\\rho(\\emptyset) = 0 =\n\\left|\\emptyset\\right|$ is clear. Now let $x\\in X$ and by induction\nhypothesis, there is a subset $X'\\subseteq X\\BSET{x}$ such that\n$\\left|X'\\right| = \\rho(X') = \\rho(X\\BSET{x})$. From {\\em (R2')} we conclude\nthat $\\rho(X)=\\rho(X\\BSET{x}) + \\alpha$ for some $\\alpha\\in\\SET{0,1}$. The\ncase $\\alpha=0$ is trivial. For the case $\\alpha=1$ we give an indirect\nargument: Assume that $\\rho(X'\\cup\\SET{x}) = \\rho(X') = \\rho(X\\BSET{x})$. Then\n$\\rho(X) = \\rho(X\\BSET{x})$ follows from {\\em (R4)}, because for every $y\\in\nX\\backslash X'$ we have $\\rho(X'\\cup\\SET{y}) = \\rho(X)$. Yet, this is a\ncontradiction to $\\rho(X) = \\rho(X\\BSET{x}) + 1$, therefore\n$\\rho(X'\\cup\\SET{x}) = \\rho(X') + 1$ follows from {\\em (R2')}, thus\n$|X'\\cup\\SET{x}|=\\rho(X'\\cup\\SET{x}) = \\rho(X)$.\n\n\\marginpar{Jan 3rd}\n\\par \\vspace*{.6\\baselineskip}\\noindent \\mbox{---\\hspace*{2.5mm}} From $\\rho$, we define the set system ${\\mathcal{I}} = \\SET{X\\subseteq E\\mid \\rho(X) = \\left|X\\right|}$. For now, let us assume that $M=(E,{\\mathcal{I}})$ is indeed a matroid. An immediate consequence of property {\\em (R5)} is that $\\rho(X) \\leq \\mathrm{rk}_{M}(X)$ for all $X\\subseteq E$.\nBy definition of $\\mathrm{rk}_{M}$, there is a subset $X'\\subseteq X$ such that $\\mathrm{rk}_{M}(X) = \\left|X'\\right| = \\rho(X') \\leq \\rho(X)$ due to {\\em (R2)}. Thus $\\rho=\\mathrm{rk}_{M}$.\n\n\\marginpar{Jan 3rd}\n\\par \\vspace*{.6\\baselineskip}\\noindent \\mbox{---\\hspace*{2.5mm}} By {\\em (R1)} we have $\\rho(\\emptyset) = 0 = \\left| \\emptyset \\right|$, thus $\\emptyset \\in {\\mathcal{I}}$, so {\\em (I1)} holds.\n\n\\marginpar{Jan 3rd}\n\\par \\vspace*{.6\\baselineskip}\\noindent \\mbox{---\\hspace*{2.5mm}} Let $X\\in {\\mathcal{I}}$. We show that $X'\\in{\\mathcal{I}}$\nfor all $X'\\subseteq X$ by induction on $\\left|X\\backslash X'\\right|$. The base case\n$X'=X$ is trivial. Now let $x\\in X\\backslash X'$. By induction hypothesis,\n$X'\\cup\\SET{x}\\in {\\mathcal{I}}$, therefore $\\rho(X'\\cup\\SET{x}) = \\left|X'\\right| + 1$.\n From {\\em (R1)} we get the inequality $\\rho(X') \\leq \\left| X' \\right|$, and from\n {\\em (R2')} we get the inequality $\\rho(X'\\cup\\SET{x}) \\leq \\rho(X') + 1$. \n Thus $\\rho(X') = \\left|X'\\right|$ follows, consequently $X'\\in {\\mathcal{I}}$, so {\\em (I2)} holds.\n\n\\marginpar{Jan 3rd}\n \\par \\vspace*{.6\\baselineskip}\\noindent \\mbox{---\\hspace*{2.5mm}} We give an indirect argument for {\\em (I3)}. Let $X,Y\\in {\\mathcal{I}}$ with $\\left|X\\right| < \\left|Y\\right|$, and assume that for all $y\\in Y$, $X\\cup\\SET{y}\\notin {\\mathcal{I}}$.\n Since $\\left| X\\right| = \\rho(X)$ and by {\\em (R2)} $\\rho$ is isotone, we can infer that\n $\\rho(X\\cup\\SET{y}) = \\rho(X)$ for all $y\\in Y$. With {\\em (R4)} we see that\n $\\rho(X\\cup Y)=\\rho(X)$, and together with {\\em (R2)} we obtain $\\rho(Y) \\leq \\rho(X\\cup Y) = \\rho(X) = \\left| X \\right| < \\left| Y \\right|$, a contradiction to $Y\\in{\\mathcal{I}}$.\n We may now conclude that $M=(E,{\\mathcal{I}})$ is a matroid.\n\\end{proof}\n\n\\studyremark{{\\em (R1)}, {\\em (R2)}, {\\em (R2')}, {\\em (R5)} is not sufficient for $\\rho$ to\nbe the rank function of a matroid! $X\\mapsto \\max \\SET{\\left| X\\cap A \\right|, \\left|X \\cap B\\right|}$ with $A = \\SET{a,b}$, $B = \\SET{c,d,e}$; $\\SET{a,c}$ and $\\SET{a,d}$ violate submodularity}\n\\subsection{Matroids Induced From Submodular Functions}\n\n\n\\begin{definition}\\marginpar{Jan 5th}\n Let $E$ be any set, $R\\subseteq \\mathbb{R}$, and let $f\\colon 2^{E} \\longrightarrow R$.\n We call the map $f$ \\deftext{non-decreasing}, if for every $X\\subseteq Y\\subseteq E$,\n the inequality $f(X) \\leq f(Y)$ holds.\n We call $f$ \\deftext{submodular}, if for all $X,Y\\subseteq E$\n the inequality\n \\( f(X\\cap Y) + f(X\\cup Y) \\leq f(X) + f(Y) \\)\n holds. \n\\end{definition}\n\n\\begin{example}\nLet $M=(E,{\\mathcal{I}})$ be a matroid. Then $\\mathrm{rk}_{M}\\colon 2^{E}\\longrightarrow \\mathbb{N}$ is a non-decreasing and submodular function.\n\\end{example}\n\n\\needspace{3\\baselineskip}\n\n\\noindent\nThe following theorem is the independent-sets version of Proposition~11.1.1 in \\cite{Ox11},\nwhich is attributed to J.~Edmonds and G.C.~Rota.\n\n\\begin{theorem}\\label{thm:fromsubmodular}\\marginpar{Jan 5th}\n Let $E$ be a finite set, and let $f\\colon 2^{E} \\longrightarrow \\mathbb{Z}$ be\n a non-decreasing, submodular function. Then $M=(E,{\\mathcal{I}})$\n where \\[\n {\\mathcal{I}} = \\SET{X\\subseteq E ~~\\middle|~~ \\forall X'\\subseteq X\\colon\\, X'\\not=\\emptyset \\Rightarrow f(X') \\geq \\left| X' \\right|}\n \\] is a matroid.\n\\end{theorem}\n\n\\begin{proof}\\marginpar{Jan 5th}\n From the definition, it is clear that $\\emptyset \\in {\\mathcal{I}}$ {\\em (I1)} as well as that \n for every $X\\in {\\mathcal{I}}$ and every $Y\\subseteq X$, $Y\\in {\\mathcal{I}}$ {\\em (I2)}. We have to show that {\\em (I3)} holds for ${\\mathcal{I}}$, too. Let $X,Y\\in {\\mathcal{I}}$ with $\\left| X \\right| < \\left| Y \\right|$. \n We give an indirect argument. \n Assume that for all $y\\in Y\\backslash X$,\\linebreak $X\\cup \\SET{y}\\notin {\\mathcal{I}}$. Since $\n \\left| X\\backslash Y \\right| + \\left|X\\cap Y \\right| = \\left| X \\right| < \\left| Y \\right| = \\left| Y\\backslash X\\right| + \\left|X\\cap Y\\right|$, we have\\linebreak $\\left| Y \\backslash X \\right| > \\left| X \\backslash Y \\right|$. For every $y\\in Y\\backslash X$, there is a subset $X_{y}\\subseteq X$ with minimal cardinality\n such that $f\\left(X_y \\cup \\SET{y}\\right) < \\left| X_{y} \\right| + 1$. Since $Y\\in {\\mathcal{I}}$\n we obtain that $X_{y} \\cap \\left(X\\backslash Y\\right) \\not= \\emptyset$. Therefore by a simple counting argument, there are $y_{1},y_{2}\\in Y\\backslash X$ such that there is some $x\\in X\\backslash Y$ with $x \\in X_{y_1}\\cap X_{y_2}$. \n Below, we first use that $f$ is non-decreasing, then that $f$ is submodular, and then the fact that $X_{y_1}\\cap X_{y_2}\\in {\\mathcal{I}}$ from {\\em (I2)} and $X\\in {\\mathcal{I}}$; finally, we use the fact that neither $X\\cup\\SET{y_1}\\in {\\mathcal{I}}$ nor\n $X\\cup\\SET{y_2}\\in{\\mathcal{I}}$ and that $f$ is integer-valued:\n \\begin{align*}\n f\\left(\n \\left(X_{y_1}\\cup X_{y_2}\\cup\\SET{y_1,y_2}\n \\right) \\BSET{x}\n \\right) & \\leq f\\left(X_{y_1}\\cup X_{y_2}\\cup\\SET{y_1,y_2}\n \\right) \\\\\n & \\leq f\\left(X_{y_1}\\cup \\SET{y_1}\n \\right) + f\\left(X_{y_2}\\cup \\SET{y_2}\n \\right) \n - f\\left(X_{y_1}\\cap X_{y_2}\n \\right) \\\\\n & \\leq f\\left(X_{y_1}\\cup \\SET{y_1}\\right) + f\\left(X_{y_2}\\cup \\SET{y_2}\\right) - \\left| X_{y_1}\\cap X_{y_2}\\right| \\\\\n & \\leq \\left| X_{y_1} \\right| + \\left| X_{y_2} \\right| - \\left| X_{y_1} \\cap X_{y_2}\\right| \\\\ & = \\left| X_{y_1} \\cup X_{y_2} \\right| \n \\\\& = \\left|\\left(X_{y_1}\\cup X_{y_2}\\cup\\SET{y_1,y_2}\\right) \\BSET{x}\\right| - 1.\n \\end{align*}\n Thus there must be a subset of minimal cardinality $C \\subseteq \\left( X_{y_1}\\cup X_{y_2}\\cup \\SET{y_1,y_2} \\right)\\BSET{x}$ such that $f(C) < \\left| C \\right|$. Then $C \\cap \\SET{y_1,y_2} = \\emptyset$ because otherwise $C$ would contradict the minimality of the cardinalities of $X_{y_1}$ and $X_{y_2}$, respectively. But then the fact that $C\\subseteq X_{y_1}\\cup X_{y_2}\\subseteq X$ would contradict $X\\in {\\mathcal{I}}$. Therefore there must be some $y\\in Y\\backslash X$ such that $X\\cup\\SET{y}\\in{\\mathcal{I}}$.\n\\end{proof}\n\n\n\\noindent If we restrict $f$ to be a map into the non-negative integers, we may simplify the \nexpression that gives ${\\mathcal{I}}$ analogously to Corollary~8.1 \\cite{We76}.\n\n\\begin{theorem}\\label{thm:submodularIndependent}\\marginpar{Jan 5th}\n Let $E$ be a finite set, and let $f\\colon 2^{E} \\longrightarrow \\mathbb{N}$ be\n a non-decreasing, submodular function. Then $M=(E,{\\mathcal{I}})$\n where \\[\n {\\mathcal{I}} = \\SET{\\vphantom{A^A}X\\subseteq E ~\\middle|~ \\forall X'\\subseteq X\\colon f(X') \\geq \\left| X' \\right|}\n \\] is a matroid. If furthermore $f(\\emptyset) = 0$, then its rank function is given by\n \\[ \\mathrm{rk}(X) = \\min \\SET{\\vphantom{A^A}f(Y) + \\left| X \\backslash Y \\right| ~\\middle|~ Y\\subseteq X}.\\]\n\\end{theorem}\n\n\\begin{proof}\\marginpar{Jan 5th}\n Let ${\\mathcal{I}}' = \\SET{X\\subseteq E \\mid \\forall X'\\subseteq X\\colon X'\\not=\\emptyset \\Rightarrow f(X') \\geq \\left| X' \\right|}$ corresponding to Theorem~\\ref{thm:fromsubmodular}.\n From the definitions, it is clear that ${\\mathcal{I}} \\subseteq {\\mathcal{I}}'$. From inspection we obtain that if $X \\in {\\mathcal{I}}' \\backslash {\\mathcal{I}}$, then $f(\\emptyset) < \\left| \\emptyset \\right| = 0$. But this is impossible for $f(\\emptyset)\\in\\mathbb{N}$. Thus ${\\mathcal{I}} = {\\mathcal{I}}'$. \n\n\n The second part of the proof follows the ideas from \\cite{Dun76} where a more general statement is proved.\\footnote{Both D.J.A.~Welsh and F.D.J.~Dunstan cite a conference abstract of the {\\em Waterloo Conference on Combinatorics 1968} by J.~Edmonds and G.C.~Rota who proved that for submodular, non-decreasing, integer-valued $f$ the rank function is given by $\\mathrm{rk}(X) = \\min\\SET{\\left| X \\right|, f(Y)-\\left| X\\backslash Y \\right|~\\middle|~ \\vphantom{A^A} Y\\subseteq X}$. Unfortunately, we were not able to get a copy of that abstract.}\nNow let us assume that we have the further property $f(\\emptyset) = 0$, we shall now prove the rank formula. We will denote the formula given in the statement of the theorem by $\\mathrm{rk}$, whereas we\nare denoting the rank formula from Definition~\\ref{def:rank} by $\\mathrm{rk}_{M}$.\nFirst, we want to show that $\\mathrm{rk}$ is non-decreasing. Let $X'\\subseteq X\\subseteq E$,\nwe do induction on $\\left| X\\backslash X' \\right|$. The base case is trivial. Now,\nlet $X\\subseteq E$, and $x\\in X$, the induction hypothesis yields that $\\mathrm{rk}(X') \\leq \\mathrm{rk}(X\\BSET{x})$. If there is a subset $Y\\subseteq X$ with $x\\in Y$, such that\n $\\mathrm{rk}(X) = f(Y) + \\left| X\\backslash Y \\right|$,\n then since $f(Y\\BSET{x}) \\leq f(Y)$ we obtain that $$\\mathrm{rk}(X\\BSET{x}) \\leq f(Y\\BSET{x}) + \\left| (X\\BSET{x})\\backslash(Y\\BSET{x}) \\right|\n \\leq f(Y) + \\left| X\\backslash Y \\right| = \\mathrm{rk}(X).$$\n Otherwise let $Y\\subseteq X\\BSET{x}$ be a subset such that\n $\\mathrm{rk}(X) = f(Y) + \\left| X\\backslash Y \\right|$. \n Then $$\\mathrm{rk}(X\\BSET{x}) \\leq f(Y) + \\left| (X\\BSET{x})\\backslash Y \\right| < f(Y) + \\left| X\\backslash Y \\right| = \\mathrm{rk}(X),$$\n thus in any case $\\mathrm{rk}(X\\BSET{x}) \\leq \\mathrm{rk}(X)$, so $\\mathrm{rk}$ is non-decreasing.\n Now, in order to show that $\\mathrm{rk}_M(X) \\leq \\mathrm{rk}(X)$ for all $X\\subseteq E$, it suffices to show that $\\mathrm{rk}(X) = \\left| X \\right|$ for all independent $X\\subseteq E$.\n Let $X\\in{\\mathcal{I}}$. By definition of ${\\mathcal{I}}$, for all\n$Y\\subseteq X$, $\\left| Y \\right| \\leq f(Y)$ holds. Thus for any $Y\\subseteq X$, we have\n\\[ \\left| X \\right| = \\left| Y \\right| + \\left|X\\backslash Y \\right| \\leq f(Y) + \\left|X\\backslash Y\\right|.\\]\nTherefore the minimum in the expression for $\\mathrm{rk}(X)$ is attained for $Y=\\emptyset$, i.e.\n \\[ \\mathrm{rk}(X) = \\min\n\\SET{\\vphantom{A^A}f(Y) + \\left| X \\backslash Y \\right| ~\\middle|~ Y\\subseteq X} = f(\\emptyset) + \\left|\nX\\backslash \\emptyset \\right| = \\left| X \\right|.\\] \nTo complete the proof that $\\mathrm{rk} = \\mathrm{rk}_M$, we have to show that for every $X\\subseteq E$,\nthere is a subset $Y\\subseteq X$ such that $Y\\in {\\mathcal{I}}$ and $\\mathrm{rk}(X) = \\left| Y \\right|$.\nLet $Z\\subseteq X$ such that $Z$ has maximal cardinality with $Z\\in {\\mathcal{I}}$.\nWe give an indirect argument and assume that $\\mathrm{rk}_M(X) = \\left| Z \\right| < \\mathrm{rk}(X) \\leq \\left| X \\right|$. Since $Z$ is maximally independent in $X$, for every\n$x\\in X\\backslash Z$ there must be a subset $Z_x\\subseteq Z$ such that\n$f(Z_x\\cup\\SET{x}) < \\left| Z_x\\cup\\SET{x} \\right| = \\left| Z_x \\right| + 1$. From $Z\\in{\\mathcal{I}}$ we may infer that $f(Z_x) \\geq \\left| Z_x \\right|$, thus we have\n$f(Z_x\\cup\\SET{x}) = \\left| Z_x \\right| = f(Z_x)$ due to $f$ being non-decreasing and integer-valued.\nWe show the auxiliary claim that for all $X'\\subseteq X\\backslash Z$, $f(Z) = f(Z\\cup X')$, by\ninduction on $\\left| X' \\right|$. The base case is trivial. For the induction step, \nlet $x'\\in X'$, and by induction hypothesis we may assume that $f(Z) = f(Z\\cup (X'\\BSET{x'}))$. Let $Z_{x'}\\subseteq Z$ such that $f(Z_{x'}\\cup\\SET{x'}) = \\left| Z_{x'} \\right| = f(Z_{x'})$ as above. Since $f$ is submodular, we obtain that $f(Z\\cup(X'\\BSET{x'})) + f(Z_{x'}\\cup\\SET{x'}) \\geq f(Z_{x'}) + f(Z\\cup X')$, and along with the previous equation this yields $f(Z\\cup(X'\\BSET{x'})) \\geq f(Z\\cup X')$. So, together with the property that $f$ is\n non-decreasing and with the induction hypothesis, \n we obtain the desired equation $f(Z\\cup X') = f(Z\\cup(X'\\BSET{x'})) = f(Z)$.\n But now, let $X'=X\\backslash Z$, then $Z\\cup X' = X$. We obtain from the auxiliary claim above, that $f(Z)=f(X)$, so\n that, by construction as a minimum, $\\mathrm{rk}(X) \\leq f(X) + \\left| X\\backslash X \\right| = f(Z) = \\left| Z \\right|$. Yet this\n contradicts $\\left| Z \\right| < \\mathrm{rk}(X)$. Thus there is an independent subset of $X$ with cardinality $\\mathrm{rk}(X)$ for every $X\\subseteq E$, and therefore $\\mathrm{rk} = \\mathrm{rk}_M$.\n \\end{proof}\n\n\\subsection{Dual Matroids}\n\n\\begin{definition}\\marginpar{Jan 5th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid. We call $X\\subseteq E$ \\deftext{spanning} in $M$, if\n\tthere is a base $B$ of $M$, such that $B \\subseteq X$.\n\\end{definition}\n\n\\begin{lemma}\\label{lem:basesminimalspanning}\\marginpar{Jan 5th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, $X\\subseteq E$.\n\tThen $X$ is a base if and only if $X$ is spanning in $M$, yet for all $x\\in X$,\n\t$X\\BSET{x}$ is not spanning in $M$.\n\\end{lemma}\n\\begin{proof}\\marginpar{Jan 5th}\n\tLet $B\\in{\\mathcal{I}}$ be a base of $M$,\n\tthen $\\mathrm{rk}(B)=\\left| B \\right|$ is maximal, so $\\mathrm{cl}(B) = E$.\n\tOn the other hand, for every $b\\in B$, $\\mathrm{rk}(B\\BSET{b}) < \\left| B \\right|$,\n\tthus $b\\notin \\mathrm{cl}(B\\BSET{b})$, so $\\mathrm{cl}(B\\BSET{b})\\not= E$.\n\tLet $X\\subseteq E$ such that $X\\notin{\\mathcal{B}}(M)$.\n\tIf further $\\mathrm{rk}(X) < \\mathrm{rk}(E)$, then $X$ clearly is not a spanning set in $M$. \n\tNow assume that $\\mathrm{rk}(X) = \\mathrm{rk}(E)$, so $X$ is spanning in $M$, and because it is not a base, $X\\notin {\\mathcal{I}}$. But then there is a base $B\\subsetneq X$ with $\\mathrm{cl}(B) = \\mathrm{cl}(X)$ (Lemma~\\ref{lem:clofbase}). So there is some $x\\in X\\backslash B$, such that $X\\BSET{x}$ still contains the base $B$ and therefore $X\\BSET{x}$ is spanning in $M$.\n\\end{proof}\n\n\\noindent Matroids allow to be axiomatized cryptomorphically by characterizing the set of bases of $M$. For full disclosure on this topic we would like to refer the reader the first chapters in \\cite{We76} and \\cite{Ox11}.\n\n\\needspace{6\\baselineskip}\n\n\\begin{theorem}\\label{thm:frombases}\\marginpar{Jan 5th}\n\tLet $E$ be a finite set, ${\\mathcal{I}}\\subseteq 2^E$. Let further\n\t\\[ {\\mathcal{B}} = \\SET{\\vphantom{A^A} X\\in{\\mathcal{I}} ~\\middle|~ \\nexists Y\\in {\\mathcal{I}}\\colon\\, X \\subsetneq Y} \\]\n\tbe the family of maximal elements of ${\\mathcal{I}}$ with regard to set-inclusion.\n\t If \n\t\\begin{enumerate}[align=parleft,leftmargin=2cm,labelsep=1.5cm]\\label{n:Bx}\n\t\t\\item[(B1)] ${\\mathcal{B}}\\not= \\emptyset$,\n\t\t\\item[(B2)] $\\forall X,Y\\in{\\mathcal{B}}\\colon\\,\\left| X \\right| = \\left| Y \\right|$, and\n\t\t\\item[(B3)] for all $X,Y\\in {\\mathcal{B}}$ and all $x\\in X\\backslash Y$, there is an element $y\\in Y\\backslash X$, such that $(X\\BSET{x})\\cup\\SET{y} \\in {\\mathcal{B}}$\n\t\\end{enumerate}\n\tholds, and if ${\\mathcal{I}} = \\SET{\\vphantom{A^A}X\\subseteq E~\\middle|~ \\exists B\\in {\\mathcal{B}}\\colon\\,X\\subseteq B}$, then $M=(E,{\\mathcal{I}})$ is a matroid.\n\\end{theorem}\n\n\\begin{proof}\\marginpar{Jan 5th}\n\tFrom {\\em (B1)} we obtain $B\\in{\\mathcal{B}}$, and clearly $\\emptyset \\subseteq B$, so\n\t$\\emptyset\\in {\\mathcal{I}}$ {\\em (I1)}. Let $X\\in{\\mathcal{I}}$, then there is some $B\\in{\\mathcal{B}}$ with $X\\subseteq B$. For $Y\\subseteq X$ we have $Y\\subseteq B$ and therefore $Y\\in{\\mathcal{I}}$ {\\em (I2)}.\n\tLet $X,Y\\in{\\mathcal{I}}$ with $\\left| X \\right| < \\left| Y \\right|$. \n\tThere are $B_X\\supseteq X$ and $B_Y\\supseteq Y$ with $B_X,B_Y\\in {\\mathcal{B}}$.\n\t\\linebreak\n\tIf $Y' = B_X\\cap(Y\\backslash X)\\not= \\emptyset$, then let $y\\in Y'$ be an arbitrary choice,\n\tand we obtain\n\t\\linebreak\n\t $X\\cup\\SET{y}\\subseteq B_X$ therefore $X\\cup\\SET{y}\\in{\\mathcal{I}}$.\n\tIf $Y'=\\emptyset$, then let $\\alpha(B_X) = \\left| (B_Y\\backslash B_X) \\backslash (Y\\backslash X)\\right|$,\n\twe prove that we can augment $X$ by induction on $\\alpha(B_X)$.\n\t Since $\\left| X \\right| < \\left| Y \\right| \\leq \\left| B_Y \\right| = \\left| B_X \\right|$, there is an element $x'\\in B_X\\backslash X$.\n\t\n\t We may use {\\em (B3)} in order to obtain the base\n\t$B_X'=(B_X\\BSET{x'})\\cup\\SET{y} \\in {\\mathcal{B}}$ where $y\\in B_Y\\backslash B_X$. If $y\\in Y\\backslash X$, then $X\\cup\\SET{y}\\subseteq B_X'$ and therefore $X\\cup\\SET{y}\\in {\\mathcal{I}}$. Otherwise $y\\in (B_Y\\backslash B_X)\\backslash(Y \\backslash X)$, then $\\alpha(B_X') = \\alpha(B_X) - 1$ and thus there is some $y\\in Y\\backslash X$ with $X\\cup\\SET{y}\\in {\\mathcal{I}}$ by the induction hypothesis. Thus\n\t${\\mathcal{I}}$ has the property {\\em (I3)}.\n\\end{proof}\n\n\n\n\\begin{definition}\\marginpar{Jan 5th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid. The \\deftext{dual matroid} of $M$ shall be the pair\n\t \\( M^{\\ast} = (E,{\\mathcal{I}}^\\ast)\\) where\\label{n:Mdual}\n\t \\[ {\\mathcal{I}}^{\\ast} = \\SET{\\vphantom{A^A} E\\backslash X ~\\middle|~ X\\subseteq E,\\, \\text{such that~}X\\text{~is spanning in~}M}. \\qedhere\\]\n\\end{definition}\n\n\\needspace{7\\baselineskip}\n\n\\begin{lemma}\\label{lem:spanningdual}\\marginpar{Jan 5th}\n Let $M=(E,{\\mathcal{I}})$ be a matroid. Then $M^{\\ast}=(E,{\\mathcal{I}}^\\ast)$ is indeed a matroid.\n\\end{lemma}\n\n\\begin{proof}\\marginpar{Jan 5th}\n\tFirst, observe that for ${\\mathcal{B}}^\\ast = \\SET{E\\backslash B ~\\middle|~\\vphantom{A^A} B\\text{ is a base of } M}$\n\twe have the set equation\n\t\\[ {\\mathcal{I}}^\\ast = \\SET{X\\subseteq E~\\middle|~\\vphantom{A^A} \\exists B'\\in{\\mathcal{B}}^\\ast\\colon\\,X\\subseteq B'}, \n\t\\]\n\tbecause the minimal spanning sets of $M$ are precisely the bases of $M$, which in turn have complements in $E$ with maximal cardinality. Since $\\emptyset\\in {\\mathcal{I}}$ implies that $M$ has at least one base, \n\twe have ${\\mathcal{B}}^\\ast \\not= \\emptyset$ {\\em (B1)}.\n\tFrom Corollary~\\ref{cor:equicardinality} we obtain that for any two $B,B'\\in {\\mathcal{B}}^\\ast$,\n\twe have $B_0,B_0'$ that are bases of $M$ with $B=E\\backslash B_0$ and $B'=E\\backslash B_0'$, therefore\n\t $\\left| B \\right| = \\left| E \\right|-\\left| B_0 \\right| = \\left| E \\right| - \\left| B_0' \\right| = \\left| B' \\right|$, so {\\em (B2)} holds.\n\t Now let $x\\in B\\backslash B' = (E\\backslash B_0)\\backslash(E\\backslash B_0') = B_0'\\backslash B_0$, then there is a\n\t $y\\in B_0\\backslash B_0' = (E\\backslash B_0')\\backslash(E\\backslash B_0) = B'\\backslash B$ such that $(B_0\\BSET{y})\\cup\\SET{x}$ is a base of $M$ (Lemma~\\ref{lem:basisexchangesymmetric}). But then\n\t \\begin{align*}\n\t E\\backslash \\left( (B_0\\BSET{y})\\cup\\SET{x}\\right) & = E\\backslash \\left( \\left( B_0\\cup\\SET{x} \\right) \\BSET{y} \\right)\\\\ & =\n\t \\left( E\\backslash(B_0\\cup\\SET{x}) \\right)\\cup\\SET{y} \\\\ & = (B\\BSET{x})\\cup\\SET{y} \\in {\\mathcal{B}}^\\ast.\n\t \\end{align*}\n\t So {\\em (B3)} holds for ${\\mathcal{B}}^\\ast$, too, and from Theorem~\\ref{thm:frombases}\n\t we obtain that $M^\\ast=(E,{\\mathcal{I}}^\\ast)$ is a matroid.\n\\end{proof}\n\n\n\\begin{corollary}\\label{cor:dualbase}\\marginpar{Jan 5th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, $B\\subseteq E$. Then\n\t$B$ is a base of $M$ if and only if $E\\backslash B$ is a base of $M^\\ast$.\n\\end{corollary}\n\\begin{proof}\\marginpar{Jan 5th}\n\tLet $(E,{\\mathcal{I}}') = M^\\ast$.\n\tIf $B$ is a base of $M$, then for all $b\\in B$, $B\\BSET{b}$ is not spanning $M$ (Lemma~\\ref{lem:basesminimalspanning}),\n\t therefore $E\\backslash B\\in {\\mathcal{I}}'$, yet $(E\\backslash B)\\cup\\SET{b}\\notin {\\mathcal{I}}'$, therefore $E\\backslash B$ is maximally independent with respect to set-inclusion, \n\tand thus it is an independent set of $M^\\ast$ \n\twith maximal cardinality (Corollary~\\ref{cor:equicardinality}), so $E\\backslash B$ is a base of $M^\\ast$.\n\tConversely, if $E\\backslash B$ is a base of $M^\\ast$, then $E\\backslash(E\\backslash B) = B$ must be minimally spanning in $M$, since otherwise\n\t$E\\backslash \\left( B\\BSET{x} \\right) \\in {\\mathcal{I}}'$ for some $x\\in B$ contradicting the maximality of $E\\backslash B$ in ${\\mathcal{I}}'$.\n\tThus $B$ is a base of $M$ (Lemma~\\ref{lem:basesminimalspanning}).\n\\end{proof}\n\n\\begin{corollary}\\label{cor:doubleast}\\marginpar{Jan 5th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid. Then $M = \\left(M^{\\ast}\\right)^{\\ast}$.\n\\end{corollary}\n\n\\begin{proof}\\marginpar{Jan 5th}\n\tBy property {\\em (I2)}, the family of independent sets of a matroid is determined by its maximal elements, which are the bases of $M$.\n\tBy Corollary~\\ref{cor:dualbase}, $B$ is base of $M$, if and only if $E\\backslash B$ is a base of $M^\\ast$, if and only if $E\\backslash(E\\backslash B) = B$\n\tis a base of $\\left( M^\\ast \\right)^\\ast$. Thus $M=\\left( M^\\ast \\right)^\\ast$.\n\\end{proof}\n\n\n\\noindent The next two lemmas can be found in J.G.~Oxley's book (\\cite{Ox11}, p.67) and yield an elegant way to characterize the rank function of the dual matroid in terms of the rank function of the primal matroid.\n\n\\begin{lemma}\\label{lem:dualAug}\\marginpar{Jan 5th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, $X,Y\\subseteq E$ with $X\\cap Y = \\emptyset$ such that\n\t$X\\in {\\mathcal{I}}$ is independent in $M$ and $Y\\in{\\mathcal{I}}^\\ast$ is independent in $M^\\ast$.\n\tThen there is a base $B\\subseteq E$ of $M$ such that $X\\subseteq B$ and $Y\\subseteq E\\backslash B$.\n\\end{lemma}\n\\begin{proof}\\marginpar{Jan 5th}\n\tLet $B$ be a base of $E\\backslash Y$ in $M$ such that $X\\subseteq B$ (Lemma~\\ref{lem:augmentation}).\n\tThen $Y\\subseteq E\\backslash B$. It remains to show that $B$ is a base of $M$.\n\t Assume that $B$ is not a base of $M$, then $\\mathrm{rk}_M(E\\backslash Y) < \\mathrm{rk}_M(E)$. \n\t But $Y\\in{\\mathcal{I}}^\\ast$, therefore $E\\backslash Y$ is spanning in $M$ -- a contradiction. \n\t Thus $B$ is the desired base of $M$.\n\\end{proof}\n\n\n\\needspace{6\\baselineskip}\n\\begin{lemma}\\label{lem:rankDual}\\marginpar{Jan 5th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid and $X\\subseteq E$.\n\tThen\n\t\\[ \\mathrm{rk}_{M^\\ast}(X) = \\left| X\\right| + \\mathrm{rk}_M(E\\backslash X) - \\mathrm{rk}_M(E) .\\]\n\\end{lemma}\n\\begin{proof}\\marginpar{Jan 5th}\n\tLet $B'_X\\subseteq X$ be a base of $X$ in $M^\\ast$,\n\tand $B_{E\\backslash X}\\subseteq E\\backslash X$ be a base of $E\\backslash X$ in $M$.\n\tThen $\\mathrm{rk}_{M^\\ast}(X) = \\left| B'_X \\right|$ and $\\mathrm{rk}_{M}(E\\backslash X) = \\left| B_{E\\backslash X} \\right|$.\n\tClearly $B'_X\\cap B_{E\\backslash X} = \\emptyset$, therefore there is a base $B$ of $M$ such that $B_{E\\backslash X} \\subseteq B$ and $B'_X \\subseteq E\\backslash B$ (Lemma~\\ref{lem:dualAug}). Since $B_{E\\backslash X}$ is a base of $E\\backslash X $ in $M$, we have that $B\\cap \\left( E\\backslash X \\right) = B_{E\\backslash X}$, and analogously,\n\t$\\left( E\\backslash B \\right)\\cap X = B'_X$. We obtain $B\\cap X = X\\backslash B'_X$ and therefore $B= B_{E\\backslash X} \\mathbin{\\dot{\\cup}} \\left( X\\backslash B'_X \\right)$, so\n\t\\[ \\mathrm{rk}_M(E) = \\left| B \\right| = \\left| B_{E\\backslash X} \\right| + \\left| X \\right| - \\left| B'_X \\right| = \\mathrm{rk}_M(E\\backslash X) + \\left| X \\right| - \\mathrm{rk}_{M^\\ast}(X), \\]\n\tand as a consequence, $\\mathrm{rk}_{M^\\ast}(X) = \\left| X \\right| + \\mathrm{rk}_M(E\\backslash X) - \\mathrm{rk}_M(E)$.\n\\end{proof}\n\n\n\\noindent The following fact will be of interest for oriented matroids in Chapter~\\ref{ch:OMs}.\nIt can be found as Proposition~2.1.11 in J.G.~Oxley's book (\\cite{Ox11}, p.68), together with the proof we present here.\n\n\\begin{lemma}\\label{lem:CircuitCocircuitOrthogonality}\\marginpar{Jan 5th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid and $M^\\ast=(E,{\\mathcal{I}}^\\ast)$ be its dual matroid. Then for every $C\\in {\\mathcal{C}}(M)$ and $D\\in {\\mathcal{C}}(M^\\ast)$,\n\twe have $\\left| C\\cap D \\right|\\not= 1$.\n\\end{lemma}\n\\begin{proof}\\marginpar{Jan 5th}\n\tWe give an indirect proof and assume that $\\SET{x} = C\\cap D$ for some $C\\in{\\mathcal{C}}(M)$ and $D\\in {\\mathcal{C}}(M^\\ast)$.\n\tSince $D\\in {\\mathcal{C}}(M^\\ast)$, we have $\\mathrm{rk}_{M^\\ast}(D) = \\left| D \\right| - 1$.\n\tWe set $H = E\\backslash D$, then by Lemma~\\ref{lem:rankDual}, we get\n $$\n\t\t\\mathrm{rk}_{M^\\ast}(D) = \\left| D \\right| - 1 = \\left| D \\right| + \\mathrm{rk}_M(H) - \\mathrm{rk}_M(E),\n\t$$\n\tand therefore $\\mathrm{rk}_M(H) = \\mathrm{rk}(E) - 1$ follows. Clearly, $\\mathrm{cl}_M(H) = H$, since otherwise there would be an element $d\\in D$ such that $d\\in \\mathrm{cl}_M(H)\\backslash H$,\n\twhich would imply that $$\\mathrm{rk}_{M^\\ast}(D\\BSET{d}) = \\left| D\\BSET{d} \\right| + \\mathrm{rk}_M \\left( H\\cup\\SET{d} \\right) - \\mathrm{rk}_M(E) = \\left| D\\BSET{d} \\right| - 1,$$ contradicting that $D\\in{\\mathcal{C}}(M^\\ast)$ is a minimally dependent set of $M^\\ast$ with respect to set-inclusion. But now we arrive at another contradiction:\n\tWe have $x\\in C\\cap D$, $x\\notin H=E\\backslash D$, and thus $C\\not\\subseteq H$, yet\n\t $\\left| C\\cap H \\right| = \\left| C \\right| - \\left| C\\cap D \\right| = \\left| C \\right| - 1$, and therefore $\\mathrm{cl}_M(C\\cap H) = C$, so we obtain the contradiction $C \\subseteq \\mathrm{cl}_M(C\\cap H) \\subseteq \\mathrm{cl}_M(H) = H$ (Lemma~\\ref{lem:clFlips}). Therefore $\\left| C\\cap D \\right|\\not= 1$ must be the case.\n\\end{proof}\n\n\n\\begin{lemma}\\label{lem:CircuitCocircuitIntersectInTwo}\\marginpar{Jan 5th}\nLet $M=(E,{\\mathcal{I}})$ be a matroid and $M^\\ast=(E,{\\mathcal{I}}^\\ast)$ be its dual matroid. Let further $C\\in {\\mathcal{C}}(M)$ be a circuit and $c,d\\in C$ with $c\\not = d$.\nThere there is some $D\\in {\\mathcal{C}}(M^\\ast)$ such that $C\\cap D = \\SET{c,d}$.\n\\end{lemma}\n\n\\noindent The proof presented here is along the lines of the proof of Lemma 2.2.3 in \\cite{BlV78}.\n\n\\begin{proof}\\marginpar{Jan 5th}\n\tSince $C\\in{\\mathcal{C}}(M)$, we have $C\\BSET{c}\\in {\\mathcal{I}}$. There is a base $B_c$ of $M$ with $C\\BSET{c}\\subseteq B_c$ (Lemma~\\ref{lem:augmentation}),\n\t and since $C\\notin {\\mathcal{I}}$, $c\\notin B_c$. Then $B'_c = E\\backslash B_c$ is a base of $M^\\ast$ with $c\\in B'_c$ (Corollary~\\ref{cor:dualbase}).\n\t Let $D'= B'_c \\cup\\SET{d}$, then $\\mathrm{rk}_{M^\\ast}(D') = \\mathrm{rk}_{M^\\ast}(E) = \\left| D' \\right| - 1$, and therefore there is a unique circuit $D\\subseteq D'$.\n\t Clearly, $d\\in D$ is an element of that circuit. Therefore $d\\in C\\cap D$. Furthermore $C\\subseteq B_c\\mathbin{\\dot{\\cup}}\\SET{c}$ and $D\\subseteq B'_c\\cup\\SET{d}=(E\\backslash B_c)\\cup\\SET{d}$\n\t yield $C\\cap D\\subseteq \\SET{c,d}$. Since $\\left| C\\cap D \\right|\\not= 1$ (Lemma~\\ref{lem:CircuitCocircuitOrthogonality}), we obtain that\n\t $C\\cap D = \\SET{c,d}$.\n\\end{proof}\n\\subsection{Minors}\n\n\\PRFR{Jan 15th}\n\\noindent\nIn this section, we introduce the natural substructures for matroids.\n\n\\begin{definition}\\label{def:Mrestriction}\\PRFR{Jan 15th}\n Let $M=(E,{\\mathcal{I}})$ be a matroid, and let $R\\subseteq E$.\n The \\deftext[restriction of M to R@restriction of $M$ to $R$]{restriction of $\\bm M$ to $\\bm R$} is the pair\\label{n:MR}\n $ M| R = (R,{\\mathcal{I}}')$ where\n \\[ {\\mathcal{I}}' = \\SET{X\\in {\\mathcal{I}}\\mid X\\subseteq R }. \\qedhere\\]\n\\end{definition}\n\n\\begin{lemma}\\PRFR{Jan 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, and let $R\\subseteq E$. Then\n\t$ M| R = (R,{\\mathcal{I}}')$ is a matroid.\n\\end{lemma}\n\n\\begin{proof}\\PRFR{Jan 15th}\n $\\emptyset\\subseteq R$ and $\\emptyset \\in {\\mathcal{I}}$ thus $\\emptyset \\in {\\mathcal{I}}'$\n\t{\\em (I1)}. \n\tLet $X\\subseteq Y\\in {\\mathcal{I}}'$, then $Y\\subseteq R$ and $Y\\in {\\mathcal{I}}$, therefore\n\t$X\\subseteq R$ and\n\t$X\\in {\\mathcal{I}}$, so $X\\in {\\mathcal{I}}'$ {\\em (I2)}.\n\tLet $X,Y\\in{\\mathcal{I}}'$ with $\\left| X \\right| < \\left| Y \\right|$. There is some\n\t$y\\in Y\\backslash X$ with $X\\cup\\SET{y}\\in {\\mathcal{I}}$, and since $X\\cup\\SET{y}\\subseteq R$,\n\t$X\\cup\\SET{y}\\in{\\mathcal{I}}'$ {\\em (I3)}.\n\\end{proof}\n\n\\begin{corollary}\\label{cor:rkRestrict}\\PRFR{Jan 15th}\n\tLet $M=(E,{\\mathcal{I}})$ and $R\\subseteq E$. Then for all $X\\subseteq R$ we have\n\t$$ \\mathrm{rk}_{M| R}(X) = \\mathrm{rk}_M(X).$$\n\\end{corollary}\n\\begin{proof}\\PRFR{Jan 15th}\n\tClear from Definition~\\ref{def:rank}.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:directSumAndRestrictionCommute}\\PRFR{Mar 7th}\n\tLet $M=(E,{\\mathcal{I}})$ and $N=(E',{\\mathcal{I}}')$ be matroids with $E\\cap E' = \\emptyset$.\n\tLet $X\\subseteq E\\cup E'$, then\n\t\\[ (M\\oplus N)| X = (M| X\\cap E)\\oplus (N| X\\cap E').\\]\n\\end{lemma}\n\\begin{proof}\\PRFR{Mar 7th}\n\tClear from Definitions~\\ref{def:directSum} and \\ref{def:Mrestriction}: the independent sets of the direct sum ${\\mathcal{I}}_\\oplus$ are\n\tdisjoint unions of independent sets of its parts, therefore the restriction of the family ${\\mathcal{I}}_\\oplus$ to subsets of $X$\n\tconsists of those\n\tdisjoint unions of the subsets of $X$, that are independent with respect to its parts.\n\\end{proof}\n\n\\needspace{4\\baselineskip}\n\n\n\\begin{definition}\\label{def:Mcontraction}\\PRFR{Jan 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, and let $C\\subseteq E$.\n\tThe \\deftext[contraction of M to C@contraction of $M$ to $C$]{contraction of $\\bm M$ to $\\bm C$} is the pair\\label{n:MC}\n\t$ M|' C = (C,{\\mathcal{I}}')$ where\n\t\\[ {\\mathcal{I}}' = \\SET{X\\subseteq C ~\\middle|~\\vphantom{A^A} \\forall B\\subseteq E\\backslash C\\colon\\,B\\in {\\mathcal{I}} \\Rightarrow B\\cup X \\in {\\mathcal{I}} }. \\qedhere\\]\n\\end{definition}\n\n\n\\begin{lemma}\\label{lem:contractionBchoice}\\PRFR{Jan 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, $C\\subseteq E$, and let $B$ be a base of $E\\backslash C$ in $M$.\n\tIf further\n\t\\begin{align*}\n\t {\\mathcal{I}}_B & = \\SET{X\\subseteq C ~\\middle|~\\vphantom{A^A} B\\cup X \\in {\\mathcal{I}} } \\text{ and }\\\\\n\t {\\mathcal{I}}' & = \\SET{X\\subseteq C ~\\middle|~\\vphantom{A^A} \\forall B'\\subseteq E\\backslash C\\colon\\,B'\\in {\\mathcal{I}} \\Rightarrow B'\\cup X \\in {\\mathcal{I}} }, \n\t \\end{align*}\n\t then ${\\mathcal{I}}' = {\\mathcal{I}}_B$.\n\\end{lemma}\n\n\\begin{proof}\\PRFR{Jan 15th}\n\tFrom the definition it is clear that ${\\mathcal{I}}' \\subseteq {\\mathcal{I}}_B$.\n\tFirst, we show that\n\t${\\mathcal{I}}_B$ does not depend on the choice of the base of $E\\backslash C$ in $M$.\n\tLet $B, B'\\subseteq E$ be any two bases of $E\\backslash C$ in $M$, and let ${\\mathcal{I}}_B$\n\tbe defined as in the lemma, and let\n\t${\\mathcal{I}}_{B'} = \\SET{X\\subseteq C \\mid B'\\cup X \\in {\\mathcal{I}} }$.\n\tIf $X\\in{\\mathcal{I}}_B$ then $B\\cup X \\in {\\mathcal{I}}$. Let $F= B\\cup B'\\cup X$, then\n\tthere is a base $B_X$ of $F$ with $B\\cup X\\subseteq B_X$ (Lemma~\\ref{lem:augmentation}).\n\tFurthermore, we already have $B_X= B\\cup X$, because both $B$ and $B'$ are independent subsets of $E\\backslash C$ with maximal cardinality, so any $\\left| B \\right| + 1$ elementary subset of $B\\cup B'$ must be dependent and therefore cannot be a subset of $B_X$. Again by Lemma~\\ref{lem:augmentation}, we obtain a base $B_X'$ of $F$ with $B'\\subseteq B_X'$.\n\tSince $\\left| B_X \\right| = \\left| B_X' \\right|$ (Corollary~\\ref{cor:equicardinality}) and the previous argument about subsets of $B\\cup B'$, we have $B_X' = B'\\cup X$, therefore $X\\in {\\mathcal{I}}_B'$. This proves ${\\mathcal{I}}_B\\subseteq {\\mathcal{I}}_B'$ for any two bases $B$ and $B'$ of $E\\backslash C$ in $M$, and therefore ${\\mathcal{I}}_B = {\\mathcal{I}}_B'$ for any two such bases.\n\n\t\\needspace{6\\baselineskip}\n\t\\noindent Let $X\\subseteq E$ and let $I\\subseteq E\\backslash C$ such that $I\\in{\\mathcal{I}}$. \n\tThen there is a base $B'$ of $E\\backslash C$ in $M$ with $I\\subseteq B'$.\n\tIf $X\\cup I\\notin {\\mathcal{I}}$, then clearly $X\\cup B'\\notin {\\mathcal{I}}$. Therefore we may write\n\t\\begin{align*}\n\t\t{\\mathcal{I}}' \\,\\,\\,\\,& =\\,\\, \\bigcap_{B'\\subseteq E\\backslash C,\\,B'\\in{\\mathcal{I}}} \\SET{X\\subseteq C~\\middle|~\\vphantom{A^A} X\\cup B'\\in {\\mathcal{I}}} \\\\\n\t\t& =\\,\\, \\bigcap_{B'\\,\\in\\, {\\mathcal{B}}_M(E\\backslash C)} \\SET{X\\subseteq C~\\middle|~\\vphantom{A^A} X\\cup B'\\in {\\mathcal{I}}} \n\n\n\n\t\t \\,\\,\\,\\, = \\,\\,\\,\\,{\\mathcal{I}}_B\n\t\\end{align*}\n\twhere $B$ is any fixed base of $E\\backslash C$ in $M$.\n\\end{proof}\n\n\\begin{lemma}\\PRFR{Jan 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, and let $C\\subseteq E$. Then\n\t$ M|' C = (C,{\\mathcal{I}}')$ is a matroid.\n\\end{lemma}\n\n\\begin{proof}\\PRFR{Jan 15th}\n\tLet $B$ be an arbitrarily fixed base of $E\\backslash C$ in $M$, then\n\t${\\mathcal{I}}' = \\SET{X\\subseteq C\\mid X\\cup B\\in {\\mathcal{I}}}$ (Lemma~\\ref{lem:contractionBchoice}).\n\tClearly $B\\cup\\emptyset = B\\in {\\mathcal{I}}$, thus $\\emptyset \\in {\\mathcal{I}}'$ {\\em (I1)}. Furthermore,\n\tif $X\\in{\\mathcal{I}}'$,then $B\\cup X \\in {\\mathcal{I}}$, therefore for any $Y\\subseteq X$, we have\n\t$B\\cup Y \\in {\\mathcal{I}}$ {\\em (I2)}. Now let $X,Y\\in{\\mathcal{I}}'$ with $\\left| X \\right| < \\left| Y \\right|$. Thus $B\\cup X \\in {\\mathcal{I}}$ and $B\\cup Y \\in {\\mathcal{I}}$ with $\\left| B\\cup X \\right| = \\left| B \\right| + \\left| X \\right| < \\left| B \\right| + \\left| Y \\right| = \\left| B\\cup Y \\right|$. There is $y\\in (B\\cup Y)\\backslash (B\\cup X) = Y\\backslash X$ such that\n\t$B\\cup X\\cup\\SET{y} \\in {\\mathcal{I}}$, and therefore $X\\cup\\SET{y}\\in {\\mathcal{I}}'$ {\\em (I3)}.\n\tThus $M|' C$ is a matroid.\n\\end{proof}\n\n\\begin{corollary}\\label{cor:rkContract}\\PRFR{Jan 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid and $C\\subseteq E$. Then for all $X\\subseteq C$\n\t\\[ \\mathrm{rk}_{M|' C} (X) = \\mathrm{rk}_M(X\\cup\\left( E\\backslash C \\right)) - \\mathrm{rk}_M(E\\backslash C).\\]\n\\end{corollary}\n\\begin{proof}\n\tImmediate consequence from Lemma~\\ref{lem:contractionBchoice} and Definition~\\ref{def:rank}.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:directSumAndContraction}\\PRFR{Mar 7th}\n\tLet $M=(E,{\\mathcal{I}})$ and $N=(E',{\\mathcal{I}}')$ be matroids with $E\\cap E' = \\emptyset$.\n\tLet $C \\subseteq E\\cup E'$. Then\n\t\\[ (M\\oplus N)|' C = (M|' C\\cap E) \\oplus (N|' C\\cap E').\\]\n\\end{lemma}\n\\begin{proof}\\PRFR{Mar 7th}\n\tDirect consequence of Definition~\\ref{def:directSum} and Corollary~\\ref{cor:rkContract}:\n\tSince the independent sets of $M\\oplus N$ are the disjoint unions of the independent sets of\n\t$M$ and $N$, it is clear that $\\mathrm{rk}_{M\\oplus N}(X) = \\mathrm{rk}_M(X\\cap E) + \\mathrm{rk}_N(X\\cap E')$\n\tholds for all $X\\subseteq E\\cup E'$\n\t(Definition~\\ref{def:rank}).\n\tThus \n\t\\begin{align*}\n\t\t \\mathrm{rk}_{(M\\oplus N)|' C}(X)& \\,\\,\\, =\\,\\,\\, \\mathrm{rk}_M\\left(\\left( X\\cap E\\right)\\cup \\left( E\\backslash C \\right)\\right)\n\t+ \\mathrm{rk}_N\\left(( X\\cap E')\\cup ( E'\\backslash C )\\right) \\\\& \\,\\,\\, \\,\\,\\, \\quad- \\mathrm{rk}_M(E\\backslash C) - \\mathrm{rk}_N(E'\\backslash C) \n\t\\\\& \\,\\,\\, =\\,\\,\\, \\mathrm{rk}_{M|' C\\cap E}(X\\cap E) + \\mathrm{rk}_{N|' C\\cap E}(X\\cap E'). \\qedhere\n\t\\end{align*}\n\\end{proof}\n\n\\PRFR{Jan 15th}\n\\noindent The operations of restriction and contraction are related by duality,\nif you do one of these operations on the dual of $M$ and then dualize the result,\nyou get the matroid you would have obtained from the other operation on $M$.\n\n\\begin{lemma}\\label{lem:restrictcontractdual}\\PRFR{Jan 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, and let $C\\subseteq E$. Then\n\t$ M|' C = \\left( M^\\ast | C \\right)^\\ast $.\n\\end{lemma}\n\n\\begin{proof}\\PRFR{Jan 15th}\n\tClearly, \t$ M|' C = \\left( M^\\ast | C \\right)^\\ast $ holds if and only if\n\t\t$ \\left( M|' C \\right)^\\ast = M^\\ast | C$ holds (Corollary~\\ref{cor:doubleast}).\n\tSince the family of independent sets of a matroid can be reconstructed from the values of its rank function, it suffices to show that for any $X\\subseteq C$ the equation\n\t\\[ \\mathrm{rk}_{\\left( M|' C \\right)^\\ast} (X) = \\mathrm{rk}_{M^\\ast | C} (X) \\]\n\tholds. First observe that for $X\\subseteq C \\subseteq E$ the set equation\n\t$\\left( E\\backslash C \\right)\\cup \\left( C\\backslash X \\right) = E\\backslash X$ holds. Now from Lemma~\\ref{lem:rankDual}, and the Corollaries \\ref{cor:rkContract} and \\ref{cor:rkRestrict} we obtain\n\t\\begin{align*}\n\t\t\\mathrm{rk}_{\\left( M|' C \\right)^\\ast} (X) & = \\left| X \\right| + \\mathrm{rk}_{M|' C} \\left( C\\backslash X \\right) - \\mathrm{rk}_{M|' C} (C) \\\\\n\t\t& = \\left| X \\right| + \\mathrm{rk}_M\\left( \\left( E\\backslash C \\right) \\cup \\left( C\\backslash X \\right) \\right) -\n\t\t\\mathrm{rk}_M \\left( E\\backslash C \\right) - \\mathrm{rk}_M (E) + \\mathrm{rk}_M \\left( E\\backslash C \\right) \\\\\n\t\t& = \\left| X \\right| + \\mathrm{rk}_M \\left( E\\backslash X \\right) - \\mathrm{rk}_M(E) = \\mathrm{rk}_{M^\\ast} (X) = \\mathrm{rk}_{M^\\ast | C} (X). \\qedhere\n\t\\end{align*}\n\\end{proof}\n\n\\begin{lemma}\\label{lem:contractrestrictcommutes}\\PRFR{Jan 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, and let $C\\subseteq E$ and $R\\subseteq E$ such that $\\left( E\\backslash C \\right)\\cap \\left( E\\backslash R \\right) = \\emptyset$. Then\n\t\\[ \\left( M| R \\right)|' \\left( C\\cap R \\right) = \\left( M|' C \\right)| (C\\cap R).\\]\n\\end{lemma}\n\\begin{proof}\\PRFR{Jan 15th}\n\tFirst, we want to establish the fact that $R\\backslash C = E\\backslash C$. Since $R\\subseteq E$, it remains to show that\n\t$\\left( E\\backslash C \\right)\\backslash \\left( R\\backslash C \\right) = \\emptyset$.\n\tFor all $x\\in \\left(E\\backslash C\\right) \\backslash \\left( R\\backslash C \\right)$ we have $x\\in E$, $x\\notin C$ and $x\\notin R$, thus $x\\in E\\backslash C$ and $x\\in E\\backslash R$. Since $\\left( E\\backslash C \\right)\\cap \\left( E\\backslash R \\right)= \\emptyset$, we conclude $\\left( E\\backslash C \\right)\\backslash \\left( E\\backslash R \\right)=\\emptyset$, so $E\\backslash C = R\\backslash C$. Furthermore, it is clear that $R\\backslash \\left( C\\cap R \\right) = R\\backslash C$ for all sets $C$ and $R$.\n\tWe give a proof of the statement of the lemma using the rank formulae from Corollaries~\\ref{cor:rkRestrict} and \\ref{cor:rkContract}.\n\tLet $X\\subseteq C\\cap R$, then\n\t\\begin{align*}\n\t\t\\mathrm{rk}_{ \\left( M| R \\right)|' \\left( C\\cap R \\right)} & = \\mathrm{rk}_{M| R} \\left( X\\cup \\left( R\\backslash C \\right) \\right) - \\mathrm{rk}_{M| R}(R\\backslash C) \\\\\n\t\t& = \\,\\,\\,\\,\\,\\,\\,\\mathrm{rk}_{M} \\left( X\\cup \\left( R\\backslash C \\right) \\right) - \\mathrm{rk}_{M}(R\\backslash C) \\\\\n\t\t& = \\,\\,\\,\\,\\,\\,\\,\\mathrm{rk}_{M} \\left( X\\cup \\left( E\\backslash C \\right) \\right) - \\mathrm{rk}_{M}(E\\backslash C) \\\\\n\t\t& = \\,\\,\\,\\,\\,\\,\\,\\mathrm{rk}_{M|' C} (X) \\quad \\quad \\,\\,\\,\\,\\,\\,\\, = \\mathrm{rk}_{ \\left( M|' C \\right)| (C\\cap R)}(X). \\qedhere\n\t\\end{align*}\n\\end{proof}\n\n\\begin{definition}\\PRFR{Mar 7th}\n\tLet $M=(E,{\\mathcal{I}})$ and $N=(E',{\\mathcal{I}}')$ be matroids.\n\tWe shall call $N$ a \\deftext[minor of $M$]{minor of $\\bm M$},\n\tif there are sets $X \\subseteq Y \\subseteq E$ such that\n\t\\[ N = \\left( M |' Y \\right)| X \\]\n\tholds.\n\\end{definition}\n\n\\begin{remark}\\label{rem:contractRestrictCommutingFormula}\\PRFR{Mar 7th}\n\tFor $M=(E,{\\mathcal{I}})$ and $X\\subseteq Y \\subseteq E$ we have $Y \\cap \\left( E\\backslash\\left( Y\\backslash X \\right) \\right)=X$ and\n\t $\\left( E\\backslash Y \\right) \\cap \\left( Y\\backslash X \\right) = \\emptyset$, \n\tso Lemma~\\ref{lem:contractrestrictcommutes} yields that\n\t\\[ \\left( M |' Y \\right)| X = \\left( M | E\\backslash\\left( Y\\backslash X \\right) \\right)|' X\n\t\\,\\,{\\mathop{\\text{~and~}}}\\,\\,\n\t\\left( M | Y \\right)|' X = \\left( M |' E\\backslash\\left( Y\\backslash X \\right) \\right)| X\n\t. \\qedhere\n\t \\] \n\\end{remark}\n\n\\begin{definition}\\PRFR{Jan 15th}\n\tLet ${\\mathcal{M}}$ be a class of matroids. Then ${\\mathcal{M}}$ shall be called a \\deftext{minor-closed class},\n\tif for every $M=(E,{\\mathcal{I}})\\in {\\mathcal{M}}$ and every $X\\subseteq E$,\n\talso $M| X \\in {\\mathcal{M}}$ and $M|' X\\in {\\mathcal{M}}$ holds.\n\\end{definition}\n\n\\begin{example}\\PRFR{Jan 15th}\n\tLet ${\\mathcal{M}}$ be the class where $M\\in {\\mathcal{M}}$ if and only if $M=\\left( E,2^E \\right)$ for any set $E$,\n\ti.e. ${\\mathcal{M}}$ is the class of all free matroids. Clearly, for every $M=\\left( E,2^E \\right)$ and every $X\\subseteq E$\n\twe have\n\t$M| X = M|' X = \\left(X,2^X\\right) \\in {\\mathcal{M}}$.\n\\end{example}\n\n\\needspace{5\\baselineskip}\n\\begin{definition}\\PRFR{Mar 7th}\n\tLet ${\\mathcal{M}}$ be a minor-closed class of matroids. A matroid $M=(E,{\\mathcal{I}})$ is called\n\t\\deftext[excluded minor]{excluded minor for $\\bm {\\mathcal{M}}$}\n\tif $M\\notin {\\mathcal{M}}$ and if for every $X\\subsetneq E$ we have both $M| X \\in {\\mathcal{M}}$ and $M|' X \\in {\\mathcal{M}}$.\n\tFurthermore, a minor-closed class of matroids ${\\mathcal{M}}$ is called \\deftextX{characterized by finitely many excluded minors}\n\tif there are only finitely many pair-wise non-isomorphic excluded minors for ${\\mathcal{M}}$.\n\\end{definition}\n\n\\begin{example}\\PRFR{Mar 7th}\nA matroid is representable over the $2$-elementary field ${\\mathbb{F}}_2$ (Definition~\\ref{def:representableMoverK})\n if and only if it has no minor isomorphic to\nthe rank-$2$ uniform matroid $\\left( \\SET{a,b,c,d},\\SET{X\\subseteq \\SET{a,b,c,d}~\\middle|~\\vphantom{A^A}\\left| X \\right|\\leq 2} \\right)$.\n(Theorem 6.5.4 \\cite{Ox11}, p.193). Thus the class of all matroids representable over ${\\mathbb{F}}_2$ is characterized by finitely many,\nor in this case, a single excluded minor.\n\\end{example}\n\n\\begin{remark}\n\tIf ${\\mathcal{M}}$ is a minor-closed class of matroids with the property that $M\\in {\\mathcal{M}} \\Leftrightarrow M^\\ast \\in {\\mathcal{M}}$ holds for all matroids $M$,\n\ti.e. ${\\mathcal{M}}$ is closed under duality;\n\tthen $N$ is an excluded minor of ${\\mathcal{M}}$ if and only if $N^\\ast$ is an excluded minor of ${\\mathcal{M}}$ (see also Lemma~\\ref{lem:contractrestrictcommutes}).\n\\end{remark}\n\n\\PRFR{Mar 7th}\n\\noindent The excluded minors for matroids representable over fields with $2$, $3$, and $4$ elements are known (\\cite{Ox11}, p.193),\nand the famous {\\em Rota's Conjecture} states that for every finite field ${\\mathbb{F}}$, the class of matroids representable over ${\\mathbb{F}}$ is\ncharacterized by finitely many excluded minors. J.~Geelen, B.~Gerards and G.~Whittle claim to have proven Rota's Conjecture and published\n an overview of their proof in \\cite{GGW14}. Furthermore, it has been shown\n that both the class of matroids representable over the field of the\n reals $\\mathbb{R}$ and the class of gammoids have the property, that\n every matroid $M$ in each respective class is a minor of an excluded minor of that class, therefore those classes cannot be characterized by\n finitely many excluded minors, because both classes are non-empty and closed under direct sums, thus they are infinite.\n The result for the class of matroids representable over $\\mathbb{R}$ \n has been proven by D.~Mayhew, M.~Newman, and G.~Whittle in \\cite{MNW09},\n and the result for the class of gammoids can be found in a paper by D.~Mayhew \\cite{Ma16}, where the excluded minor constructed for\n an arbitrary gammoid is also an excluded minor for the class of matroids representable over $\\mathbb{R}$.\n \n\\subsection{Matroids Representable Over a Field}\n\n\\PRFR{Jan 15th}\n\\noindent A quite natural class of matroids arises from the notion of linear independence. We only give \na short introduction here. Those readers, who are interested in the classes of matroids representable over some given field ${\\mathbb{F}}$, shall hereby be referred to J.G.~Oxley's book \\cite{Ox11}.\n\n\\begin{definition}\\label{def:Mmu}\\PRFR{Jan 15th}\n\tLet ${\\mathbb{K}}$ be a field, $E$ and $C$ be finite sets. Let $\\mu\\in {\\mathbb{K}}^{E\\times C}$ be an $E\\times C$-matrix over ${\\mathbb{K}}$. The \\deftext[matroid represented over a field]{matroid represented by $\\bm \\mu$ over $\\bm{\\mathbb{K}}$} is the pair\\label{n:matMmu} $M(\\mu) = (E,{\\mathcal{I}})$ where\n\t\\[ {\\mathcal{I}} = \\SET{X\\subseteq E~\\middle|~\\vphantom{A^A} {\\mathrm{idet}~} \\left( \\mu | X \\right) = 1}. \\qedhere\\]\n\\end{definition}\n\n\\begin{lemma}\\PRFR{Jan 15th}\nLet ${\\mathbb{K}}$ be a field, $E$ and $C$ be finite sets. Let $\\mu\\in {\\mathbb{K}}^{E\\times C}$. Then\n$M(\\mu)$ is a matroid.\n\\end{lemma}\n\n\\noindent The proof is essentially elementary linear algebra.\n\n\\begin{proof} Let $(E,{\\mathcal{I}}) = M(\\mu)$. \\PRFR{Jan 15th}\n\tIt is clear from Definition~\\ref{def:idet} that for $X\\subseteq E$, the equality ${\\mathrm{idet}~} \\left( \\mu| X \\right) = 1$ holds if and only if the set\n\t$V_X = \\SET{\\mu_x\\mid x\\in X}$ is linear independent in the vector space ${\\mathbb{K}}^C$ with the further property\n\t $\\left| V_X \\right| = \\left| X \\right|$. Thus $\\emptyset\\in {\\mathcal{I}}$ {\\em (I1)}. \n\t For every $Y\\subseteq X\\in{\\mathcal{I}}$, we\n\t have that $V_Y = \\SET{\\mu_y\\mid y\\in Y}$ is linear independent in ${\\mathbb{K}}^C$ with $\\left| V_Y \\right| = \\left| Y \\right|$, thus $Y\\in{\\mathcal{I}}$ {\\em (I2)}. Let $X,Y\\in{\\mathcal{I}}$ with $\\left| X \\right| < \\left| Y \\right|$, and let $V_X$, $V_Y$ be defined as above. Since $\\left| V_Y \\right| > \\left| V_X \\right|$ and $V_Y$ is linear independent in ${\\mathbb{K}}^C$, we have that\n\t ${\\mathrm{span}}_{{\\mathbb{K}}^C}(V_X) \\subsetneq {\\mathrm{span}}_{{\\mathbb{K}}^C}\\left( V_X\\cup V_Y \\right)$. Therefore, there is some $\\mu_y\\in V_Y$ with $\\mu_y\\notin {\\mathrm{span}}_{{\\mathbb{K}}^C}(V_X)$, and consequently, $V' = V_X\\cup\\SET{\\mu_y}$\n\t is linear independent in ${\\mathbb{K}}^C$ with $\\left| V' \\right| = \\left| X \\right| + 1$, thus\n\t $X\\cup\\SET{y}\\in {\\mathcal{I}}$ {\\em (I3)}.\n\\end{proof}\n\n\n\\begin{corollary}\\PRFR{Jan 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, ${\\mathbb{K}}$ be a field, $E$ and $C$ be finite sets, and\n\t$\\mu\\in {\\mathbb{K}}^{E\\times C}$ be a matrix. For all $R\\subseteq E$,\n\t\\[ M(\\mu)| R = M\\left( \\mu | R\\right).\\]\n\\end{corollary}\n\n\n\\begin{remark}\\label{rem:colops}\\PRFR{Jan 15th}\n\tIt is a well-known fact from linear algebra that the following operations on $\\mu\\colon E\\times C\\longrightarrow \\mathbb{K}$ do not\n\tchange linear dependency between rows, and therefore do not alter the matroid $M(\\mu)$:\n\t\\begin{enumerate}\\renewcommand{\\theenumi}{{\\em (\\roman{enumi})}}\\renewcommand{\\labelenumi}{\\theenumi}\n\t\t\\item Interchanging two columns $c_1,c_2\\in C$, i.e. if $\\nu(e,c) = \\begin{cases} \\mu(e,c) &\\quad \\text{if~} c\\notin\\SET{c_1,c_2},\\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\mu(e,c_2) &\\quad \\text{if~} c=c_1,\\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\mu(e,c_1) &\\quad \\text{if~} c=c_2,\\end{cases}$\n\t\t\t\t\tthen $M(\\mu) = M(\\nu)$.\n\t\t\\item Adding a multiple of one column to another column, i.e. for $c_1,c_2\\in C$ with $c_1\\not= c_2$ and $\\alpha\\in \\mathbb{K}$, i.e.\n\t\t\\hfill\n\t\tif $\\nu(e,c) = \\begin{cases} \\mu(e,c) &\\quad \\text{if~} c\\not=c_2,\\\\\n\t\t\t\t\t\t\t\t\t \\mu(e,c_2) + \\alpha\\cdot\\mu(e,c_1) &\\quad \\text{if~} c=c_2,\\end{cases}^{\\vphantom{X}}$\\linebreak\n\t\tthen $M(\\mu) = M(\\nu)$.\n\t\t\\item Multiplying a column $c_1\\in C$ with $\\alpha\\in \\mathbb{K}\\BSET{0}$, i.e. \\\\\n\t\tif $\\nu(e,c) = \\begin{cases} \\mu(e,c) &\\quad \\text{if~} c\\not=c_1,\\\\\n\t\t\t\t\t\t\t\t\t \\alpha\\cdot\\mu(e,c_1) &\\quad \\text{if~} c=c_1,\\end{cases}^{\\vphantom{X}}\\quad\\quad$\n\t\tthen $M(\\mu) = M(\\nu)$.\n\t\\end{enumerate}\n\tFurthermore, if $B\\subseteq E$ is a base of $M(\\mu)$, then we can use Gau\u00df-Jordan elimination steps\\footnote{This procedure is commonly refered to as ''pivoting in the ordered basis $B$`` in the context of linear programming. Careful pivoting is the foundation of the simplex algorithm for solving linear optimization problems.} in order to obtain an injective map\n\t$\\iota\\colon B\\longrightarrow C$ and a matrix $\\nu$, which has the properties $M(\\nu) = M(\\mu)$ and for all $b\\in B$ and all $c\\in C$,\n\t$\\nu(b,c) = \\begin{cases} 1 & \\text{if~} c = \\iota(b),\\\\ 0 & \\text{otherwise}. \\end{cases}^{\\vphantom{X}}_{\\vphantom{X}}$\n\n\t\\noindent\n\tFrom the matrix $\\nu$, we can easily read some important properties of $M(\\mu)$. Let $e\\in E\\backslash B$, \n\tthen the unique circuit contained in $B\\cup\\SET{e}$ consists of $e$ and the elements $b'\\in B$ where $\\nu(e,\\iota(b'))\\not= 0$.\n\tIf $B'\\subseteq B$, then $\\mathrm{cl}_{M(\\nu)}(B')$ consists of $B'$ and all $e\\in E\\backslash B$ which have the property that $\\nu(e,c)=0$ holds for all $c\\in C\\backslash\\left( \\iota[B'] \\right)$.\n\\end{remark}\n\n\\begin{definition}\\label{def:representableMoverK}\\PRFR{Jan 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, ${\\mathbb{K}}$ be a field. We say that $M$ is \\deftext[matroid representable over a field]{representable over $\\bm {\\mathbb{K}}$}, if there is a finite set $C$\n\tand a matrix $\\mu\\in {\\mathbb{K}}^{E\\times C}$, such that $M = M(\\mu)$.\n\\end{definition}\n\n\n\\begin{lemma}\\label{lem:contractequalspivot}\\PRFR{Jan 15th}\n Let $E,C$ be finite sets, and $\\mu\\in {\\mathbb{K}}^{E\\times C}$ be a matrix.\n Let $e\\in E$ and $c\\in C$, such that $\\mu(e,c) \\not= 0$.\n Let\n \\[ \\nu \\colon \\left( E\\BSET{e} \\right)\\times \\left( C\\BSET{c} \\right) \\longrightarrow \\mathbb{K},\\,(f,d)\\mapsto \\mu(f,d) - \\frac{\\mu(e,d)}{\\mu(e,c)}\\cdot \\mu(f,c)\\]\n be the matrix obtained by carrying out a Gau\u00df-Jordan elimination step with the pivot index $(e,c)$ and then deleting the corresponding row and column.\n Then $$M(\\mu)|'\\left( E\\BSET{e} \\right) = M(\\nu).$$\n\\end{lemma}\n\\begin{proof}\n\tLet $\\nu'\\in \\mathbb{K}^{E\\times C}$ where for all $(f,d)\\in E\\times C$\n\t\\[ \\nu'(f,d) = \\begin{cases} \\mu(f,d) - \\frac{\\mu(e,d)}{\\mu(e,c)}\\cdot \\mu(f,c) & \\quad \\text{if~} d\\not= c, \\\\\n\t \t\t\t\t\t\t\t\\frac{\\mu(f,c)}{\\mu(e,c)} & \\quad \\text{if~} d=c. \\\\\n\t\t\t\t\t\\end{cases}\n\t\\]\n\tSince $\\nu'$ arises from $\\mu$ by elementary column operations, we have $M(\\mu) = M(\\nu')$ (Remark~\\ref{rem:colops}). Furthermore,\n\t$\\nu'(e,c) = 1$ and $\\nu'(e,d) = 0$ for all $d\\in C\\BSET{c}$. Let $E'\\subseteq E\\BSET{e}$ and $C'\\subseteq C\\BSET{c}$ with $\\left| E' \\right| = \\left| C' \\right|$. Then\n\t\\begin{align*}\n\t\\det \\left( \\nu'| \\left( E'\\cup\\SET{e} \\right)\\times \\left( C'\\cup\\SET{c} \\right) \\right)&\\,\\, =\\,\\,\n\t\t\\sigma \\cdot \\det \\left( \\nu' | E'\\times C' \\right)\n\t\t\\,\\, = \\,\\, \\sigma \\cdot \\det \\left( \\nu | E'\\times C' \\right)\n\t\\end{align*}\n\tfor some $\\sigma\\in \\SET{-1,1}$. Thus for $X\\subseteq E\\BSET{e}$ $${\\mathrm{idet}~} \\left( \\nu'| \\left( X\\cup\\SET{e} \\right)\\right) = {\\mathrm{idet}~} \\left( \\nu | X \\right),$$\n\tand consequently $X$ is independent in $M(\\nu)$, if and only if $X\\cup\\SET{e}$ is independent in $M(\\nu') = M(\\mu)$. Therefore\n\t$M(\\nu) = M(\\mu) |' \\left( E\\BSET{e} \\right)$.\n\\end{proof}\n\n\\begin{remark}\\label{rem:contraction}\\PRFR{Feb 15th}\n\tLet $M(\\mu)$ be a matroid for some $\\mu\\in \\mathbb{R}^{E\\times C}$. A straightforward consequence of Lemma~\\ref{lem:contractequalspivot} is\n\tthat for $X\\subseteq E$, we can pivot in a base $B$ of $M(\\mu)$ with the property that $B\\backslash X$ is a base of $E\\backslash X$ --\n\twhich exists due to Lemma~\\ref{lem:augmentation} -- and then restrict the resulting matrix $\\nu$ to $X\\times C_0$ where\n\t$C_0 = \\SET{c\\in C\\mid \\forall b'\\in B\\backslash X\\colon\\, \\nu(b',c) = 0}$. Then $M(\\mu)|' X = M\\left( \\nu| X\\times C_0 \\right)$.\n\\end{remark}\n\n\\begin{lemma}\\label{lem:standardMatrixNu}\\PRFR{Feb 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid that is representable over ${\\mathbb{K}}$, such that for some $n,r\\in \\mathbb{N}$, $E=\\dSET{e_1,e_2,\\ldots,e_n}$\n\tand $B_0 = \\dSET{e_1,e_2,\\ldots,e_r}$ is a base of $M$. Then there is a matrix $\\nu\\in {\\mathbb{K}}^{E\\times B_0}$ such that\n\t$\\nu| B_0\\times B_0$ is the identity matrix for $B_0$ over ${\\mathbb{K}}$.\n\\end{lemma}\n\\begin{proof}\\PRFR{Feb 15th}\n\tThis is basic linear algebra.\n\tLet $\\mu\\in {\\mathbb{K}}^{E\\times C}$ be a matrix with $M=M(\\mu)$. Then the row vectors $\\SET{\\mu_b \\mid b\\in B_0}$ form \n\ta basis of a sub-vector space $V\\subseteq {\\mathbb{K}}^C$, and since $B_0$ is a base of $M$, we have that $\\mu_e\\in V$ for all\n\t$e\\in E$. Thus every $\\mu_e$ has a unique representation as linear combination of vectors from $\\SET{\\mu_b \\mid b\\in B_0}$,\n\tand we can set $\\nu(e,b)$ to be the coefficient of $\\mu_{b}$ with respect to the linear combination representing $\\mu_e$,\n\tfor all $e\\in E$ and $b\\in B_0$. Since a change of basis in a vector space does not affect linear dependency, we have\n\t$M(\\mu) = M(\\nu)$.\n\\end{proof}\n\n\n\n\n\n\\begin{remark}\\label{rem:stdRep}\\PRFR{Feb 15th}\n\tAn immediate consequence of Lemma~\\ref{lem:standardMatrixNu} is, that\n\tevery matroid $M=(E,{\\mathcal{I}})$ representable over ${\\mathbb{K}}$ with $r=\\mathrm{rk}_M(E)$ has a matrix $\\mu\\in {\\mathbb{K}}^{E\\times \\SET{1,2,\\ldots,r}}$\n\t such that $M=M(\\mu)$ and such that for some base $B\\subseteq E$ of $M$,\n\tthe matrix $\\mu|(B\\times \\SET{1,2,\\ldots,r})$ resembles an identity matrix --- up to renaming of the rows.\n\tThus we may consider $\\mu^\\top$ to be that identity matrix in apposition with a matrix $A^\\top \\in {\\mathbb{K}}^{\\SET{1,2,\\ldots,r}\\times (E\\backslash B)}$,\n\ti.e. that $\\mu = \\left( I_r ~~ A^\\top \\right)^\\top$. A matrix of this form is called \\deftext[standard representation over ${\\mathbb{K}}$]{standard representation}. If $\\mu$ is a standard representation, then\n\t $\\nu = \\left( -A ~~ I_{|E|-r} \\right)^\\top$ has the property that $M^\\ast = M(\\nu)$\n\tand further, that for all $e,f\\in E$, $\\langle \\mu_e, \\nu_f \\rangle = 0$ (Corollary~1, \\cite{We76}, p.~143).\n\tThus for every field ${\\mathbb{K}}$ the family of matroids representable over ${\\mathbb{K}}$ is closed under duality.\n\\end{remark}\n\\subsection{Ranked Cyclic Flats Axioms}\n\n\\studyremark{See Welsh Chapter 8, might give easier proof. If this is still needed...}\n\nThe following axiomatization of matroids has been by J.~Bonin and A.~deMier in\n\\cite{BdM08}. Since it is closely connected to Mason's\n$\\alpha$-criterion\\footnote{See Theorem \\ref{thm:AlphaCriterion} in section\n\\ref{sec:StrictGammoids}.} it is a particularly interesting axiom system for\nthe study of gammoids.\n\n\\begin{theorem}\n\tLet $E$ be a finite set, $\\zeta \\subseteq 2^{E}$ and $\\rho \\colon \\zeta\n\t\\longrightarrow \\mathbb{N}$ a map. There is a matroid $M$ on the ground set $E$ with\n\trank function $\\mathrm{rk} \\colon 2^{E}\\longrightarrow \\mathbb{N}$ such that $\\zeta = {\\mathcal{Z}}(M)$\n\tand $\\rho(Z) = \\mathrm{rk}(Z)$ for all $Z\\in \\zeta$, if and only if\n\t$(\\zeta,\\rho)$ has the properties\n\t\\begin{enumerate}\n\t\t\\item[(Z0)] $(\\zeta,\\subseteq)$, ordered by ordinary set-inclusion,\n\t\t\t\t\t is a lattice\\footnote{Since $\\zeta$ is finite, $(\\zeta,\\subseteq)$ is a complete lattice with lattice zero $\\bigwedge \\zeta$.} with meet $\\wedge$ and join $\\vee$,\n\t\t\\item[(Z1)] $\\rho( \\bigwedge \\zeta ) = 0$,\n\t\t\\item[(Z2)] $0 < \\rho(Y) - \\rho(X) < |Y\\backslash X|$ for all $X,Y \\in \\zeta$ with $X\\subsetneq Y$,\n\t\t\\item[(Z3)] for all $X,Y\\in \\zeta$,\n\t\t\\[ \\rho(X) + \\rho(Y) \\geq \\rho(X\\vee Y) + \\rho(X\\wedge Y) + |(X\\cap Y)\\backslash (X\\wedge Y)| .\\]\n\t\\end{enumerate}\n\\end{theorem}\n\n\\noindent We give a different proof than the one found in \\cite{BdM08}, which uses the property that cyclic flats are unions of closures of circuits and shows that {\\em (Z0) -- (Z3)} implies the circuit axioms for minimally dependent subsets of $E$.\n\n\\begin{proof}\n\tFirst, we show that the rank function on the family ${\\mathcal{Z}}$ of cyclic flats of $M$ has the properties {\\em (Z0) -- (Z3)}.\n\n\tTODO!\n\\end{proof}\n\n\n\\section{Single Element Extensions}\n\nH.H.~Crapo has exhaustively studied extensions of matroids by single elements in \\cite{C65}.\n\n\n\\begin{definition}\\PRFR{Feb 15th}\n Let $M=(E,{\\mathcal{I}})$ be a matroid, $A,B\\subseteq E$.\n Then $A$ and $B$ are a \\deftext[modular pair in M@modular pair in $M$]{modular pair in $\\bm M$},\n whenever $$\\mathrm{rk}(A\\cap B) + \\mathrm{rk}(A\\cup B) = \\mathrm{rk}(A) + \\mathrm{rk}(B)$$ holds.\n A modular pair is called \\deftext[trivial modular pair]{trivial},\n if $A\\subseteq B$ or $B\\subseteq A$.\n\\end{definition}\n\n\\begin{example}\\label{ex:indepModPairs}\\PRFR{Feb 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid and $A,B\\subseteq E$ such that $A\\cup B \\in {\\mathcal{I}}$.\n\tThen $A$ and $B$ are a modular pair in $M$, since\n\t\\[ \\mathrm{rk}(A\\cup B) + \\mathrm{rk}(A\\cap B) = \\left| A\\cup B \\right| + \\left| A\\cap B \\right| = \\left| A \\right| + \\left| B \\right| = \\mathrm{rk}(A) + \\mathrm{rk}(B).\n\t\\qedhere \\]\n\\end{example}\n\n\\begin{lemma}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, $A,B\\subseteq E$.\n\tThen $A$ and $B$ are a modular pair in $M$ if and only if there is a base $X$ of $A\\cup B$\n\tsuch that $X\\cap A$ is a base of $A$ and $X\\cap B$ is a base of $B$. \n\n\\end{lemma}\n\\begin{proof}\n\tLet $X$ be a base of $A\\cup B$ such that $X\\cap A$ is a base of $A$ and $X\\cap B$ is a base of $B$.\n\tThen $X\\cap A \\cap B$ is a base of $A\\cap B$: Since $\\mathrm{rk}(A) + \\mathrm{rk}(B) \\geq \\mathrm{rk}(A\\cup B) + \\mathrm{rk}(A\\cap B)$\n\tholds by submodularity of the rank function,\n\tand since $\\left| X\\cap A\\cap B \\right| \\leq \\mathrm{rk}(A\\cap B)$ holds by {\\em (I2)} and Definition~\\ref{def:rank},\n\twe obtain\n\t\\begin{align*}\n\t\\left| X\\cap A \\cap B \\right| & \\leq \\mathrm{rk}(A\\cap B) \\leq \\mathrm{rk}(A) + \\mathrm{rk}(B) - \\mathrm{rk}(A\\cup B) \\\\\n\t& = \\left| X\\cap A \\right| + \\left| X\\cap B \\right| - \\left| X \\right| = \\left| X\\cap A\\cap B \\right|.\n\t\\end{align*}\n\tIt has been shown in Example~\\ref{ex:indepModPairs} that $X\\cap A$ and $X\\cap B$ are a modular pair in $M$.\n\tThus $A$ and $B$ are a modular pair in $M$, because $\\mathrm{rk}(A) = \\mathrm{rk}(X\\cap A)$, $\\mathrm{rk}(B) = \\mathrm{rk}(X\\cap B)$,\n\t$\\mathrm{rk}(A\\cap B) = \\mathrm{rk}(X\\cap A\\cap B)$, and $\\mathrm{rk}(A\\cup B) = \\mathrm{rk}(X)$ holds.\n\tNow let $A,B\\subseteq E$ such that there is no base $X$ of $A\\cup B$, for which both $X\\cap A$ is a base of $A$\n\tand $X\\cap B$ is a base of $B$. Then for all bases $X$ of $A\\cup B$, for which $X\\cap A$ is a base of $A$,\n\tand for which $X\\cap A \\cap B$ is a base of $A\\cap B$,\n\tthere is some $b\\in B\\backslash \\mathrm{cl}(X\\cap B)$, i.e. $\\mathrm{rk}(B) > \\mathrm{rk}(X\\cap B)$. Lemma~\\ref{lem:augmentation} guarantees that there is\n\ta base $X$ of $A\\cup B$ with $\\mathrm{rk}(X\\cap A) = \\mathrm{rk}(A)$ and $\\mathrm{rk}(X\\cap A\\cap B) = \\mathrm{rk}(A\\cap B)$. Thus we obtain that\n\t\\( \\mathrm{rk}(A) + \\mathrm{rk}(B) > \\left| X\\cap A \\right| + \\left| X\\cap B \\right| = \\left| X \\right| + \\left| X\\cap A\\cap B \\right| = \\mathrm{rk}(A\\cup B) + \\mathrm{rk}(A\\cap B) \\)\n\tholds, which\n\timplies that $A$ and $B$ are not a modular pair in $M$.\n\\end{proof}\n\n\\begin{definition}\\PRFR{Jan 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, and let $C \\subseteq {\\mathcal{F}}(M)$ be a set of flats of $M$.\n\tWe call $C$ a \\deftext[modular cut of M@modular cut of $M$]{modular cut of $\\bm M$}, if $C$ has the properties\n\tthat\n\t\\begin{enumerate}\\renewcommand{\\theenumi}{(\\roman{enumi})}\\renewcommand{\\labelenumi}{\\theenumi}\n\t\\item for all $A,B\\in{\\mathcal{F}}(M)$ the implication\n\t$$\\mathrm{rk}(A) + \\mathrm{rk}(B) = \\mathrm{rk}(A\\cap B) + \\mathrm{rk}(A\\cup B) \\quad \\Longrightarrow \\quad A\\cap B\\in C$$\n\tholds, and\n\t\\item for all $X,Y\\in {\\mathcal{F}}(M)$ with $X\\subseteq Y$, the implication $X\\in C \\Rightarrow Y\\in C$ holds.\n\\end{enumerate}\n The \\deftextX{class of all modular cuts of $\\bm M$} shall be denoted by\\label{n:MM}\n \\[ {\\mathcal{M}}(M) = \\SET{C\\subseteq {\\mathcal{F}}(M)\\mid C\\text{ is a modular cut of }M}. \\qedhere\\]\n\\end{definition}\n\n\\begin{definition}\\PRFR{Jan 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid and let $e\\notin E$. The \\deftextX{class of single element extensions of $\\bm M$ by $\\bm e$}\n\t is defined to be\\label{n:XMe}\n\t\\[ {\\mathcal{X}}(M,e) = \\SET{(E\\cup\\SET{e},{\\mathcal{I}}') ~\\middle|~ {\\mathcal{I}}' \\subseteq 2^{E\\cup\\SET{e}}\\colon\\,\\,\n\t{\\mathcal{I}}'\\cap 2^E = {\\mathcal{I}} \n\t{\\mathop{\\text{~and~}}} (E\\cup\\SET{e},{\\mathcal{I}}')\\text{ is a matroid}}.\\]\n\tLet $N=(F,{\\mathcal{J}})$ be a matroid. $N$ shall be called \\deftext[single element extension of M@single element extension of $M$]{single element extension of $\\bm M$}, if $F\\backslash E = \\SET{f}$ and $N\\in {\\mathcal{X}}(M,f)$.\n\\end{definition}\n\n\\noindent We convince ourselves that $N$ is indeed an extension of $M$ in the sense that $N$ behaves exactly like $M$ with respect to subsets of $E$.\n\n\\needspace{3\\baselineskip}\n\\begin{lemma}\\label{lem:extension_rk}\\PRFR{Jan 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, $e\\notin E$, $N\\in {\\mathcal{X}}(M,e)$, and let $X\\subseteq E$.\n\tThen $\\mathrm{rk}_{M}(X) = \\mathrm{rk}_{N}(X)$.\n\\end{lemma}\n\\begin{proof}\n\tLet $N=(E\\cup\\SET{e},{\\mathcal{I}}')$. Since ${\\mathcal{I}}' \\cap 2^{E} = {\\mathcal{I}}$, $\\SET{Y\\subseteq X\\mid Y\\in {\\mathcal{I}}} = \\SET{Y\\subseteq X\\mid Y\\in {\\mathcal{I}}'}$. Thus\n\t\\[ \\mathrm{rk}_{M}(X) = \\max\\SET{\\,\\left| Y \\right|\\,\\, \\vphantom{A^A}~\\middle|~ Y\\subseteq X,\\, Y\\in{\\mathcal{I}}} =\n\t\\max\\SET{\\,\\left| Y \\right|\\,\\, \\vphantom{A^A}~\\middle|~ Y\\subseteq X,\\, Y\\in{\\mathcal{I}}'} = \\mathrm{rk}_N(X). \\qedhere\\]\n\\end{proof}\n\n\\noindent Now we have all the definitions that we need in order to present H.H.~Crapo's results\n \\cite{C65}:\n\n \\needspace{6\\baselineskip}\n\n\\begin{theorem}\\label{thm:crapo}\\PRFR{Jan 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid and $e\\notin E$. Then there is a bijection\n\t\\[ \\phi \\colon {\\mathcal{X}}(M,e) \\longrightarrow {\\mathcal{M}}(M) \\]\n\twhich maps the single element extension $N$ to\n\tthe modular cut \n\t\\[ \\phi(N) = \\SET{F\\in{\\mathcal{F}}(M)\\mid e \\in \\mathrm{cl}_N(F)}. \\]\n\\end{theorem}\n\n\\begin{proof}\\PRFR{Jan 15th}\n\tFirst, we show that $\\phi$ is well-defined.\n\tLet $N \\in {\\mathcal{X}}(M,e)$ be a single element extension of $M$. \n\tWe have to prove that $\\phi(N)$ is indeed a modular cut of $M$.\n\tLet $F\\in \\phi(N)$, and $G\\in {\\mathcal{F}}(M)$ with $F\\subseteq G$, then $e\\in \\mathrm{cl}_{N}(F) \\subseteq \\mathrm{cl}_{N}(G)$, thus $G\\in \\phi(N)$.\n\tNow, let $A,B\\in \\phi(N)$ such that\n\t\\[ \\mathrm{rk}_M(A) + \\mathrm{rk}_M(B) = \\mathrm{rk}_M(A\\cap B) + \\mathrm{rk}_M(A\\cup B). \\]\n\tWe give an indirect argument for $A\\cap B\\in \\phi(N)$.\n\tAssume that $A\\cap B \\notin \\phi(N)$. Then $e\\notin \\mathrm{cl}_{N}(A\\cap B)$ thus\n\t$\\mathrm{rk}_{N}( (A\\cap B) \\cup \\SET{e}) > \\mathrm{rk}_{N}(A\\cap B)$. By Lemma~\\ref{lem:extension_rk},\n\twe have the equation \\[ \\mathrm{rk}_N(A) + \\mathrm{rk}_N(B) = \\mathrm{rk}_N(A\\cap B) + \\mathrm{rk}_N(A\\cup B). \\]\n\tFurthermore, $A,B\\in \\phi(N)$,\n\ttherefore $e\\in \\mathrm{cl}_{N}(A)$, $e\\in \\mathrm{cl}_{N}(B)$, and $e\\in \\mathrm{cl}_{N}(A\\cup B)$, so\n\t$\\mathrm{rk}_{N}(A\\cup\\SET{e}) = \\mathrm{rk}_{N}(A)$, $\\mathrm{rk}_{N}(B\\cup\\SET{e}) = \\mathrm{rk}_{N}(B)$, and\n\t$\\mathrm{rk}_{N}(A\\cup B\\cup\\SET{e}) = \\mathrm{rk}_{N}(A\\cup B)$. This yields\n\t\\begin{align*}\n \\mathrm{rk}_N(\\left(A\\cap B\\right) \\cup \\SET{e}) & >\n\t \\mathrm{rk}_{N}(A\\cap B) \\\\&\n\t \t= \\mathrm{rk}_{N}(A) + \\mathrm{rk}_N(B) - \\mathrm{rk}_{N}(A\\cup B)\n\t \\\\ & = \\mathrm{rk}_{N}(A\\cup\\SET{e}) + \\mathrm{rk}_{N}(B\\cup\\SET{e}) - \\mathrm{rk}_{N}(A\\cup B\\cup\\SET{e}),\\\\\n\t\\end{align*}\n\twhich contradicts {\\em (R3)}, the submodularity of $\\mathrm{rk}_{N}$, which guarantees\n\t\\[ \\mathrm{rk}_N\\left(A\\cup\\SET{e}\\right) + \\mathrm{rk}_N\\left(B\\cup\\SET{e}\\right) \\geq \\mathrm{rk}_{N}(A\\cup B\\cup\\SET{e}) + \\mathrm{rk}_{N}(\\left(A\\cap B\\right)\\cup\\SET{e}).\\]\n\t Thus $A\\cap B\\in \\phi(N)$, so $\\phi(N)$ is indeed a modular cut of $M$.\n\n\t\\par \\vspace*{.6\\baselineskip}\\noindent \\mbox{---\\hspace*{2.5mm}} \\PRFR{Jan 15th}\n\tNow, we show that $\\phi$ is injective. Let $N,N'\\in {\\mathcal{X}}(M,e)$ with $N\\not= N'$. \n\tWithout loss of generality we may assume that there is a set $X\\subseteq E\\cup\\SET{e}$ \n\twhich is independent in $N$, yet dependent in $N'$.\n\t Since $N$ coincides \n\twith $M$ on $2^{E}$, we obtain that $e\\in X$ and that $X\\BSET{e}\\in {\\mathcal{I}}$ is independent in $N$, $N'$ and $M$.\n\tLet $F=\\mathrm{cl}_{M}(X\\BSET{e})\\supseteq X\\BSET{e}$.\n\t Then $e\\in\\mathrm{cl}_{N'}(X\\BSET{e}) \\subseteq \\mathrm{cl}_{N'}(F)$ so $F\\in\\phi(N')$, but\n\t $e\\notin \\mathrm{cl}_{N}(F) = \\mathrm{cl}_{M}(F) = F$, so $F\\notin\\phi(N)$. Thus $\\phi(N)\\not= \\phi(N')$.\n\n \\par \\vspace*{.6\\baselineskip}\\noindent \\mbox{---\\hspace*{2.5mm}} \\PRFR{Jan 15th}\n It remains to show that $\\phi$ is surjective. Let $C$ be a\nmodular cut of $M$. We define $N=(E\\cup\\SET{e},{\\mathcal{I}}')$ such that \\[\n{\\mathcal{I}}' = {\\mathcal{I}} \\cup \\SET{X\\cup \\SET{e} \\mid X\\in{\\mathcal{I}},\\,\\mathrm{cl}_M(X)\\notin C}.\\]\nAssume for a moment that $N$ is a matroid, then $\\phi(N) = C$: Let $F\\in\n{\\mathcal{F}}(M)$ and let $X\\subseteq F$ be a base of $F$ in $M$. If $F\\in C$,\nthen $X\\cup\\SET{e}\\notin {\\mathcal{I}}'$, thus $e\\in \\mathrm{cl}_{N}(F)$, and therefore\n$F\\in\\phi(N)$. If $F\\notin C$, then $X\\cup\\SET{e}\\in {\\mathcal{I}}'$ and $e\\notin\n\\mathrm{cl}_{N}(F)$, therefore $F\\notin \\phi(N)$.\n\n\\par \\vspace*{.6\\baselineskip}\\noindent \\mbox{---\\hspace*{2.5mm}} \\PRFR{Jan 15th}\nWe show that $N$ is indeed a\nmatroid by explicating D.J.A.~Welsh's sketch on p.~319 \\cite{We76}: Observe that the map \\[ \\rho\\colon 2^{E\\cup\\SET{e}} \\longrightarrow \\mathbb{N},\\,X\\mapsto \\max\\SET{ \\left| Y \\right| ~\\middle|~ Y\\subseteq X,\\, Y\\in {\\mathcal{I}}'} \\]\nsatisfies the equation \n\\[\n\t\\rho(X) = \\begin{cases}\n\t\t\t\\mathrm{rk}_M(X) & \\quad \\text{if } e\\notin X,\\\\\n\t\t\t\\mathrm{rk}_M(X\\BSET{e}) + 1 & \\quad \\text{if } e\\in X,\\,\\mathrm{cl}_M(X\\BSET{e})\\notin C,\\\\\n\t\t\t\\mathrm{rk}_M(X\\BSET{e}) & \\quad \\text{if }e\\in X,\\,\\mathrm{cl}_M(X\\BSET{e})\\in C.\n\t\\end{cases} \n\\] Furthermore, we see that \\[\n{\\mathcal{I}}' = \\SET{X\\subseteq E\\cup\\SET{e}\\vphantom{A^A}~\\middle|~ \\forall Y\\subseteq X\\colon \\rho(Y) \\geq \\left| Y \\right|}, \\]\nwhich is the family of independent sets of a matroid obtained from $\\rho$ by Theorem~\\ref{thm:submodularIndependent}, whenever $\\rho$ is non-negatively integer-valued, non-decreasing, and submodular. Clearly, $\\rho$ is non-negatively integer-valued. Furthermore, $\\rho$ restricted to $2^E$ \nis $\\mathrm{rk}_M$, thus $\\rho$ is non-decreasing and submodular on $2^E$. Let $X,Y\\subseteq E\\cup\\SET{e}$ such that $e\\in X$ and $e\\in Y$.\n If $\\mathrm{cl}_M(X\\BSET{e})\\in C \\Leftrightarrow \\mathrm{cl}_M(Y\\BSET{e})\\in C$, then\n$\\rho(X) = \\mathrm{rk}_M(X\\BSET{e}) + \\alpha$ and $\\rho(Y) = \\mathrm{rk}_M(Y\\BSET{e}) + \\alpha$ for the same value of $\\alpha\\in \\SET{0,1}$, thus $\\rho$ is non-decreasing because $\\mathrm{rk}_M$ is non-decreasing. Otherwise, if $X\\subseteq Y$\n and $\\mathrm{cl}_M(X\\BSET{e})\\in C \\not\\Leftrightarrow \\mathrm{cl}_M(Y\\BSET{e})\\in C$, then $\\mathrm{cl}_M(X)\\notin C$ whereas $\\mathrm{cl}_M(Y)\\in C$, because $C$ is closed under super-flats and $\\mathrm{cl}_M$ preserves set-inclusion \n(Lemma~\\ref{lem:clFlips}). \n But then $\\mathrm{cl}_M(X\\BSET{e})\\not= \\mathrm{cl}_M(Y\\BSET{e})$, so $\\mathrm{rk}_M(\\mathrm{cl}_M(X)) < \\mathrm{rk}_M(\\mathrm{cl}_M(Y))$,\nthus $\\rho(X) = \\mathrm{rk}_M(X\\BSET{e}) + 1 \\leq \\mathrm{rk}_M(Y\\BSET{e}) = \\rho(Y)$.\nTherefore $\\rho$ is non-decreasing on its whole domain.\n Now let $A,B\\subseteq E\\cup\\SET{e}$, we have to show that the submodular inequality \n \\begin{align}\n \\rho(A) + \\rho(B) \\geq \\rho(A\\cap B) + \\rho(A\\cup B)\\label{eq:rhoSubmod}\n \\end{align}\n holds. Clearly $$\\rho(A) + \\rho(B) = \\mathrm{rk}_M(A\\BSET{e}) + \\mathrm{rk}_M(B\\BSET{e}) + \\alpha$$\nfor some $\\alpha\\in\\SET{0,1,2}$ and analogously $$\\rho(A\\cap B) + \\rho(A\\cup B) = \\mathrm{rk}_M((A\\cap B)\\BSET{e}) + \\mathrm{rk}_M((A\\cup B)\\BSET{e}) + \\beta$$ for some $\\beta\\in \\SET{0,1,2}$. \nSince $\\mathrm{rk}_M$ is a submodular function,\nwe may deduce inequality~(\\ref{eq:rhoSubmod}) from $\\alpha \\geq \\beta$\nas well as from $$\\beta - \\alpha \\leq \\mathrm{rk}_M(A\\BSET{e}) + \\mathrm{rk}_M(B\\BSET{e}) - \\mathrm{rk}_M((A\\cap B)\\BSET{e}) - \\mathrm{rk}_M((A\\cup B)\\BSET{e}).$$\n If $\\beta=2$, then $\\mathrm{cl}_M((A\\cup B )\\BSET{e})\\notin C$, therefore $\\mathrm{cl}_M(A\\BSET{e})\\notin C$ and $\\mathrm{cl}_M(B\\BSET{e})\\notin C$ because $C$ is closed under super-flats. So in this case, $\\alpha = 2 \\geq \\beta$. If $\\beta = 0$ then clearly $\\alpha \\geq \\beta$, too.\n So the submodular inequality (\\ref{eq:rhoSubmod}) holds due to $\\alpha \\geq \\beta$ unless\n $\\beta=1$ and $\\alpha = 0$.\n In this case, if $$\\mathrm{rk}_M(A\\BSET{e}) + \\mathrm{rk}_M(B\\BSET{e}) - \\mathrm{rk}_M((A\\cap B)\\BSET{e}) - \\mathrm{rk}_M((A\\cup B)\\BSET{e}) \\geq 1,$$ then \n(\\ref{eq:rhoSubmod}) follows as mentioned above.\nWe close our argumentation by showing that for $\\beta = 1$ and $\\alpha = 0$,\n $$\\mathrm{rk}_M(A\\BSET{e}) + \\mathrm{rk}_M(B\\BSET{e}) - \\mathrm{rk}_M((A\\cap B)\\BSET{e}) - \\mathrm{rk}_M((A\\cup B)\\BSET{e}) = 0$$ can never be the case.\n There are two ways that lead to $\\beta =1$. Assume that $e\\notin A\\cap B$, then $\\mathrm{cl}_M((A\\cup B)\\BSET{e})\\notin C$.\n If $e\\in A$, then $\\mathrm{cl}_M(A\\BSET{e})\\notin C$ follows, thus $\\alpha \\geq 1$; similarly if $e\\in B$. Thus $e\\in A\\cap B$\n is implied by $\\beta = 1$ and $\\alpha = 0$. Consequently, $\\mathrm{cl}_M((A\\cap B) \\BSET{e})\\notin C$ is necessary for $\\beta = 1$.\n Furthermore, for $\\alpha = 0$ it is necessary that $\\mathrm{cl}_M(A\\BSET{e})\\in C$\n and $\\mathrm{cl}_M(B\\BSET{e})\\in C$. But then $\\mathrm{cl}_M(A\\BSET{e})$ and $\\mathrm{cl}_M(B\\BSET{e})$ cannot be a modular pair in $M$,\n because $C$ is closed under intersections of modular pairs yet $C$ does not contain the intersection of these two flats.\n This yields $$\\mathrm{rk}_M(A\\BSET{e}) + \\mathrm{rk}_M(B\\BSET{e}) - \\mathrm{rk}_M((A\\cap B)\\BSET{e}) - \\mathrm{rk}_M((A\\cup B)\\BSET{e})\\not= 0 .$$\nSo all premises for Theorem~\\ref{thm:submodularIndependent} are satisfied, $N$ is a matroid and therefore $\\phi$ is surjective.\n\\end{proof}\n\n\\noindent The next lemma summarizes how the family of flats behaves when a matroid is extended. \n\n\n\\begin{figure}[htb]\n\\begin{center}\n\n\t\\includegraphics[width=\\textwidth]{flatsOfSingleElementExtension}\n\\end{center}\n\\caption{Construction of the lattice of flats of a single element extension from the lattice of flats of the original matroid and the corresponding modular cut (Lemma~\\ref{lem:flatsOfExtension}).} \n\\end{figure}\n\n\\begin{lemma}\\label{lem:flatsOfExtension}\\PRFR{Feb 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, $e\\notin E$, and $N\\in {\\mathcal{X}}(M,e)$.\n\tFurthermore, let $C\\in{\\mathcal{M}}(M)$ be the modular cut of $M$ where\n\t\\[ C = \\SET{F\\in {\\mathcal{F}}(M) ~\\middle|~ e\\in \\mathrm{cl}_N(F)}. \\]\n\tThen \n\t\\begin{align*}\n\t {\\mathcal{F}}(N)\\,\\, =\\,\\, &\\left( {\\mathcal{F}}(M) \\backslash C \\right) \\,\\, \\cup \\,\\, \\SET{F\\cup\\SET{e} ~\\middle|~ F\\in C} \n\t \t\t\\,\\,\n\t \t\t\\\\ & \\cup \\,\\, \\SET{F\\cup\\SET{e} ~\\middle|~ F\\in{\\mathcal{F}}(M)\\backslash C,\\, \\forall x\\in E\\backslash F\\colon\\,\\mathrm{cl}_M(F\\cup\\SET{x})\\notin C}\n\n\t\t\\\\ = \\,\\, & \\left( {\\mathcal{F}}(M) \\backslash C \\right) \\,\\, \\cup \\,\\, \\SET{F\\cup\\SET{e} ~\\middle|~ F\\in{\\mathcal{F}}(M),\\,F\\in C \\Leftrightarrow \n\t\t\\mathrm{cl}_N(F\\cup\\SET{e})\\BSET{e} \\in C} \n\t.\\end{align*}\n\\end{lemma}\n\n\n\\needspace{3\\baselineskip}\n\\begin{proof}\\PRFR{Feb 15th}\n\tFirst, we show that the second equation holds, which is implied by the equation\n\t\\begin{align*} \\SET{F\\cup\\SET{e} ~\\middle|~ F\\in C} \n\t \t\t\\,\\,\n\t \t\t& \\cup \\,\\, \\SET{F\\cup\\SET{e} ~\\middle|~ F\\in{\\mathcal{F}}(M)\\backslash C,\\, \\forall x\\in E\\backslash F\\colon\\,\\mathrm{cl}_M(F\\cup\\SET{x})\\notin C}\n\t \t\t\\\\&\n\t \t\\!= \\, \\SET{F\\cup\\SET{e} ~\\middle|~ F\\in{\\mathcal{F}}(M),\\,F\\in C \\Leftrightarrow \n\t\t\\mathrm{cl}_N(F\\cup\\SET{e})\\BSET{e} \\in C} . \n\t\\end{align*}\n\tIf $F\\in C$, then $F\\in{\\mathcal{F}}(M)$ and $e\\in \\mathrm{cl}_N(F)$, so $\\mathrm{cl}_N(F\\cup\\SET{e}) = F\\cup\\SET{e}$ \n\tand therefore $\\mathrm{cl}_N(F\\cup\\SET{e})\\BSET{e} = F \\in C$.\n\tIf $F\\notin C$ and for all $x\\in E\\backslash F$ we have $\\mathrm{cl}_M(F\\cup\\SET{x})\\notin C$, i.e. whenever $F\\notin C$ is not covered by a flat $G\\in C$ in\n\t ${\\mathcal{F}}(M)$, then $\\mathrm{cl}_N(F\\cup\\SET{e}) = F\\cup\\SET{e}$, since otherwise $G = \\mathrm{cl}_N(F\\cup\\SET{e})\\BSET{e}$ would be\n\t a maximal subset of $E$ with rank $\\mathrm{rk}_N(G) = \\mathrm{rk}_N(F) + 1 = \\mathrm{rk}_M(F) + 1$, and therefore we would have $G\\in{\\mathcal{F}}(M)$.\n\t Furthermore, there would have to be some element\n\t $g\\in G\\backslash F$, so we would have found a flat $G\\in C$ and with $\\mathrm{cl}_M(F\\cup\\SET{g}) = G$, contradicting the assumption.\n\t Thus we have $\\mathrm{cl}_N(F\\cup\\SET{e})\\BSET{e} = \\left( F\\cup\\SET{e} \\right)\\BSET{e} = F \\notin C$.\n\t Therefore we obtain\n\t \\begin{align*} \\SET{F\\cup\\SET{e} ~\\middle|~ F\\in C} \n\t \t\t\\,\\,\n\t \t\t& \\cup \\,\\, \\SET{F\\cup\\SET{e} ~\\middle|~ F\\in{\\mathcal{F}}(M)\\backslash C,\\, \\forall x\\in E\\backslash F\\colon\\,\\mathrm{cl}_M(F\\cup\\SET{x})\\notin C}\n\t \t\t\\\\&\n\t \t\\!\\subseteq \\, \\SET{F\\cup\\SET{e} ~\\middle|~ F\\in{\\mathcal{F}}(M),\\,F\\in C \\Leftrightarrow \n\t\t\\mathrm{cl}_N(F\\cup\\SET{e})\\BSET{e} \\in C} . \n\t\\end{align*}\n\tNow let $F'\\in \\SET{F\\cup\\SET{e} ~\\middle|~ F\\in{\\mathcal{F}}(M),\\,F\\in C \\Leftrightarrow \n\t\t\\mathrm{cl}_N(F\\cup\\SET{e})\\BSET{e} \\in C}$ and $F= F'\\BSET{e}$.\n\tIf $F\\in C$, then clearly $F'\\in \\SET{F\\cup\\SET{e} ~\\middle|~ F\\in C}$.\n\tIf $F\\notin C$, we give an indirect argument and assume that\n\t$$F'\\notin\\SET{F\\cup\\SET{e} ~\\middle|~ F\\in{\\mathcal{F}}(M)\\backslash C,\\, \\forall x\\in E\\backslash F\\colon\\,\\mathrm{cl}_M(F\\cup\\SET{x})\\notin C}.$$\n\tSo there is some $x\\in E\\backslash F$ such that $\\mathrm{cl}_M(F\\cup\\SET{x}) \\in C$. Let $G = \\mathrm{cl}_M(F\\cup\\SET{x})$,\n\tthen $\\mathrm{cl}_N(G) = G\\cup\\SET{e}$, thus $\\mathrm{rk}_N(G\\cup\\SET{e}) = \\mathrm{rk}_N(G) = \\mathrm{rk}_M(G) = \\mathrm{rk}_M(F) + 1 = \\mathrm{rk}_N(F) + 1$.\n\tThus $\\mathrm{cl}_N(F\\cup\\SET{e}) = \\mathrm{cl}_N(G) = G\\cup\\SET{e}$, so $\\mathrm{cl}_N(F\\cup\\SET{e})\\BSET{e} = G \\in C$, contradicting the assumption\n\tthat $F'$ is a member of the right-hand side set. Consequently, the second equation of the lemma holds.\n\n\t\\noindent\n\tWe show the inclusion of the left-hand side of the first equation in the right-hand side of the first equation.\n\tLet $X\\in{\\mathcal{F}}(N)$, we have to treat two cases. If $e\\notin X$, then the defining property of $X\\in{\\mathcal{F}}(N)$ is that\n\tthe strict inequality\n\t $\\mathrm{rk}_N(X\\cup\\SET{y}) > \\mathrm{rk}_N(X)$ holds for all $y\\in \\left( E\\cup\\SET{e} \\right)\\backslash X$. \n\t This together with $\\mathrm{rk}_M = \\mathrm{rk}_N|_{2^E}$\n\t implies that $X\\in{\\mathcal{F}}(M)$. Furthermore, $X = \\mathrm{cl}_N(X)$ implies $e\\notin X$ and therefore $X\\notin C$, so $X\\in{\\mathcal{F}}(M)\\backslash C$.\n\tNow assume that $e\\in X$, and let $X' = X\\BSET{e}$. If $e\\in \\mathrm{cl}_N(X')$, then clearly $X'\\in C$ and\n\tso \\linebreak\n\t $X\\in \\SET{F\\cup\\SET{e} ~\\middle|~ F\\in C}$.\n\t If otherwise $e\\notin \\mathrm{cl}_N(X')$, then we must have $\\mathrm{rk}_N(X') = \\mathrm{rk}_N(X) - 1$,\n\t and $X'\\in {\\mathcal{F}}(N)$ because $\\mathrm{cl}_N(X') \\subsetneq \\mathrm{cl}_N(X) = X = X'\\cup\\SET{e}$.\n\t As a consequence, we obtain that $X'\\in {\\mathcal{F}}(M)$ and $X'\\notin C$. Assume that for some $y\\in E\\backslash X'$,\n\t $\\mathrm{cl}_M(X'\\cup\\SET{y})\\in C$, then $e\\in \\mathrm{cl}_N(X'\\cup\\SET{y})$ so $X = \\mathrm{cl}_N(X')\\cup\\SET{e}$ would be a proper subset of the flat\n\t $\\mathrm{cl}_N(X'\\cup\\SET{y})$ of rank $\\mathrm{rk}_N(X') + 1$, but $\\mathrm{rk}_N(X) = \\mathrm{rk}_N(X') + 1$, which is impossible. Therefore, \n\t for all $y\\in E\\backslash X'$ we have $\\mathrm{cl}_M(X'\\cup\\SET{y})\\notin C$. Thus we obtain\n\t $$ X\\in \\SET{F\\cup\\SET{e} ~\\middle|~ F\\in{\\mathcal{F}}(M)\\backslash C,\\, \\forall x\\in E\\backslash F\\colon\\,\\mathrm{cl}_M(F\\cup\\SET{x})\\notin C}.$$\n\n\n\t \\noindent\n\t Finally, we show the inclusion of the right-hand side of the first equation in the left-hand side of the first equation.\n\t Let $X\\in {\\mathcal{F}}(M) \\backslash C$, then $e\\notin \\mathrm{cl}_N(X)$, so $\\mathrm{cl}_N(X) = \\mathrm{cl}_M(X) = X$, thus $X\\in{\\mathcal{F}}(N)$.\n\t Let $X\\in C$, and let $X' = X\\cup\\SET{e}$. Then $\\mathrm{cl}_N(X) =$\\linebreak$ \\mathrm{cl}_M(X) \\cup\\SET{e} = X\\cup\\SET{e}$, \n\t and therefore $X\\cup\\SET{e}\\in{\\mathcal{F}}(N)$. Now let $X\\notin C$ and for all $y\\in E\\backslash X$,\n\t $\\mathrm{cl}_M(X\\cup\\SET{y})\\notin C$. Let $G = \\mathrm{cl}_N(X\\cup\\SET{e})$.\n\t Assume that we have the proper inclusion $X\\cup\\SET{e} \\subsetneq G$,\n\t then $\\mathrm{rk}_N(G) = \\mathrm{rk}_N(X) + 1$ yields that there is some \\linebreak $g\\in G\\backslash\\left( X\\cup\\SET{e} \\right)$ such that\n\t $\\mathrm{cl}_N(X\\cup\\SET{g}) = G$. This leads us to the contradiction $G\\BSET{e} = \\mathrm{cl}_M(X\\cup\\SET{g}) \\in C$. \n\t Therefore we must have $X\\cup\\SET{e} = G\\in{\\mathcal{F}}(N)$.\n\\end{proof}\n\n\n\\section{Theorems of Hall, Rado, Ore, and Perfect}\n\n\\PRFR{Jan 15th}\nD.J.A.~Welsh gives the following very elegant generalization of the \ntheorems of Rado and Hall in \\cite{We71}. From this generalization, the theorems of Hall, Rado, Ore, and Perfect follow as an easy corollary each. Before we present the theorem, we need some definitions.\n\n\\begin{definition}\\PRFR{Jan 15th}\n\tLet $I$ and $E$ be sets. A \\deftext{family of subsets} of $E$ indexed by $I$ is\n\ta map $A_{\\bullet}\\colon I\\longrightarrow 2^{E}$ with domain $I$, such that for every\n\t$i\\in I$ the image $A_{i}$ is a subset of $E$.\n\tWe denote such a family by writing $(A_i)_{i\\in I} \\subseteq E$, or shorter\\label{n:Afam}\n\t$(A_i)_{i\\in I}$ whenever $E$ is clear from the context. We call $(A_i)_{i\\in I}$\n\t\\deftext{finite} if $I$ is finite. Further, we call $(A_i)_{i\\in I}$ a \\deftext{family of non-empty subsets}, if for all $i\\in I$, $A_{i} \\not= \\emptyset$.\n\\end{definition}\n\n\\begin{definition}\\PRFR{Jan 15th}\n\tLet $I$, $E$ be sets, and let ${\\mathcal{A}} = (A_i)_{i\\in I} \\subseteq E$ be a family of subsets of $E$. A \\deftext{system of representatives} is a map $x_{\\bullet}\\colon I\\longrightarrow E$ such that there is a bijection $\\sigma \\colon I\\longrightarrow I$ with \n\t$x_{i} \\in A_{\\sigma(i)}$ for all $i\\in I$. We will denote such a family\n\tby writing \n\t\\linebreak\n\t$(x_i)_{i\\in I} \\in {\\mathcal{A}}$. A system of representatives is called\n\t\\deftext{system of distinct representatives}, if $x_{\\bullet}$ is an injective map.\n\tA \\deftext{transversal} of ${\\mathcal{A}}$ is a subset $T\\subseteq E$ such that \n\tthere is a bijection $\\sigma\\colon T\\longrightarrow I$ with $t\\in A_{\\sigma(t)}$ for all $t\\in T$. A \\deftext{partial transversal} of ${\\mathcal{A}}$ is a subset $P\\subseteq E$ such that\n\tthere is an injection $\\iota \\colon P\\longrightarrow I$ with $t\\in A_{{\\iota(t)}}$ for all $t\\in P$.\n\tIf $P$ is a partial transversal of ${\\mathcal{A}}$, we define the \\deftext[defect of a partial transversal]{defect} of $P$ to be $\\left| I \\right| - \\left| P \\right|$,\n\ti.e. the cardinality of those indices in $I$ that are not in the image of the corresponding $\\iota$.\n\\end{definition}\n\n\\begin{theorem}\\label{thm:radohall}\\PRFR{Jan 15th}\n Let ${\\mathcal{A}}=(A_i)_{i\\in I} \\subseteq E$ be a finite family of non-empty subsets of $E$,\n and let $\\mu\\colon 2^{E} \\longrightarrow \\mathbb{N}$ be a map with the properties that\n \\begin{enumerate}\\renewcommand{\\theenumi}{(\\roman{enumi})}\\renewcommand{\\labelenumi}{\\theenumi}\n \t\\item for all $X \\subseteq Y\\subseteq E$, $\\mu(X) \\leq \\mu(Y)$, and\n \t\\item for all $X,Y\\subseteq E$, $\\mu(X) + \\mu(Y) \\geq \\mu(X\\cap Y) + \\mu(X\\cup Y)$.\n \\end{enumerate}\n Then there is a system of representatives $(x_i)_{i\\in I} \\in {\\mathcal{A}}$ with the property that\n \\begin{enumerate}\n \t\\item[(1)] for all $J\\subseteq I$, $\\mu\\left(\\SET{x_i \\mid i\\in J}\\right) \\geq \\left| J\\right|$\n\\end{enumerate}\nif and only if ${\\mathcal{A}}$ has the property that\n\\begin{enumerate}\n\t\\item[(2)] for all $J\\subseteq I$, $\\mu \\left(\\bigcup_{i\\in J} A_i\\right) \\geq \\left| J\\right|$.\n\\end{enumerate}\n\\end{theorem}\n\n\\noindent This proof of the theorem follows the course of \\cite{We71} --- a very nice version of which can be found on p.100 of \\cite{We76} --- and it focuses more on details than brevity.\n\n\\begin{proof}\\PRFR{Jan 15th}\n\tLet $(x_{i})_{i\\in I} \\in {\\mathcal{A}}$ be such a system of representatives, that\n\t{\\em(1)} holds, and let $\\sigma\\colon I\\longrightarrow I$ be a permutation\n\tthat has the property $x_{i}\\in A_{\\sigma(i)}$ for all $i\\in I$. Let $J\\subseteq I$, then\n\t$\\SET{x_{\\sigma^{-1}(i)} ~\\middle|~ i \\in J} \\subseteq \\bigcup_{{i\\in J}} A_{i}$.\n\tBy {\\em (i)} $\\mu$ is non-decreasing, therefore\n\t\\[ \\left| J \\right| = \\left| \\sigma^{{-1}}[J] \\right| \\leq \n\t\\mu\\left(\\SET{x_i ~\\middle|~ i\\in \\sigma^{-1}[J]}\\right) \\leq \\mu \\left(\\bigcup_{i\\in J} A_i\\right).\\]\n\tFor the converse implication, we employ induction on the integer vector $v = \\left( \\left|A_{i}\\right| \\right)_{i\\in I}$. The base case is $v_{i} = 1$ for all $i\\in I$ where every $A_{i}$ is a singleton set, thus for any system of representatives $(x_i)_{i\\in I} \\in {\\mathcal{A}}$, we have $A_{i} = \\SET{x_{\\sigma^{-1}(i)}}$ for all $i\\in I$.\n\tTherefore, $\\SET{x_i ~\\middle|~ i\\in \\sigma^{-1}[J]} = \\bigcup_{i\\in J} A_{i}$ and the equivalence is obvious. For the induction step, let $i'\\in I$ such that $\\left| A_{i'} \\right| > 1$. In this case, we claim that there is some $x\\in A_{i'}$, such that\n\tthe derived family\n\t${\\mathcal{A}}' = (A'_i)_{i\\in I}$ where $A'_{i} = A_{i}$ if $i\\not= i'$, and $A'_{i'} = A_{i'}\\BSET{x}$\n\tstill has the property {\\em (2)}. Assume that this claim is false, then for any\n\t$\\dSET{x,y}\\subseteq A_{i'}$ there are $J_{x},J_{y}\\subseteq I\\BSET{i'}$ such that\n\t\\begin{align*}\n\t\t\\mu\\left( \\left(A_{i'}\\BSET{x}\\right) \\cup \\bigcup_{i\\in J_x} A_i \\right) &\n\t\t\\leq \\left|J_x \\right| < \\left|J_x\\right| + 1 \\text{, and}\\\\\n\t\t\\mu\\left( \\left(A_{i'}\\BSET{y}\\right) \\cup \\bigcup_{i\\in J_y} A_i \\right) &\n\t\t\\leq \\left|J_y \\right| < \\left|J_y\\right| + 1 .\\\\\n\t\\end{align*}\n\tWe use the submodularity {\\em (ii)} of $\\mu$ in order to obtain that\n\t\\begin{align*}\n\t\t\\mu\\left( \\left(A_{i'}\\BSET{x}\\right) \\cup \\bigcup_{i\\in J_x} A_i \\right) + \\mu\\left( \\left(A_{i'}\\BSET{y}\\right) \\cup \\bigcup_{i\\in J_y} A_i \\right) & \\geq \\mu(B_{\\cap}) + \n\t\t\\mu\\left( A_{i'} \\cup \\bigcup_{i\\in J_x\\cup J_y} A_i \\right)\n\t\\end{align*}\n\twhere \\[ B_{\\cap} = \\left( \\left(A_{i'}\\BSET{x}\\right) \\cup \\bigcup_{i\\in J_x} A_i \\right) \\cap \n\t\\left( \\left(A_{i'}\\BSET{y}\\right) \\cup \\bigcup_{i\\in J_y} A_i \\right).\\]\n\tClearly, $\\bigcup_{i\\in J_x \\cap J_y} A_{i} \\subseteq B_{\\cap}$, and since $\\mu$ is non-decreasing due to property {\\em (i)},\n\twe obtain that\n\t\\begin{align*}\n\t\\mu(B_{\\cap}) + \n\t\t\\mu\\left( A_{i'} \\cup \\bigcup_{i\\in J_x\\cup J_y} A_i \\right) & \\geq\n\t\t\\mu\\left(\\bigcup_{i\\in J_x\\cap J_y} A_i\\right) + \\mu\\left( A_{i'} \\cup \\bigcup_{i\\in J_x\\cup J_y} A_i \\right).\n\t\\end{align*}\n\tWe now may use property {\\em (2)} with $J = J_{x}\\cup J_{y} \\cup \\SET{i'}$, and $J=J_{x}\\cap J_{y}$, respectively. We add the respective inequalities and obtain\n\t\\begin{align*}\n\t\\mu\\left( A_{i'} \\cup \\bigcup_{i\\in J_x\\cup J_y} A_i \\right) + \\mu\\left( \\bigcup_{i\\in J_x\\cap J_y} A_i \\right) & \\geq \\left( \\left| J_x \\cup J_y \\right| + 1\\right) +\n\t\\left| J_x \\cap J_y \\right| = \\left| J_x \\right| + \\left| J_y \\right| + 1.\n\t\\end{align*}\n\tYet, this yields\n\t\\begin{align*}\n\t\\mu\\left( \\left(A_{i'}\\BSET{x}\\right) \\cup \\bigcup_{i\\in J_x} A_i \\right) + \\mu\\left( \\left(A_{i'}\\BSET{y}\\right) \\cup \\bigcup_{i\\in J_y} A_i \\right) & \\geq \\left| J_x \\right| + \\left|J_y\\right| + 1\n\t\\end{align*}\n\twhich contradicts\n\t\\begin{align*}\n\t\\mu\\left( \\left(A_{i'}\\BSET{x}\\right) \\cup \\bigcup_{i\\in J_x} A_i \\right) + \\mu\\left( \\left(A_{i'}\\BSET{y}\\right) \\cup \\bigcup_{i\\in J_y} A_i \\right) & \\leq \\left| J_x \\right| + \\left|J_y\\right|.\n\t\\end{align*}\n\tThus the claim holds, and since $\\left|A'_{i'}\\right| < v_{i'}$, we may use the induction hypothesis on ${\\mathcal{A}}'$ which guarantuees the existence of a system of representatives $(x_i)_{i\\in I}$ with\n\tproperty {\\em (1)}. Every such $(x_i)_{i\\in I}$ is also a system of representatives of\n\t${\\mathcal{A}}$, therefore $(x_i)_{i\\in I}$ with {\\em (1)} exists.\n\\end{proof}\n\n\\begin{corollary}[Hall]\\label{cor:Hall}\\PRFR{Jan 15th}\n Let ${\\mathcal{A}} = (A_i)_{i\\in I}$ be a finite family of sets, then ${\\mathcal{A}}$ has a transversal if and only if for all $J\\subseteq I$,\n$$\\left| \\bigcup_{{i\\in J}} A_{i} \\right| \\geq \\left| J \\right|.$$\n\\end{corollary}\n\n\\begin{proof} \\PRFR{Jan 15th}\n\tApply Theorem~\\ref{thm:radohall} with $\\mu(X) = \\left| X \\right|$ and $E = \\bigcup_{{i\\in I}} A_{i}$.\n\\end{proof}\n\n\\begin{corollary}[Rado]\\label{cor:Rado}\\PRFR{Jan 15th}\n Let $M=(E,{\\mathcal{I}})$ be a matroid, and let ${\\mathcal{A}} = (A_i)_{i\\in I}$ be a finite family of subsets of $E$, then ${\\mathcal{A}}$ has a transversal which is independent in $M$ if and only if for all $J\\subseteq I$,\n$$\\mathrm{rk}_M \\left( \\bigcup_{{i\\in J}} A_{i}\\right) \\geq \\left| J \\right|.$$\n\\end{corollary}\n\n\\begin{proof}\\PRFR{Jan 15th}\n\tApply Theorem~\\ref{thm:radohall} with $\\mu(X) = \\mathrm{rk}_M(X)$.\n\\end{proof}\n\n\\begin{corollary}[Ore]\\PRFR{Jan 15th} Let ${\\mathcal{A}} = (A_i)_{i\\in I}$ be a finite family of sets, and $d\\in \\mathbb{N}$, then ${\\mathcal{A}}$ has a partial transversal $T$ with defect $\\leq d$ if and only if for all $J\\subseteq I$,\n$$\\left| \\bigcup_{{i\\in J}} A_{i} \\right| \\geq \\left| J \\right| - d.$$\n\\end{corollary}\n\\begin{proof}\\PRFR{Jan 15th}\n\tApply Theorem~\\ref{thm:radohall} with $\\mu(X) = \\left| X \\right| + d$ and $E = \\bigcup_{{i\\in I}} A_{i}$.\n\\end{proof}\n\n\\needspace{9\\baselineskip}\n\\begin{corollary}[Perfect]\\PRFR{Jan 15th} Let $M=(E,{\\mathcal{I}})$ be a matroid, $d\\in \\mathbb{N}$, and let ${\\mathcal{A}} = (A_i)_{i\\in I}$ be a finite family of subsets of $E$, then ${\\mathcal{A}}$ has a partial transversal $T$ with defect $ \\leq d$ which is independent in $M$ if and only if for all $J\\subseteq I$,\n$$\\mathrm{rk}_M \\left( \\bigcup_{{i\\in J}} A_{i}\\right) \\geq \\left| J \\right| - d.$$\n\\end{corollary}\n\\begin{proof}\\PRFR{Jan 15th}\n\tApply Theorem~\\ref{thm:radohall} with $\\mu(X) = \\mathrm{rk}_M(X) + d$.\n\\end{proof}\n\n\n\\subsection{Matroids Induced by Bipartite Graphs}\n\n\\PRFR{Jan 15th}\nIn this section, we describe how matchings in bipartite graphs can be used to induce\na matroid on one color-class from a matroid given on the other color-class. The class of\ntransversal matroids consists of those matroids, which can be obtained from a free matroid by bipartite matroid induction.\n\n\n\\begin{definition}\\PRFR{Jan 15th}\n Let $D = (V,A)$ be a digraph, and let $M\\subseteq \\binom{V}{2}$ be a set of unordered pairs of vertices of $D$. We call $M$ a \\deftext[matching in D@matching in $D$]{matching in $\\bm D$},\n if the sets in $M$ are pair-wise disjoint, and if for every $\\dSET{x,y} \\in M$, there is an arc\n $(x,y)\\in A$ or $(y,x)\\in A$.\n\\end{definition}\n\n\\begin{definition}\\PRFR{Jan 15th}\n\tLet $A,B$ be sets with $A\\cap B = \\emptyset$ and $\\Delta\\subseteq A\\times B$.\n\tWe call the digraph \\label{n:DABD}$D=(A\\mathbin{\\dot{\\cup}} B, \\Delta)$ the \\index{bipartite graph}\\deftext{directed bipartite graph} for\n\t$\\Delta$ from $A$ to $B$.\n\\end{definition}\n\n\\PRFR{Jan 15th}\n\\noindent If there are no isolated vertices in the directed bipartite graph\n $(A\\mathbin{\\dot{\\cup}} B,\\Delta)$, then the partition of its vertices into $A$ and $B$ \n can be deduced from $\\Delta$. Thus it is reasonable to identify the directed bipartite\n graph $(A\\mathbin{\\dot{\\cup}} B,\\Delta)$ with its arcs $\\Delta$.\n\n \\begin{definition}\\PRFR{Jan 15th}\\label{def:arcSystemDelta}\n \tLet $A,B$ be finite sets with $A\\cap B = \\emptyset$,\n \t and let $\\Delta\\subseteq A\\times B$. The\n \t\\deftext[arc system of ABD@arc system of $(A\\mathbin{\\dot{\\cup}} B,\\Delta)$]{arc system of $\\bm{(A\\mathbin{\\dot{\\cup}} B,\\Delta)}$} shall be \n \tdenoted by ${\\mathcal{A}}_{\\Delta}$. It is defined to be the family\\label{n:ArcSystem} ${\\mathcal{A}}_{\\Delta} = (A_i)_{i\\in B} \\subseteq A$ where\n \t\\[\n \t\tA_b = \\SET{a\\in A \\mid (a,b) \\in \\Delta}\n \t\\]\n \tfor every $b\\in B$.\n \\end{definition}\n\n\n \\begin{theorem}\\label{thm:bipartiteInduction}\\PRFR{Jan 15th}\n \tLet $D,E$ be finite sets with $D\\cap E = \\emptyset$, $M=(E,{\\mathcal{I}})$ be a matroid,\n \tand let $\\Delta\\subseteq D\\times E$. Furthermore, let $N=(D,{\\mathcal{I}}')$ be \n \tsuch that ${\\mathcal{I}}'\\subseteq 2^{D}$ with the defining property\n \tthat for all $X\\subseteq D$, $X\\in {\\mathcal{I}}'$ if and only if $X$ is a partial transversal of the arc system\n \t${\\mathcal{A}}_\\Delta$ such that there is an injective map $\\iota\\colon X\\longrightarrow E$ with\n \t$x\\in A_{\\iota(x)}$ for all $x\\in X$ with the additional property that\n \t$\\SET{\\iota(x)\\mid x\\in X}$ is independent in $M$.\n \tThen $N$ is a matroid.\n \\end{theorem}\n\n \\noindent The following proof is based on the proof in \\cite{We76}, p.119.\n \\begin{proof}\\PRFR{Jan 15th}\n \tLet $\\mu\\colon 2^{D}\\longrightarrow \\mathbb{N}$ be the map where \\[ \\mu(X) =\n \t\\mathrm{rk}_{M}\\left( \\SET{e\\in E\\mid \\exists x\\in X\\colon\\, x\\in A_e} \\right) \\]\n \tfor every $X\\subseteq D$. Clearly, $\\mu(\\emptyset) =\\mathrm{rk}_M (\\emptyset) =\n \t0$ and $\\mu$ is non-decreasing and submodular. By\n \tTheorem~\\ref{thm:submodularIndependent}, the set \\[ {\\mathcal{I}}'' =\n \t\\SET{X\\subseteq D \\mid \\forall X'\\subseteq X\\colon \\mu(X') \\geq \\left| X'\n \t\\right|}\\] defines a matroid $N' = (D,{\\mathcal{I}}'')$. We show that ${\\mathcal{I}}' =\n \t{\\mathcal{I}}''$. Let $X\\in{\\mathcal{I}}'$, and let $\\iota\\colon X\\longrightarrow E$ an\n \tinjective map such that $\\iota[X]\\in {\\mathcal{I}}$ and for all $x\\in X$, $x\\in\n \tA_{\\iota(x)}$. Then, clearly, for all $X'\\subseteq X$, $\\iota[X']\n \t\\subseteq \\SET{e\\in E\\mid \\exists x'\\in X'\\colon\\, x'\\in A_e}$.\n \tTherefore for all $X'\\subseteq X$, $$ \\left| X' \\right| =\n \t\\mathrm{rk}_{M}\\left(\\iota[X']\\right) \\leq \\mathrm{rk}_{M}\\left( \\SET{e\\in E\\mid \\exists\n \tx'\\in X'\\colon\\, x'\\in A_e}\\right) = \\mu(X'), $$ thus $X\\in {\\mathcal{I}}''$.\n \tConversely, assume that $X\\in{\\mathcal{I}}''$. We flip around the arc system and and consider the\n \tfollowing family of subsets of $E$: Let ${\\mathcal{B}}_{X} = (B_i)_{{i\\in X}}\n \t\\subseteq E$ be the family of subsets of $E$ where for $x\\in X$, the\n \tsubset \\( B_{x} = \\SET{e\\in E\\mid x\\in A_e} \\) consists of all elements\n \tof $E$ that $x$ can pair with in $\\Delta$. By Rado's Theorem\n \t(Corollary~\\ref{cor:Rado}), the family ${\\mathcal{B}}_{X}$ has a transversal\n \t$Y\\subseteq E$ that is independent in $M$, if and only if for all\n \t$X'\\subseteq X$ we have the inequality \\[ \\mathrm{rk}_M \\left( \\bigcup_{{x'\\in X'}} B_{x'}\\right) \\geq\n \t\\left| X' \\right|. \\] But \\( \\bigcup_{x'\\in X'} B_{x'} = \\bigcup_{x' \\in\n \tX'} \\SET{e\\in E\\mid x'\\in A_e} = \\SET{e\\in E\\mid \\exists x'\\in\n \tX'\\colon\\, x'\\in A_e} \\), which gives that $\\mathrm{rk}_M \\left( \\bigcup_{{x'\\in\n \tX'}} B_{x'}\\right) = \\mu(X')$. By definition, $X\\in {\\mathcal{I}}''$ implies that\n \tfor all $X'\\subseteq X$, $\\mu(X') \\geq \\left| X' \\right|$. Thus we may\n \tinfer that there is an $M$-independent transversal $Y$ of ${\\mathcal{B}}_{X}$.\n \tThis gives rise to a bijective map $\\sigma\\colon Y\\longrightarrow X$ such that for every\n \t$y\\in Y$ we have $y\\in B_{\\sigma(y)}$. Yet $y\\in B_{\\sigma(y)}$ implies that\n \t$\\sigma(y) \\in A_{y}$. Therefore there is an injective map\n \t$\\tilde\\iota\\colon X\\longrightarrow E$ with $\\tilde\\iota(x) =\n \t\\sigma^{-1}(x)$ which witnesses that $X$ is a partial transversal of\n \t${\\mathcal{A}}_{\\Delta}$, and therefore $X\\in {\\mathcal{I}}'$.\n \\end{proof}\n\n \\begin{remark}\\PRFR{Jan 15th}\n Obviously, the premise $D\\cap E = \\emptyset$ in Theorem~\\ref{thm:bipartiteInduction} may be dropped,\n since we may give the elements of $D$ distinct names $D'$, apply the construction, and then rename the elements of the so obtained matroid back to $D$.\n \\end{remark}\n\n \\begin{definition}\\PRFR{Jan 15th}\n Let $D,E$ be finite sets, $\\Delta\\subseteq D\\times E$, and let $M_0=(E,{\\mathcal{I}})$ be a matroid. The \\deftext[matroid induced by D from M@matroid induced by $\\Delta$ from $M$]{matroid induced by $\\bm\\Delta$ from $\\bm{M_{\\bm 0}}$} shall be the pair\\label{n:MDM0}\n $M(\\Delta,M_0) = (D,{\\mathcal{I}}_{\\Delta,M_0})$ where ${\\mathcal{I}}_{\\Delta,M_0} \\subseteq 2^{D}$\nconsists of all partial transversals $X$ of the family ${\\mathcal{A}}_{\\Delta}$ that can be admissibly injected into an independent set of $M_{0}$, i.e. there is an injective map $\\iota\\colon X\\longrightarrow E$ such that for all $x\\in X$, $x\\in A_{\\iota(x)}$ and $\\iota[X] \\in {\\mathcal{I}}$.\n \\end{definition}\n\\subsection{Transversal Matroids}\\label{sec:shortTransversalMatroids}\n\n\\PRFR{Jan 15th}\n\\noindent In this section, we provide the definition of transversal matroids and forestall\nthose important properties of transversal matroids, that we need in order to develop the theory of\nroutings in directed graphs --- which, in turn, is essential for defining gammoids. Further properties of transversal matroids are treated in Section~\\ref{sec:TransversalMatroids}.\n\n\\begin{definition}\\PRFR{Jan 15th}\n\tLet $E,I$ be a finite sets, and ${\\mathcal{A}} = (A_i)_{i\\in I} \\subseteq E$ be a family of subsets. The \\deftext[transversal matroid presented by A@transversal matroid presented by ${\\mathcal{A}}$]{transversal matroid presented by $\\bm{{\\mathcal{A}}}$} shall be the pair\\label{n:MAEIA}\n\t$M({\\mathcal{A}}) = (E,{\\mathcal{I}}_{\\mathcal{A}})$ where ${\\mathcal{I}}_{\\mathcal{A}} \\subseteq 2^{E}$ with\n\tthe property that for all $X\\subseteq E$, $X\\in {\\mathcal{I}}_{{\\mathcal{A}}}$ if and only if\n\t$X$ is a partial transversal of ${\\mathcal{A}}$.\n\\end{definition}\n\n\\begin{corollary}\\label{cor:transversalMatroid}\\PRFR{Jan 15th}\n\tLet $E,I$ be a finite sets, and ${\\mathcal{A}} = (A_i)_{i\\in I} \\subseteq E$ be a family of subsets.\n\tThen $M({\\mathcal{A}}) = (E,{\\mathcal{I}}_{\\mathcal{A}})$ is a matroid.\n\\end{corollary}\n\n\\begin{proof}\\PRFR{Jan 15th}\n\tLet $M_0 = (I,2^I)$ be the free matroid on $I$, and let\n\t$\\Delta = \\SET{(e,i)\\in E\\times I\\mid e\\in A_i}$.\n\tThen $M({\\mathcal{A}}) = M(\\Delta,M_0)$ is the matroid induced by $\\Delta$ from $M_{0}$,\n\twhich is a matroid by Theorem~\\ref{thm:bipartiteInduction}.\n\\end{proof}\n\n\\noindent\nWe just proved that the maximal partial transversals are bases of a matroid $M({\\mathcal{A}})$.\n\n\\begin{corollary}\\label{cor:maximalpartialtransversals}\\PRFR{Jan 15th}\n\tLet $E,I$ be finite sets, and let ${\\mathcal{A}} = (A_i)_{i\\in I} \\subseteq E$ be a family of subsets.\n\tTwo maximal partial transversals $S,T\\subseteq E$ of ${\\mathcal{A}}$ have\n\t$\\left| S \\right| = \\left| T \\right|$.\n\\end{corollary}\n\n\\begin{definition}\\PRFR{Jan 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid. We call $M$ \\deftext{transversal matroid}, if\n\tthere is a finite family of subsets ${\\mathcal{A}} = (A_i)_{i\\in I} \\subseteq E$, such that\n\t$M = M({\\mathcal{A}})$, i.e. $M$ is the transversal matroid presented by ${\\mathcal{A}}$.\n\\end{definition}\n\n\\section{Directed Graphs}\n\n\\PRFR{Jan 22nd}\nIn this section, we present the basic definitions and properties of directed\ngraphs used in the course of this work. We aim to be consistent regarding terminology with \nthe monograph {\\em Digraphs: Theory, Algorithms, and Applications} by J.~Bang-Jensen\nand G.~Gutin \\cite{BJG09}, although we may divert from it in technical details since we do not need \nthe full\ngenerality of \\cite{BJG09}. \n\n\\begin{definition}\\label{def:directedGraph}\\PRFR{Jan 22nd}\n A pair $D = (V,A)$\\label{n:DVA} is called \\deftext{directed graph}, or shorter\n \\deftext{digraph}, whenever $V$ is a finite set and $A\\subseteq V\\times V$.\n Every $v\\in V$ is called \\deftext{vertex} of $D$ and every $a = (u,v)\\in A$ is\n called \\deftext{arc} of $D$. Furthermore, $u$ is called the \\deftext{tail} of\n the arc $a$ and $v$ is called the \\deftext{head} of $a$. We also say that $a=(u,v)$ is an arc that \\deftextX{leaves} $\\bm u$ \n and \\deftextX{enters} $\\bm v$, or shorter that $a$ \\deftextX{goes from} $\\bm u$ \\deftextX{to} $\\bm v$.\n Furthermore, $u$ and $v$ are the \\deftext[end vertices of an arc]{end vertices} of $a$, and we say that $u$ and $v$ are \\deftext{incident} with $a$.\n Two vertices that are incident with the same arc are called \\deftext{adjacent}.\n An arc $a = (u,v)$\n with $u=v$ is called a \\deftext[loop (digraph)]{loop}.\n\\end{definition}\n\n\\input{Text\/Ex\/131_Digraph.tex}\n\n\\PRFR{Jan 22nd}\n\\noindent Clearly, we can construct another directed graph $D^{\\mathrm{opp}}$ from any\ndirected graph $D$ by swapping heads and tails of all arcs of $D$, thus\neffectively reorienting all arcs to their opposite direction.\n\n\\begin{definition} \\PRFR{Jan 22nd}\n Let $D = (V,A)$ be a digraph. The \\deftext{opposite digraph} is defined to\n be the unique directed graph \\( D^{\\mathrm{opp}} = (V, A^{\\mathrm{opp}}) \\)\\label{n:Dopp} with the property\n \\[ (u,v)\\in A^{\\mathrm{opp}} \\Leftrightarrow (v,u)\\in A.\\]\n\\end{definition}\n\n\\noindent It is easy to see that $(D^{\\mathrm{opp}})^{\\mathrm{opp}} = D$.\n\n\\input{Text\/Ex\/132_Digraph_Opp.tex}\n\n\\begin{definition} \\PRFR{Jan 22nd}\n Let $D = (V,A)$ be a digraph, $x\\in V$. We call $x$ a \\deftext{source} in\n $D$ if $x$ is never the head of an arc in $D$. Analogously, we call $x$ a\n \\deftext{sink} in $D$ if $x$ is never the tail of an arc in $D$.\n\\end{definition}\n\n\\noindent From this definition it is clear that $x$ is a source in $D$ if and\nonly if $x$ is a sink in $D^{\\mathrm{opp}}$, and analogously, $x$ is a sink in $D$\nif and only if $x$ is a source in $D^{\\mathrm{opp}}$.\n\n\\input{Text\/Ex\/133_Digraph_Source_Sink.tex}\n\n\\begin{definition}\\PRFR{Jan 22nd}\n Let $D = (V,A)$ be a digraph, the \\deftextX{outer-extension operator in $\\bm D$}\n shall be the map\\label{n:DclD}\n \\[ \\DclD{\\bullet}{D}\\colon 2^V \\longrightarrow 2^V,\\,X\\mapsto \\DclD{X}{D} \\]\n where\n \\[ \\DclD{X}{D} = X \\cup \\SET{v\\in V\\mid \\exists x\\in X\\colon\\,(x,v)\\in A}.\\]\n We call $\\DclD{X}{D}$ the \\deftext[outer extension of X in D@outer extension of $X$ in $D$]{outer extension of $\\bm X$ in $\\bm D$}.\n If the digraph is clear from the context, we write $\\Dcl{\\bullet}$ for $\\DclD{\\bullet}{D}$. The \\deftextX{outer-margin operator in $\\bm D$} is defined to be the map\\label{n:partD}\n \\[ \\partial_D \\bullet \\colon 2^V \\longrightarrow 2^V,\\,X\\mapsto \\partial_D X\\]\n where \\[\\partial_D X = \\left.\\DclD{X}{D} \\right\\backslash X.\\] We call $\\partial_D X$ the \\deftext[outer margin of X in D@outer margin of $X$ in $D$]{outer margin of $\\bm X$ in $\\bm D$}. Again, if no confusion can occur, we write $\\partial \\bullet$ as a shorthand for $\\partial_D \\bullet$.\n\\end{definition}\n\n\\input{Text\/Ex\/135_Digraph_Outer_Margin.tex}\n\n\\needspace{6\\baselineskip}\n\\begin{definition}\\PRFR{Jan 22nd}\n Let $D = (V,A)$ be a digraph, $\\mathbb{N} \\ni n>0$, and \\label{n:walk}$w=(w_i)_{i=1}^n \\in\n V^{n}$. Then $w$ is a \\deftext{walk} in $D$, if for all\n $i\\in\\SET{1,2,\\ldots, n-1}$ there is an arc $(w_i,w_{i+1})\\in A$. The\n \\deftext{start vertex} --- or \\deftext{initial vertex} --- of $w$ is $w_{1}$, and the \\deftext[end vertex of a walk]{end vertex}\n --- or \\deftext{terminal vertex} --- of\n $w$ is denoted by $w_{-1} = w_{n}$. The set of vertices \\deftext[visited by w@visited by $w$]{visited\n by $\\bm w$} is denoted by \\( \\left|w\\right| = \\SET{w_1,w_2,\\ldots,w_n}.\\) \n The set of all arcs \\deftext[traversed by w@traversed by $w$]{traversed by $\\bm w$} is denoted by\n \\( \\left| w\\right|_A = \\SET{(w_i,w_{i+1})\\mid i=1,2,\\ldots,n-1}.\\)\n The set of all\n walks in $D$ is denoted by\\label{n:PbfD}\n \\[ {\\mathbf{W}}(D) = \n \\SET{w \\in \\bigcup_{n=1}^{\\infty} V^n ~\\middle|~ w\\text{~is a walk in~}D}.\\]\n The \\deftext[length of $w$]{length} of the walk $w=(w_i)_{i=1}^n$ is $n$. A walk $w$ is\n called \\deftext[trivial walk]{trivial}, if its length is $1$. We say that a walk $w$ is\n a \\deftext[path]{path}, if no vertex is visited twice by $w$, i.e. if\n $w_{i}=w_{j}$ already implies $i=j$. The family of paths in $D$ is denoted by\n \\label{n:simplePath}\n \\[ {\\mathbf{P}} (D) = \\SET{p\\in {\\mathbf{W}}(D) \\mid p \\text{~is a path}}.\\]\n Furthermore, for all $u,v\\in V$, we shall denote the set of all walks from $u$ to $v$ in $D$ by\\label{n:PathUV}\n \\[ {\\mathbf{W}}(D; u,v) = \\SET{w\\in {\\mathbf{W}}(D) \\mid w_1=u {\\mathop{\\text{~and~}}} w_{-1}=v} \\]\n and the set of all paths from $u$ to $v$ in $D$ by\\label{n:SPathUV}\n \\[ {\\mathbf{P}}(D; u,v) = \\SET{p\\in {\\mathbf{P}}(D) \\mid p_1 = u {\\mathop{\\text{~and~}}} p_{-1} = v}. \\qedhere\\]\n\\end{definition}\n\n\\noindent Instead of $w=(w_i)_{i=1}^n=(w_1,w_2,\\ldots,w_n)$ we shall also write $w_{1}w_{2}\\ldots w_{n}$. Furthermore, we set the convention that $(w_{1}w_{2}\\ldots w_{n})^i$ shall denote the walk consisting of $i$-iterations of the visited-vertex sequence $w_{1}w_{2}\\ldots w_{n}$, i.e. $(abc)^{3}$ shall denote the non-path walk $abcabcabc$. \n\n\n\n\\begin{definition}\\PRFR{Jan 22nd}\n Let $D= (V,A)$ be a digraph, and let $w=(w_i)_{{i=1}}^{n}\\in {\\mathbf{W}}(D)$ and $q=(q_i)_{i=1}^{m}\\in{\\mathbf{W}}(D)$ be walks.\n Then we say that \\deftext[compatible walks]{$\\bm w$ is compatible with $\\bm q$}, if $w_{n}=q_{1}$.\n In that case, we define the \\deftext[concatenation of w and q@concatenation of $w$ and $q$]{concatenation of $\\bm w$ and $\\bm q$} to be the walk\\label{n:pdotq}\n \\( w.q = w_{1}w_{2}\\ldots w_{n}q_{2}q_{3}\\ldots q_{m} \\).\n\\end{definition}\n\n\\input{Text\/Ex\/134_Digraph_paths.tex}\n\n\\begin{definition}\\PRFR{Jan 22nd}\n Let $D=(V,A)$ be a digraph. A walk $w=(w_i)_{i=1}^n \\in {\\mathbf{W}}(D)$ is called \\deftext{cycle walk}, or shorter, \\deftextX{cycle},\n if $w_1 = w_n$ and $w_2 w_3 \\ldots w_n$ is a path.\n\\end{definition}\n\n\\noindent Observe that there is no {\\em ``empty walk''}, thus the trivial walks are not considered to be cycles.\n\n\\begin{definition}\\PRFR{Jan 22nd}\n Let $D=(V,A)$ be a digraph. $D$ shall be called \\deftext{acyclic digraph}, if every walk $w\\in{\\mathbf{W}}(D)$ is a path, i.e. whenever\n \\( {\\mathbf{W}}(D) = {\\mathbf{P}}(D) \\).\n\\end{definition}\n\n\\begin{corollary}\\PRFR{Jan 22nd}\n Let $D=(V,A)$ be a digraph. Then $D$ is acyclic if and only if there is no cycle\n walk $w\\in{\\mathbf{W}}(D)$.\n\\end{corollary}\n\n\\subsection{Routings and Transversals}\n\n\\PRFR{Jan 22nd}\nIn this section, we introduce a correspondence between families of pair-wise vertex disjoint paths in digraphs and\ntransversals of certain families of sets, which will be valuable for the study of gammoids.\n\n\\needspace{6\\baselineskip}\n\\begin{definition}\\PRFR{Jan 22nd}\n\tLet $D = (V,A)$ be a digraph, and $X,Y\\subseteq V$. A \\deftext{routing} from $X$ to $Y$ in $D$ is a family of paths $R\\subseteq {\\mathbf{P}}(D)$ such that\n\t\\begin{enumerate}\\renewcommand{\\theenumi}{(\\roman{enumi})}\\renewcommand{\\labelenumi}{\\theenumi}\n\t\t\\item for each $x\\in X$ there is some $p\\in R$ with $p_{1}=x$,\n\t\t\\item for all $p\\in R$ the end vertex $p_{-1}\\in Y$, and\n\t\t\\item for all $p,q\\in R$, either $p=q$ or $\\left|p\\right|\\cap \\left|q\\right| = \\emptyset$\n \n\t\\end{enumerate}\n\tWe will write $R\\colon X\\double{\\rightarrow} Y$ in $D$ as a shorthand for ``$R$ is a routing from $X$ to $Y$\n in $D$'', and if no confusion is possible, \\label{n:routing}\n we just write $X\\double{\\rightarrow} Y$ instead of $R$ and $R\\colon X\\double{\\rightarrow} Y$.\n A routing $R$ is called \\deftext{linking} from $X$ to $Y$, if it is a routing onto $Y$, i.e. whenever $Y = \\SET{p_{-1}\\mid p\\in R}$.\n\\end{definition}\n\n\n\\input{Text\/Ex\/154_Routing}\n\n\\begin{remark}\\label{rem:straighteningWalks}\\PRFR{Jan 22nd}\n We defined a routing to consist of paths only, but for most of what we are concerned with when using the concept of a routing in $D$,\n the property that the walks of $R$ are indeed paths is not crucial. Let $R'\\subseteq {\\mathbf{W}}(D)$ be a family of walks such that \n for all $w,q\\in R'$ the implication $\\left| w \\right|\\cap \\left| q \\right|\\not= \\emptyset \\Rightarrow w=q$ holds.\n Now let $w\\in R' \\backslash {\\mathbf{P}}(D)$\n be a non-path walk in $R'$. Then there is a vertex $v\\in \\left| w \\right|$ such that\n $w = w_0 v p' v w_1$ where $w_0,w_1,p'\\in {\\mathbf{W}}(D)$ such that $vp'v$ is a cycle walk.\n Clearly $\\hat{w} = w_0 v w_1$ has less such cycle sub-walks than $w$ and $\\left| w_0 v w_1 \\right|\\subseteq \\left| w \\right|$,\n thus we may iteratively straighten out any cycles in the walks from $R'$ without changing the \n start and the end vertices. The property, that the family of walks consists of pair-wise vertex \n disjoint walks, remains intact throughout the procedure.\n The result of this process is a family of paths which is a linking\n from $\\SET{w_1\\mid w\\in R'}$ onto $\\SET{w_{-1}\\mid w\\in R'}$ in $D$.\n\\end{remark}\n\n\\noindent The straightening out of cycle walks in routings is a special case of the following construction.\n\n\\needspace{4\\baselineskip}\n\\begin{definition}\\PRFR{Mar 7th}\n Let $D=(V,A)$ be a digraph, $w \\in {\\mathbf{W}}(D)$ be a walk. Then $w$ is called \\deftext[essential path in $D$]{essential path in $\\bm D$}, if for all $w'\\in {\\mathbf{W}}(D; w_1,w_{-1})$ with $\\left| w' \\right| \\subseteq \\left| w \\right|$ we have\n $\\left| w' \\right| = \\left| w \\right|$.\n Let $R\\subseteq {\\mathbf{P}}(D)$ be a routing, then $R$ is called \\deftext[essential routing in $D$]{essential routing in $\\bm D$},\n if $p$ is an essential path in $D$ for all $p\\in R$.\n\\end{definition}\n\n\\begin{lemma}\\PRFR{Mar 7th}\n Let $D=(V,A)$ be a digraph and let $R\\subseteq {\\mathbf{P}}(D)$ be a routing from $X$ to $Y$ in $D$. Then\n there is an essential routing from $X$ to $Y$ in $D$.\n\\end{lemma}\n\\begin{proof}\\PRFR{Mar 7th}\n We show this by induction on the number of paths in $R$ that are not essential.\n In the base case, $R$ itself is an essential routing from $X$ to $Y$.\n Now let $p\\in R$ be a path that is not essential in $D$. Then there is a path $p'\\in {\\mathbf{P}}(D,p_1,p_{-1})$ \n with $\\left| p' \\right|\\subsetneq \\left| p \\right|$, such that $\\left| p' \\right|$ is $\\subseteq$-minimal. \n Such $p'$ is an essential path. \n Then $\\left( R\\BSET{p} \\right)\\cup\\SET{p'}$ is a routing from $X$ to $Y$ in $D$ with fewer non-essential paths,\n so by induction hypothesis there is an essential routing from $X$ to $Y$ in $D$.\n\\end{proof}\n\n\\input{Text\/Ex\/155_Bipartite_Linking}\n\n\\noindent The following connection between the linkings in directed graphs and the transversals of a set system, which define linkings in a bipartite graph that can be deduced from the digraph, has first been pointed out by A.W.~Ingleton and M.J.~Piff in \\cite{IP73}. But first, we need to clarify how to deduce the correct family of sets given a digraph and a set of targets.\n\n\\needspace{7\\baselineskip}\n\\begin{definition}\\label{def:linkageSystem}\\PRFR{Jan 22nd}\n Let $D=(V,A)$ be a digraph, and let $T\\subseteq V$ be a set of vertices.\n The \\deftext[linkage system of D to T@linkage system of $D$ to $T$]{linkage system of $\\bm D$ to $\\bm T$} -- denoted by\n ${\\mathcal{A}}_{D,T}$ -- is defined to be the family\\label{n:ADT}\n \\[ {\\mathcal{A}}_{D,T} = \\left(A^{(D,T)}_i\\right)_{i\\in V\\backslash T} \\subseteq V\\]\n where for $v\\in V\\backslash T$\n \\[ A^{(D,T)}_{v} = \\SET{w\\in V\\mid (v,w)\\in A} \\cup \\SET{v}. \\qedhere\\]\n\\end{definition}\n\n\\begin{lemma}\\label{lem:ADTtransversals}\\PRFR{Jan 22nd}\n Let $D=(V,A)$ be a digraph, $T \\subseteq V$.\n Every maximal partial transversal of ${\\mathcal{A}}_{D,T}$ is a transversal of ${\\mathcal{A}}_{D,T}$.\n\\end{lemma}\n\n\\begin{proof}\\PRFR{Jan 22nd}\n Clearly, $V\\backslash T$ is a transversal of ${\\mathcal{A}}_{D,T}$, and therefore\n $\\mathrm{rk}_{M({\\mathcal{A}}_{D,T})} = \\left| V\\backslash T \\right|$.\n Let $P$ be a maximal partial transversal of ${\\mathcal{A}}_{D,T}$,\n then $P$ is a base of $M({\\mathcal{A}}_{D,T})$ and thus $\\left| P \\right| = \\left| V\\backslash T \\right|$ due to the equicardinality of bases\n {\\em(B2)}. Therefore\n \n every injective map $\\iota\\colon P\\longrightarrow V\\backslash T$ with $p\\in A_{\\iota(p)}^{(D,T)}$ for all $p\\in P$ is a bijection that witnesses that $P$ is a transversal of ${\\mathcal{A}}_{D,T}$.\n\\end{proof}\n\n\\noindent The following lemma has been named {\\em The Fundamental Lemma} by A.W.~Ingleton and M.J.~Piff \\cite{IP73}, who used it as the key to proving that strict gammoids are precisely the duals of transversal matroids. We are going to use it in order to show augmentation properties of routings in digraphs, too.\n\n\n\\begin{lemma}\\label{lem:linkage}\\PRFR{Jan 22nd}\n Let $D=(V,A)$ be a digraph, $S,T \\subseteq V$.\n Then there is a linking from $S$ to $T$ in $D$, if and only if\n $V\\backslash S$ is a transversal of the linkage system ${\\mathcal{A}}_{D,T}$.\n\\end{lemma}\n\n\\noindent The proof presented here can be found on p.217 \\cite{We76}, where the lemma is called\n {\\em The Linkage Lemma}.\n\n\\begin{proof}\\PRFR{Jan 22nd}\n Assume that $R\\colon S\\double{\\rightarrow} T$ is a linking in $D$. We construct the bijective map\n $\\sigma\\colon V\\backslash S \\longrightarrow V\\backslash T$ such that for $v\\in V\\backslash S$, the image\n \\[ \\sigma(v) = \\begin{cases} u & \\quad \\textit{(a)}\n \\,\\, \\text{if }\\exists p\\in R\\colon\\,(u,v)\\in\\left| p\\right|_A, \\text{ and} \\\\\n v & \\quad \\textit{(b)} \\,\\, \\text{otherwise.}\n \\end{cases}\\]\n The map $\\sigma$ is well-defined because $R$ consists of pair-wise vertex\n disjoint paths in $D$; and whenever $v\\in T$, then either $v\\in S$\n in which case $v$ is not part of the domain of $\\sigma$, or there is a\n non-trivial path $p\\in R$ that ends in $v$. Then $\\sigma(v) \\notin T$\n since otherwise $R$ could not be onto $T$ as every path has precisely one\n end vertex. From the definition of ${\\mathcal{A}}_{D,T}$ and the construction of\n $\\sigma$ it is clear, that for every $v\\in V\\backslash S$, $v \\in\n A^{(D,T)}_{\\sigma(v)}$. Assume that $\\sigma$ is not injective, thus there\n are $v,w\\in V\\backslash S$ with $v\\not=w$, yet $\\sigma(v) = \\sigma(w)$. This is\n not possible if $v$ and $w$ are in the same case of $\\sigma$. Thus without\n loss of generality we may assume that $\\sigma$ maps $v$ through case {\\em\n (a)} and $w$ through case {\\em (b)}. Thus $\\sigma(v) = \\sigma(w) = w$, and\n $(w,v)\\in \\left| p\\right|_{A}$ for some $p\\in R$. Since for $w$ case {\\em\n (b)} holds, we can infer that $w = p_{1}$ is the initial vertex of a path in $R$.\n But then $w\\in S$ which is not part of the domain of $\\sigma$. Therefore\n no such $v,w\\in V$ exist and $\\sigma$ is an injective map. Since $R$ is a\n linking, $\\left| S \\right| = \\left| T \\right|$ and $\\left| V\\backslash S \\right|\n = \\left| V \\backslash T\\right| < \\infty$, thus $\\sigma$ is a bijection and $V\\backslash\n S$ is indeed a transversal of ${\\mathcal{A}}_{D,T}$.\n\n\\PRFR{Jan 22nd}\n \\noindent \n Conversely, assume that $V\\backslash S$ is a transversal of ${\\mathcal{A}}_{D,T}$. Thus\n there is a bijection $\\sigma\\colon V\\backslash S \\longrightarrow V\\backslash T$ such that for\n all $v\\in V\\backslash S$, $v\\in A^{(D,T)}_{\\sigma(v)}$. We can construct a\n linking $R\\colon X\\double{\\rightarrow} Y$ from $\\sigma$ in the following way: for\n $v\\in S\\cap T$, we can let the trivial path $v \\in R$. For $v\\in T\\backslash S$,\n there is some $k\\in \\mathbb{N}$ such that $\\sigma^{k}(v) \\notin V\\backslash S$:\n assume that for every $k\\in \\mathbb{N}$, $\\sigma^{k}(v)\\in V\\backslash S$, then\n $\\SET{\\sigma^{k}(v)\\vphantom{A^A}~\\middle|~ k\\in \\mathbb{N}} \\subseteq V\\backslash S$, yet $V\\backslash S$ is\n finite. Thus there must be some $k_0,k_1\\in \\mathbb{N}$ with $k_0 < k_{1}$ and\n $\\sigma^{k_0}(v) = \\sigma^{k_1}(v)$. Now let $k_{0},k_{1}\\in \\mathbb{N}$ be\n integers with $k_{0} < k_{1}$ and $\\sigma^{k_0}(v) = \\sigma^{k_1}(v)$\n such that $k_{0}$ is smallest possible. Clearly $v\\notin V\\backslash T$, so\n $k_0 > 0$. But then $\\sigma$ is a bijection, therefore the pre-images of\n $\\sigma^{k_0}(v)$ and $\\sigma^{k_1}(v)$ coincide. Now we have\n $\\sigma^{k_0-1}(v) = \\sigma^{k_1-1}(v)$ which contradicts the minimality\n of $k_{0}$. Thus the trajectory of $v$ under repetitions of $\\sigma$ has\n no cycle and therefore must be finite. Let $k\\in \\mathbb{N}$ such that\n $\\sigma^{k}(v) \\notin V\\backslash S$. The range of $\\sigma$ yields that $\\sigma^{k}(v) \\in V\\backslash\n T$ and therefore $\\sigma^{k}(v)\\in S\\backslash T$. The construction of\n ${\\mathcal{A}}_{D,T}$ guarantees that for every $i\\in\\SET{0,1,\\ldots,k-1}$ there\n is an arc $(\\sigma^i(v),\\sigma^{i+1}(v))\\in A$. Since $\\sigma$-trajectories have no cycles, we can add the path\n $\\sigma^{k}(v)\\sigma^{k-1}(v)\\ldots\\sigma(v)v \\in R$. All paths obtained\n from the above constructions are pair-wise vertex disjoint,\n because $\\sigma$ is a bijection of finite sets, and so $R$ is indeed a\n linking from $S$ to $T$ in $D$.\n\\end{proof}\n\n\\needspace{2\\baselineskip}\n\\noindent We can extend Lemma~\\ref{lem:linkage} to routings in the natural way.\n\n\\begin{lemma}\\label{lem:routage}\\PRFR{Jan 22nd}\n Let $D=(V,A)$ be a digraph, $S,T \\subseteq V$.\n Then there is a routing from $S$ to $T$ in $D$, if and only if\n there is some $T'\\subseteq T$ such that\n $V\\backslash (S\\cup T')$ is a transversal of the linkage system ${\\mathcal{A}}_{D,T}$.\n\\end{lemma}\n \n\\begin{proof}\\PRFR{Jan 22nd}\nEvery routing $R\\colon S\\double{\\rightarrow} T$ in $D$ consists of a linking from\n$S$ to $T_R = \\SET{p_{-1}\\mid p\\in R}$ and a set of unused targets\n$T'=T\\backslash T_{R}$, and thus for every $t'\\in T'$, we may add the trivial path $t'$\nto $R$ and obtain the linking $R'\\colon S\\cup T' \\double{\\rightarrow} T$ \nwhere $R' = R\\cup \\SET{t'\\in {\\mathbf{P}}(D)\\mid t'\\in T'}$. Therefore, $R$ induces the \ntransversal $V\\backslash \\left( S \\cup T' \\right)$ of ${\\mathcal{A}}_{D,T}$ by Lemma~\\ref{lem:linkage}.\nConversely, let $T'\\subseteq T$ such that $V\\backslash(S\\cup T')$ is a transversal of ${\\mathcal{A}}_{D,T}$. By Lemma~\\ref{lem:linkage} there is a linking $R\\colon S\\cup T' \\double{\\rightarrow} T$ in $D$. Then\n$R' = \\SET{p\\in R \\mid p_1 \\in S}$ is a routing from $S$ to $T$ in $D$.\n\\end{proof}\n\n\\subsection{Menger's Theorem}\n\n\\PRFR{Jan 22nd}\nF.~G\u00f6ring published an intriguingly short and beautiful proof of Menger's Theorem \\cite{Go00}. \nIn this section,\nwe present a slightly more verbose variant of this proof, which is transformed \ninto the context\nof this work, along with two required yet straightforward definitions.\n\n\n\\begin{definition}\\PRFR{Jan 22nd}\n Let $D=(V,A)$ be a digraph, $S,T\\subseteq V$. \n A set $X\\subseteq V$ is called \\deftext[S-T-separator@$S$-$T$-separator]{$\\bm{S}$-$\\bm{T}$-separator} in $D$, if for\n every $p\\in{\\mathbf{P}}(D)$ with $p_{1}\\in S$ and $p_{-1}\\in T$, $\\left|p\\right|\\cap X\\not= \\emptyset$.\n\\end{definition}\n\n\\PRFR{Jan 22nd}\n\\noindent It is easy to see that straightening out cycle paths from walks (Remark~\\ref{rem:straighteningWalks}) yields paths using a subset of the original vertices,\nthus if $X$ is an $S$-$T$-separator, then for all $w\\in {\\mathbf{W}}$ with $w_1 \\in S$ and $w_{-1}\\in T$ we also have $\\left| w \\right|\\cap X\\not=\\emptyset$.\n\n\\begin{example}\\label{ex:outermarginseparates}\\PRFR{Jan 22nd}\n Let $D=(V,A)$ be a digraph, $S\\subseteq V$. Then $\\partial S$ is a minimal\n $S$-$(V\\backslash S)$-separator in $D$: Since $S\\cap (V\\backslash S) = \\emptyset$, \n any walk from $s\\in S$ to $t\\in V\\backslash S$ must use an\n arc that starts in $S$ but ends outside of $S$, and therefore it must visit an element of the outer margin $\\partial S$. \n Now let $v\\in \\partial S$, then there is some $u\\in S$ such that $(u,v)\\in A$. So $uv\\in {\\mathbf{P}}(D)$ \n is a path from $S$ to $V\\backslash S$, yet $\\partial S \\cap \\left| uv \\right| = \\SET{v}$, therefore\n $\\partial S\\BSET{v}$ is not an $S$-$(V\\backslash S)$-separator; thus $\\partial S$ is a minimal $S$-$(V\\backslash S)$-separator in $D$.\n\\end{example}\n\n\\PRFR{Jan 22nd}\n\\noindent Clearly, both $S$ and $T$ are $S$-$T$-separators in every digraph $D$. Furthermore,\nevery $S$-$T$-separator in $D$ is an $S'$-$T'$-separator for every $S'\\subseteq S$ and $T'\\subseteq T$.\n\n\\begin{definition}\\PRFR{Jan 22nd}\n Let $D=(V,A)$ be a digraph, $S,T\\subseteq V$. A routing $Y\\double{\\rightarrow} T$\n in $D$ is called \\index{connector}\\deftext[S-T-connector@$S$-$T$-connector]{$\\bm{S}$-$\\bm{T}$-connector} in $D$,\n whenever $Y\\subseteq S$.\n\\end{definition}\n\n\\begin{theorem}[Menger's Theorem \\cite{Me27,Go00}]\\label{thm:MengerGoering}\\index{Menger's Theorem}\\PRFR{Jan 22nd}\nLet $D=(V,A)$ be a digraph, $S,T\\subseteq V$ subsets of vertices of $D$,\nand $k \\in \\mathbb{N}$ the minimal cardinality of an $S$-$T$-separator in $D$.\nThere is an $S$-$T$-connector $R\\colon Y\\double{\\rightarrow} T$ that consists of\n$k$ paths.\n\\end{theorem}\n\n\\needspace{7\\baselineskip}\n\n\\vspace*{-\\baselineskip}\n\\begin{wrapfigure}{r}{7.5cm}\n\\vspace{1.25\\baselineskip}\n\\begin{centering}~\n\\includegraphics{MengersThm}\n\\end{centering}%\n\\vspace*{-.5\\baselineskip}\n\\end{wrapfigure}\n~\n \n\n\\begin{proof}\\PRFR{Jan 22nd}\nBy induction on $\\left|A\\right|$. If $A=\\emptyset$, then there are only trivial paths in\n${\\mathbf{P}}(D)$. Thus $S\\cap T$ is a minimal $S$-$T$-separator. Clearly,\n$\\SET{v \\in {\\mathbf{P}}(D) ~\\middle|~ v\\in S\\cap T}$ is a routing from $S\\cap T$ to $T$ in $D$.\n\n\n\\PRFR{Jan 22nd}\n\\noindent\nFor the induction step, let $(v,w)=a\\in A$. The theorem holds for\n$D'=(V,A\\BSET{a})$ by induction hypothesis, if $D'$ has no $S$-$T$-separator\n$X'$ with $\\left|X'\\right|1$ and $\\SET{r,s}\\cap \\left| p' \\right| = \\emptyset$ for\n\tall $p'\\in R\\BSET{p}$. Let $q = p_1p_2\\ldots p_{n-1}$ be the path that arises when the vertex $s$ is chopped off of $p$.\n\t Then $R' = R\\BSET{p}\\cup\\SET{q}$ is a routing from $X$ to $(T\\BSET{s})\\cup \\SET{r}$ in $D_{r\\leftarrow s}$.\n\tOtherwise, we have that $(r,s) \\notin \\bigcup_{p\\in R} \\left| p \\right|_A$. Let $Q=\\SET{r,s}\\cap \\left( \\bigcup_{p\\in R} \\left| p \\right| \\right)$ be \n\tthe criterion for a case analysis. If $Q=\\emptyset$, then $R$ is obviously a routing in $D_{r\\leftarrow s}$, \n\tbecause $D$ and $D_{r\\leftarrow s}$ coincide on $V\\BSET{r,s}$. \n\tThen no path $p\\in R$ has $p_{-1}=s$, thus $R$ even is a routing from $X$ \n\tto $T\\BSET{s}$ in $D_{r\\leftarrow s}$. If $Q=\\SET{s}$, then there is a path $p\\in R$ with $p_{-1}=s$, yet no path of $R$ visits $r$, \n\ttherefore $R\\BSET{p}\\cup\\SET{pr}$ is the desired routing in $D_{r\\leftarrow s}$.\n\tIf $Q=\\SET{r}$, then no path in $R$ visits $s$, and there is a path $p=(p_i)_{i=1}^{n}$ that visits $r = p_j$ with $j\\in\\SET{1,2,\\ldots,n}$. Then $R\\BSET{p}\\cup\\SET{p_1p_2\\ldots p_j}$\n\tis the desired routing in $D_{r\\leftarrow s}$.\n\tIf $Q=\\SET{r,s}$, then there are two paths $p,q\\in R$ with $p\\not= q$ such that $s\\in \\left| p \\right|$ and $r\\in \\left| q \\right|$.\n\tLet $q=(q_i)_{i=1}^m$, and let $1\\leq j \\leq m$ such that $q_j = r$. \n\tSince $s$ is a sink in $D$, we have $p_{-1}=s$. Let $p' = pq_{j+1}q_{j+2}\\ldots q_m$ be\n\tthe path in $D_{r\\leftarrow s}$ that first follows $p$ and then follows the end of $q$. We have\n\t$\\left| p' \\right|_A\\subseteq A_{r\\leftarrow s}$ since $(r,q_{j+1})\\in \\left| q \\right|_A \\subseteq A$\n\tthus $(s,q_{j+1})\\in A_{r\\leftarrow s}$, and the digraphs $D$ and $D_{r\\leftarrow s}$ have the same arcs on $V\\BSET{r,s}$. \n\tFurthermore, let $q'= q_1q_2\\ldots q_j$, clearly $q'\\in {\\mathbf{P}}(D_{r\\leftarrow s})$, \n\tthus $(R\\BSET{p,q})\\cup\\SET{p',q'}$ is the desired routing in $D_{r\\leftarrow s}$.\n\n\\noindent\n\tThe second implication of the lemma follows from the first implication together with the fact, that in the situation of the lemma where the operand $s$ is a sink of $D$,\n\t$\\left( D_{r \\leftarrow s} \\right)_{s\\leftarrow r} = D$ holds.\n\\end{proof}\n\n\\begin{theorem}\\label{thm:gammoidRepresentationWithBaseTerminals}\\PRFR{Jan 22nd}\n\tLet $M=(E,{\\mathcal{I}})$ be a gammoid, and $B$ a base of $M$.\n\tThen there is a digraph $D=(V,A)$, \n\tsuch that \\[ M = \\Gamma(D,B,E)\\]\n\tand every $b\\in B$ is a sink in $D$.\n\\end{theorem}\n\\begin{proof}\\PRFR{Jan 22nd}\n\tLet $D'=(V,A')$ be a digraph and $T' \\subseteq A'$ such that $M = \\Gamma(D',T',E)$.\n\tWe may assume that $\\left| T' \\right| = \\left| B \\right| = \\mathrm{rk}_M(E)$ and that all $t'\\in T'$ are\n\tsinks in $D'$ (Corollary~\\ref{cor:wlogNiceDigraph}).\n\tLet $R\\colon B \\double{\\rightarrow} T'$ a linking of $B$ onto $T'$ in $D'$.\n\tWe prove the statement by induction on $\\left| \\bigcup_{p\\in R} \\left| p \\right| \\right|$.\n\tIn the base case we have $\\left| \\bigcup_{p\\in R} \\left| p \\right| \\right| = \\left| B \\right|$, and\n\ttherefore every path $p\\in R$ is trivial. Thus $B = T$ and $D=D'$ is the desired digraph.\n\tNow let $\\left| \\bigcup_{p\\in R} \\left| p \\right| \\right| > \\left| B \\right|$,\n\tthus there is a non-trivial path $p=(p_i)_{i=1}^{n} \\in R$ where $n>1$. Let $s = p_{n}$ and let\n\t$r = p_{n-1}$. The vertex $s$ is a sink in $D'$ since $s\\in T'$, and clearly\n\t $(r,s)\\in \\left| p \\right|_A\\subseteq A'$. Since \\linebreak\n\t $\\left| B \\right| = \\left| T' \\right|$, $r\\notin T'$.\n\t The proof of Lemma~\\ref{lem:MasonsFundamental} yields\n\t that $R' = R\\BSET{p}\\cup\\SET{p_1p_2\\ldots p_{n-1}}$ is\n\ta linking of $B$ onto $\\left( T'\\BSET{s} \\right)\\cup\\SET{r}$\n\t in $D'_{r\\leftarrow s}$ with $\\left| \\bigcup_{p\\in R'} \\left| p \\right| \\right| < \\left| \\bigcup_{p\\in R} \\left| p \\right| \\right|$.\n\t Furthermore Lemma~\\ref{lem:MasonsFundamental} implies that \n\t $\\Gamma(D',T',E) = \\Gamma(D'_{r\\leftarrow s},\\left( T'\\BSET{s} \\right)\\cup\\SET{r}, E)$\n\t and the existence of the digraph $D$ follows\n\tfrom the induction hypothesis for the linking $R'$ with respect to the representation \n\t $(D'_{r\\leftarrow s}, \\left( T'\\BSET{s} \\right)\\cup\\SET{r},E)$.\n\\end{proof}\n\n\n\n\n\\subsection{Number of Vertices Needed to Represent a Gammoid}\n\n\\PRFR{Mar 7th}\n\\noindent\nIn the paper {\\em Representative Sets and Irrelevant Vertices: New Tools for Kernelization} \\cite{KW12}, S.~Kratsch and M.~Wahlstr\u00f6m proved \nthe following upper bound result regarding the number of vertices in a given digraph, that suffice to be considered in order to find certain $S$-$T$-separators of minimal cardinality.\nThis bound may be used to derive a bound on the number of vertices needed\nin order to represent a gammoid on a ground set of given cardinality.\n\n\\needspace{4\\baselineskip}\n\n\\begin{theorem}[\\cite{KW12}, Theorem~3]\\label{thm:upperBoundSizeOfV}\\PRFR{Mar 7th}\n\tLet $D=(V,A)$ be a digraph, $E,T\\subseteq V$, and $r>0$ be the cardinality of a minimal $E$-$T$-separator in $D$.\n\tThere is a set $Z\\subseteq V$ with $E\\cup T \\subseteq Z$ and\n\t $\\left| Z \\right| = O(\\left| E \\right|\\cdot\\left| T \\right|\\cdot r)$\n\tsuch that for all $X\\subseteq E$ and $Y\\subseteq T$ there is a minimal $X$-$Y$-separator $S$ in $D$ with $S\\subseteq Z$.\n\tThe set $Z$ can be found in randomized polynomial time with failure probability $O(2^{-n})$.\n\\end{theorem}\n\n\\PRFR{Mar 7th}\n\\noindent\nFor the proof, see \\cite{KW12}.\\footnote{The actual proof starts on page 24.} \nThe statement that $E\\cup T \\subseteq Z$ is not part of the original theorem in \\cite{KW12}, as well as the condition $r>0$,\nbut it is easy to see that these modifications are valid, since $\\left| E\\cup T \\right| = O(\\left| E \\right|\\cdot\\left| T \\right|\\cdot r)$ for $r\\geq 1$.\n\n\n\\begin{remark}\\label{rem:boundForZ}\\PRFR{Mar 7th}\n\t In \\cite{KW12}, the authors only give the $O$-behavior of the size of $Z$ in Theorem~\\ref{thm:upperBoundSizeOfV},\n\t but it is possible to derive the factor hidden in the $O$-notation by inspecting their proof and the proof of\n\t Lemma~4.1 \\cite{Marx09} by D.~Marx.\n\tWe obtain\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\n\t \\[ \\left| Z \\right| \\leq \\binom{r}{1} \\cdot \\binom{\\left| E \\right|}{1} \\cdot \\binom{\\left| T \\right|}{1} + \\left| E \\right| + \\left| T \\right| = r\\cdot \\left| E \\right|\\cdot \\left| T \\right| + \\left| E \\right| + \\left| T \\right|. \\qedhere \\]\n\t\n\\end{remark}\n\n\n\\begin{corollary}\\label{cor:upperBoundOnV}\\PRFR{Mar 7th}\n\tLet $M=(E,{\\mathcal{I}})$ be a gammoid.\n\tThere is a representation $(D,T,E)$ of $M$ with $D=(V,A)$, such that\n\t\\[ \\left| V \\right| = O\\left( \\left| E \\right| \\cdot \\mathrm{rk}_M(E)^2 \\right) \\leq O\\left( \\left| E \\right|^3 \\right).\\]\n\\end{corollary}\n\\begin{proof}\\PRFR{Mar 7th}\n\tLet $(D,T,E)$ be a representation of $M$ where $\\left| T \\right| = \\mathrm{rk}_M(E)$ and $D=(V,A)$ (Lemma~\\ref{lem:rankequalsTcard}).\n\tLet $Z'\\subseteq V$ be a subset of $V$ as in the consequent \n\tof Theorem~\\ref{thm:upperBoundSizeOfV}.\n\n\n\tLet $D'=(Z',A')$ be the digraph,\n\twhere for all $x,y\\in Z'$, there is an arc\n\t\\[ (x,y)\\in A' \\quad\\Longleftrightarrow\\quad \\exists p\\in {\\mathbf{P}}(D; x,y) \\colon\\, \\left| p \\right|\\cap Z' = \\SET{x,y}.\\]\n\tThus there is an arc leaving $y\\in Z'$ and entering $z\\in Z'$ in $D'$ if there is a path from $y$ to $z$ in $D$ that never visits\n\tanother vertex of $Z'$.\n\tLet $p=(p_i)_{i=1}^n \\in {\\mathbf{P}}(D)$ be a path of length $n$ from $p_1\\in E$ to $p_{-1}\\in T$.\n\tLet $I' = \\SET{i\\in \\mathbb{N} ~\\middle|~ 1\\leq i\\leq n {\\mathop{\\text{~and~}}} p_i \\in Z'} = \\dSET{i_1,i_2,\\ldots,i_k}$ with $i_1 < i_2 < \\ldots < i_k$.\n\tThen let $p' = p_{i_1}p_{i_2}\\ldots p_{i_k}$, i.e. $p'$ is the path consisting of the vertices visited by $p$ that are in $Z'$.\n\tObserve that\n\t$p_1 = p'_1$, $p_{-1} = p'_{-1}$, and $p'\\in {\\mathbf{P}}(D')$ holds.\n\tLet $R\\colon X\\double{\\rightarrow} T$ be a routing from $X\\subseteq E$ to $T$ in $D$,\n\tand let\n\t $R' = \\SET{p'\\in {\\mathbf{P}}(D') ~\\middle|~ p\\in R}$ be the set of paths in $D'$ that consists of all\n\t$p'$ derived from $p\\in R$ as described above. By construction of $D'$, we see that $R'$ is a \n\trouting from $X$ to $T$ in $D'$. Thus every independent set $X\\subseteq E$\n\tof $M = \\Gamma(D,T,E)$ is also an independent set of $N = \\Gamma(D',T,E)$.\n\tNow assume that there is some $X\\subseteq E$ that is independent in $N$, but not in $M$.\n\tThen $D$ would have an $X$-$T$-separator $S$ with $\\left| S \\right| < \\left| X \\right|$, and by Theorem~\\ref{thm:upperBoundSizeOfV}\n\twe may assume that $S\\subseteq Z'$ holds. Thus $S$ would be an $X$-$T$-separator of $D'$, too, contradicting the assumption \n\tthat $X$ is independent with respect to $N$. Therefore every independent set of $N$ is also independent with respect to $M$.\n\tConsequently, $M = N$. Thus $(D',T,E)$ is a representation of $M$ using \n\tonly $O\\left( \\left| E \\right| \\cdot \\mathrm{rk}_M(E)^2 \\right)$ vertices.\n\\end{proof}\n\n\\begin{remark}\\label{rem:upperBoundForV}\\PRFR{Mar 7th}\n\tIn the light of Remark~\\ref{rem:boundForZ} we obtain that if $M=(E,{\\mathcal{I}})$ is a gammoid,\n\tthere is a representation $(D,T,E)$ where $D=(V,A)$ such that $\\left| T \\right| = \\mathrm{rk}_M(E)$\n\tand such that \\[ \\left| V \\right| \\,\\,\\, \\leq \\,\\,\\, \\mathrm{rk}_M(E)^2 \\cdot \\left| E \\right| + \\mathrm{rk}_M(E) + \\left| E \\right| \\,\\,\\, \\leq \\,\\,\\, 2 \\left| E \\right|^3. \\qedhere\\]\n\\end{remark}\n\n\n\\subsection{Duality Respecting Representations}\n\n\\begin{definition}\\label{def:dualityRespectingRepr}\\PRFR{Jan 22nd}\n\tLet $(D,T,E)$ be a representation of a gammoid. We say that $(D,T,E)$ is a \\deftext{duality respecting representation},\n\tif \n\t\\[ \\Gamma(D^{\\mathrm{opp}},E\\backslash T,E) = \\left( \\Gamma(D,T,E) \\right)^\\ast . \\qedhere \\]\n\\end{definition}\n\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[scale=1.1]{DualityAndNonDualityRespecting}\n\\end{center}\n\\caption{Non-duality respecting and duality respecting representations of $U$. \\label{fig:drnondr}}\n\\end{figure}\n\\begin{example}\nConsider the uniform matroid $U=\\left( \\SET{a,b,c}, \\SET{\\emptyset,\\SET{a},\\SET{b},\\SET{c}} \\right)$\nand the digraphs $D_1=\t\\left( \\SET{a,b,c},\\SET{(a,b),(b,c)} \\right)$ and $D_2=\\left( \\SET{a,b,c},\\SET{(a,c),(b,c)} \\right)$\n(Fig.~\\ref{fig:drnondr}).\nClearly $\\Gamma(D_1,\\SET{c},\\SET{a,b,c}) = U = \\Gamma(D_2,\\SET{c},\\SET{a,b,c})$,\nbut $\\Gamma(D_1^{{\\mathrm{opp}}}, \\SET{a,b}, \\SET{a,b,c}) \\not= U^\\ast$, since there is no routing from $\\SET{b,c}$ to $\\SET{a,b}$ in $D_1^{\\mathrm{opp}}$.\nOn the other hand, such a routing exists in $D_2^{\\mathrm{opp}}$, and indeed we have $U^\\ast = \\Gamma(D_2^{\\mathrm{opp}},\\SET{a,b},\\SET{a,b,c})$. Therefore duality respecting\nrepresentations exist, but not all representations have this property.\n\\end{example}\n\n\\needspace{6\\baselineskip}\n\\begin{lemma}\\label{lem:sourcesinkrepresentation}\\PRFR{Jan 22nd}\n\tLet $M=(E,{\\mathcal{I}})$ be a gammoid, and $B\\subseteq E$ a base of $M$.\n\tThere is a digraph $D=(V,A)$ such that the sinks of $D$ are precisely the elements of $B$,\n\tthe sources of $D$ are precisely the elements of $E\\backslash B$, and\n\tsuch that $M = \\Gamma(D,B,E)$.\n\\end{lemma}\n\\begin{proof}\\PRFR{Jan 22nd}\n\tThere is a digraph $D'=(V',A')$ such that $M=\\Gamma(D',B,E)$ (Theorem~\\ref{thm:gammoidRepresentationWithBaseTerminals}),\n\tand without loss of generality we may assume that the only sinks in $D'$ are the elements of $B$, and that all sources in $D'$ are\n\telements of $E$ --- since sources not in $E$ and sinks not in $B$ cannot be part of a path that belongs to any routing \n\tfrom $X\\subseteq E$ to $B$ in $D'$, and\n\ttherefore may be dropped from $D'$ without changing the represented gammoid.\n\tClearly, we can give each $e \\in V'\\cap \\left( E\\backslash B \\right)$ a new name -- say $e''$ -- in $D'$, yielding a digraph \\linebreak $D'' =(V'',A'')$\n\twhere $V''\\cap E = B$. Then we can add the elements $E\\backslash B$ back to $D''$ as isolated vertices, and after that,\n\twe add an arc leaving $e$ and entering its renamed copy $e''$ for every $e\\in E\\backslash B$.\n\tLet us denote the digraph that we just constructed by \\linebreak $D=(V,A)$\n\twhere $V = V''\\mathbin{\\dot{\\cup}} \\left( E\\backslash B \\right)$ and $A = A'' \\cup \\SET{(e,e'')\\mid e\\in E\\backslash B}$.\n\tClearly, each routing $R\\colon X\\double{\\rightarrow} B$ with $X\\subseteq E$ in $D'$ induces the routing\n\t$R'' = \\SET{ p_1p_1''p_2\\ldots p_n \\mid p_1p_2\\ldots p_n\\in R}$ from $X$ to $B$ in $D$; and conversely,\n\teach routing $Q''\\colon X\\double{\\rightarrow} B$ with $X\\subseteq E$ in $D$ induces the routing\n\t$Q = \\SET{p_1p_3p_4\\ldots p_n\\mid p_1p_2\\ldots p_n\\in Q''}$ from $X$ to $B$ in $D'$. Therefore, $\\Gamma(D,B,E) = \\Gamma(D',B,E) = M$.\n\\end{proof}\n\n\n\\begin{lemma}\\label{lem:dualityrespectingrepresentation}\\PRFR{Jan 22nd}\n\tLet $(D,T,E)$ be a representation of a gammoid with $T\\subseteq E$,\n\tand such that every $e\\in E\\backslash T$ is a source of $D$, and every $t\\in T$ is a sink of $D$.\n Then $(D,T,E)$ is a duality respecting representation.\n\\end{lemma}\n\\begin{proof}\\PRFR{Jan 22nd}\n\tWe have to show that the bases of $N = \\Gamma(D^{\\mathrm{opp}}, E\\backslash T, E)$ are precisely the complements of the\n\tbases of $M = \\Gamma(D,T,E)$ (Corollary~\\ref{cor:dualbase}).\n\n\tLet $B\\subseteq E$ be a base of $M$, then there is a linking $L\\colon B\\double{\\rightarrow} T$ in $D$, and since $T$ consists of sinks,\n\twe have $\\SET{x\\in{\\mathbf{P}}(D)~\\middle|~x\\in T\\cap B} \\subseteq L$.\n\tFurther, let $L^{\\mathrm{opp}} = \\SET{ p_n p_{n-1} \\ldots p_1 \\mid p_1 p_2 \\ldots p_n \\in L}$.\n\tThen $L^{\\mathrm{opp}}$ is a linking\n\tfrom $T$ to $B$ in $D^{\\mathrm{opp}}$ which routes $T\\backslash B$ to $B\\backslash T$. The special property of $D$, that $E\\backslash T$ consists of sources and that $T$ consists of sinks,\n\timplies, that for all $p\\in L$, we have $\\left| p \\right|\\cap E = \\SET{p_1,p_{-1}}$.\n\tObserve that thus\n $$R = \\SET{p\\in L^{\\mathrm{opp}} \\mid p_{1}\\in T \\backslash B} \\cup \\SET{x\\in {\\mathbf{P}}(D^{\\mathrm{opp}})\\mid x\\in E\\backslash \\left( T\\cup B \\right)}$$\n\tis a linking from $E\\backslash B=\\left( T\\mathbin{\\dot{\\cup}} \\left( E\\backslash T \\right) \\right)\\backslash B$ onto $E\\backslash T$ in $D^{\\mathrm{opp}}$, thus $E\\backslash B$ is a base of $N$. An analog argument yields that for every base $B'$ of $N$,\n\t$E\\backslash B'$ is a base of $M$. Therefore $\\Gamma(D^{\\mathrm{opp}}, E\\backslash T, E) = \\left( \\Gamma(D,T,E) \\right)^\\ast$.\n\\end{proof}\n\n\\begin{corollary}\\label{cor:dualityrespectingrepresentation}\\PRFR{Jan 22nd}\n\tLet $M=(E,{\\mathcal{I}})$ a gammoid. Then there is a duality respecting representation $(D,T,E)$ with\n\t$\\Gamma(D,T,E) = M$. Consequently, $M^\\ast$ is a gammoid if and only if $M$ is a gammoid.\n\\end{corollary}\n\\begin{proof}\n\tImmediate consequence of Lemmas~\\ref{lem:sourcesinkrepresentation} and \\ref{lem:dualityrespectingrepresentation}.\n\\end{proof}\n\n\n\n\n\\PRFR{Mar 7th}\n\\noindent Unfortunately, the property of a representation to be duality respecting is not preserved by the digraph pivot operation.\nThus we cannot take a duality respecting representation, pivot in a base as in the proof of Theorem~\\ref{thm:gammoidRepresentationWithBaseTerminals} and then expect that the resulting representation is still duality respecting.\n\n\\needspace{9\\baselineskip}\n\n\\vspace*{-\\baselineskip}\n\\begin{wrapfigure}{r}{5cm}\n\\vspace{1.6\\baselineskip}\n\\vspace*{-\\baselineskip}\n\\begin{centering\n\\includegraphics[width=5cm]{pivotnotcompatiblewithdualityrespecting}\n\\end{centering}%\n\\vspace*{-2\\baselineskip}\n\\end{wrapfigure}\n~\n \n\n\\begin{longexample}\\label{ex:pivotBreaksDualityRespecting}\\PRFR{Mar 7th}\nConsider the gammoid $M$ on the ground set $E=\\dSET{a,b,c,d,e,f,g,h}$ represented by the digraph $D=(V,A)$ with the vertex set\n $V=E\\mathbin{\\dot{\\cup}}\\dSET{x,y}$\nand the arcs $A = $ $\\left( \\SET{e,f,g,h}\\times\\SET{x,y} \\right) \\cup\\left( \\SET{x}\\times\\SET{a,b,d} \\right) \\cup \\left( \\SET{y}\\times \\SET{a,c,d} \\right)$\n together with the target set $T=\\SET{a,b,c,d}$,\ni.e. we have $M=\\Gamma(D,T,E)$. The bases of $M$ are the set $T=\\SET{a,b,c,d}$, the sets of the form $X\\cup\\SET{y}$ where\n$X\\subseteq T$ with $\\left| X \\right| = 3$ and $y\\in\\SET{e,f,g,h}$, and the sets of the form $X\\cup Y$ where $X\\subseteq T$ with\n$\\left| X \\right|= 2$ and $Y\\subseteq \\SET{e,f,g,h}$ with $\\left| Y \\right| = 2$. Clearly, $(D,T,E)$ is duality respecting (Lemma~\\ref{lem:dualityrespectingrepresentation}). Observe that there is only\none routing that links $\\SET{a,b,c,h}$ to $T$ -- up to symmetries of $D$ that stabilize $E$ -- namely $R=\\SET{a,b,c,hxd} \\subseteq {\\mathbf{P}}(D)$.\n\\end{longexample}\n\n\\needspace{9\\baselineskip}\n\n\\vspace*{-\\baselineskip}\n\\begin{wrapfigure}{l}{5cm}\n\\vspace{1.6\\baselineskip}\n\\vspace*{-\\baselineskip}\n\\begin{centering\n\\includegraphics[width=5cm]{pivotnotcompatiblewithdualityrespecting2}\n\\end{centering}%\n\\vspace*{-2\\baselineskip}\n\\end{wrapfigure}\n~\n \n\n\\noindent If we use the routing $R$ together with the procedure \ndescribed in the proof of Theorem~\\ref{thm:gammoidRepresentationWithBaseTerminals}, \nwe obtain the digraph $D'$ depicted to the left. The vertex $h$ is now a sink in $D'$, but $d$ is not a source in $D'$, \ntherefore Lemma~\\ref{lem:dualityrespectingrepresentation} is not applicable to $D'$.\nThere are two routings that link the base $B' = \\SET{a,b,e,h}$ to the target set $T'=\\SET{a,b,c,h}$ in $D'$,\n$R_1' = \\SET{a,b,exdc,h}$ and $R_2' = \\SET{a,b,eyxdc,h}$. Therefore every routing from $B$ to $T'$\nuses the vertex $d$ as an inner vertex of some path, so the construction from the proof of \nLemma~\\ref{lem:dualityrespectingrepresentation} breaks at this point.\n Let $B^\\ast = E\\backslash B' = \\SET{c,d,f,g}$ and $T^\\ast = E\\backslash T' = \\SET{d,e,f,g}$. There is no routing from $B^\\ast$ to $T^\\ast$ in $\\left( D' \\right)^{\\mathrm{opp}}$ because $c$ can only be linked to $d$, and \n therefore $\\mathrm{rk}_{\\Gamma\\left( \\left( D' \\right)^{\\mathrm{opp}}, T^\\ast, E \\right)}\\left( \\SET{c,d} \\right) = 1$, thus $B^\\ast$ is \n not independent in $\\Gamma\\left( \\left( D' \\right)^{\\mathrm{opp}}, T^\\ast, E \\right)$. Consequently, $(D',T',E)$ is not a\n duality respecting representation of $M$. There are two obvious ways to modify $D'$ such that the resulting digraph is again duality \n respecting, but both methods introduce another arc. If we would like to use Lemma~\\ref{lem:dualityrespectingrepresentation} as it is \n stated, we could rename $d$ with $x$, add a new $d$-vertex and the arc $(d,x)$ to $D'$, effectively forcing $d$ to be a source again.\n Or we could add the arc $(x,c)$ to $D'$ --- which corresponds to adding the arc $(x,c)$ to $D$ --- then\n $d$ is no longer on any essential path from $x$ to any $t\\in T'$. This would imply that for every $X\\subseteq E$ and\n every routing from $X$ to $T'$\n that uses $d$ as\n an inner vertex there is a routing $R_{-d}$ from $X$ to $T'$ that omits $d$ entirely. This routing $R_{-d}$ could be used\n in the construction from the proof of Lemma~\\ref{lem:dualityrespectingrepresentation}, which yields a routing $R^\\ast$\n linking $E\\backslash X$ to $T^\\ast$\n in the opposite digraph.\n\n\\noindent ~\n \\strut\\hfill $\\blacktriangle$\n\n\n\\subsection{Complexity-Bounded Classes of Gammoids}\n\n\\PRFR{Mar 27th}\n\\noindent\nIn this section, we introduce three measures of complexity for gammoids that are related to a class of certain\nrepresentations, and examine the corresponding classes of matroids\nwith a bounded complexity measure.\n\n\n\\begin{definition}\\label{def:standardRepresentation}\\PRFR{Mar 27th}\n\tLet $M$ be a gammoid and $(D,T,E)$ with $D=(V,A)$ be a representation of $M$.\n\tThen $(D,T,E)$ is a \\deftext[standard representation of a gammoid]{standard representation of $\\bm M$},\n\tif $(D,T,E)$ is a duality respecting representation, $T\\subseteq E$, every $t\\in T$ is a sink in $D$,\n\tand every $e\\in E\\backslash T$ is a source in $D$.\n\\end{definition}\n\n\\begin{remark}\\PRFR{Mar 27th}\\label{rem:standardRepresentation}\n\tLemmas~\\ref{lem:sourcesinkrepresentation} and \\ref{lem:dualityrespectingrepresentation} guarantee that every gammoid $M$ has\n\ta standard representation.\n\\end{remark}\n\n\\begin{definition}\\PRFR{Mar 27th}\n\tLet $M$ be a gammoid. The \\deftext[arc-complexity of a gammoid]{arc-complexity of $\\bm M$}\n\tis defined to be \\label{n:ArcCompl}\n\t\\[ \\mathrm{C}_A(M) = \\min\\SET{\\vphantom{A^A} {\\left| A \\right|} ~\\middle|~ \\left( (V,A), T, E \\right)\\text{~is a standard representation of~}M}.\\]\n\tThe \\deftext[vertex-complexity of a gammoid]{vertex-complexity of $\\bm M$}\n\tis defined to be \\label{n:VertexCompl}\n\t\\[ \\mathrm{C}_V(M) = \\min\\SET{\\vphantom{A^A} {\\left| V \\right|} ~\\middle|~ \\left( (V,A), T, E \\right)\\text{~is a standard representation of~}M}.\n\t\\qedhere \\]\n\\end{definition}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{lemma}\\label{lem:arcCkVDualityAndMinors}\\PRFR{Mar 27th}\n\tLet $M=(E,{\\mathcal{I}})$ be a gammoid, $X\\subseteq E$. Then\n\t\\[ \\mathrm{C}_A(M| X) \\leq \\mathrm{C}_A(M), \\quad \\mathrm{C}_A(M|' X) \\leq \\mathrm{C}_A(M), \\quad \\mathrm{C}_A(M) = \\mathrm{C}_A(M^\\ast)\\]\n as well as\n \\[ \\mathrm{C}_V(M| X) \\leq \\mathrm{C}_V(M), \\quad \\mathrm{C}_V(M|' X) \\leq \\mathrm{C}_V(M), \\quad \\mathrm{C}_V(M) = \\mathrm{C}_V(M^\\ast).\\]\n\\end{lemma}\n\\begin{proof}\\PRFR{Mar 27th}\n\tLet $(D,T,E)$ be a standard representation of $M$. Then $(D^{\\mathrm{opp}},E\\backslash T, E)$ is a standard representation of $M^\\ast$:\n\tBy Definition~\\ref{def:dualityRespectingRepr} we have\n\t $M^\\ast = \\Gamma(D^{\\mathrm{opp}}, E\\backslash T, E)$, and since every sink of $D$ is a source of $D^{\\mathrm{opp}}$\n\tand every source of $D$ is a sink of $D^{\\mathrm{opp}}$, the set $E\\backslash T$ consists of sinks of $D^{\\mathrm{opp}}$, \n\tand the set $T= E\\backslash\\left( E\\backslash T \\right)$ consists of sources of $D^{\\mathrm{opp}}$.\n\tConsequently, $\\mathrm{C}_A(M^\\ast) \\leq \\mathrm{C}_A(M)$ and $\\mathrm{C}_V(M^\\ast) \\leq \\mathrm{C}_V(M)$. It follows that $\\mathrm{C}_A(M) = \\mathrm{C}_A(M^\\ast)$,\n\tas well as $\\mathrm{C}_V(M) = \\mathrm{C}_V(M^\\ast)$, \n\tsince $M=\\left( M^\\ast \\right)^\\ast$ (Corollary~\\ref{cor:doubleast}).\n\n\t\\noindent\n\tNow let $(D,T,E)$ be a standard representation of $M$ where $D=(V,A)$ such that $\\left| A \\right| = \\mathrm{C}_A(M)$.\n\tIf $T\\subseteq X$, then $(D,T,X)$ is a standard representation of $M| X$ and therefore\n\t$\\mathrm{C}_A(M| X) \\leq \\mathrm{C}_A(M)$.\n\tOtherwise let $Y = T\\backslash X$, and let $B_0\\subseteq X$ be a set of maximal cardinality such that\n\tthere is a routing $R_0\\colon B_0\\double{\\rightarrow} Y$ in $D$. Let $D'=(V,A')$ be the digraph that arises from $D$ \n\tby a sequence of pivot operations as they are described in the proof of Theorem~\\ref{thm:gammoidRepresentationWithBaseTerminals}\n\twith respect to the routing $R_0$.\n\tObserve that every $b\\in B_0$ is a sink in $D'$ and that $\\left| A' \\right| = \\left| A \\right|$.\n\tWe argue that $(D',(T\\cap X) \\cup B_0, X)$ is a standard representation of $M| X$:\n\tLet $Y_0 = \\SET{p_{-1}~\\middle|~ p\\in R_0}$ be the set of targets that are entered by the routing $R_0$.\n\tIt follows from the proof of Theorem~\\ref{thm:gammoidRepresentationWithBaseTerminals} that the triple\n\t$(D',(T\\cap X) \\cup B_0 \\cup \\left( Y\\backslash Y_0 \\right), E)$ is a representation of $M$. The chain of pivot operations \n\twe carried out on $D$ preserves all those sources and sinks of $D$, which are not visited by a path $p\\in R_0$.\n\tSo we obtain that every $e\\in E\\backslash\\left( T\\cup B_0 \\right)$ is a source in $D'$, and that every $t\\in T\\cap X$\n\tis a sink in $D'$. Thus the set $T' = (T \\cap X)\\cup B_0$ consists of sinks in $D'$, and the set\n\t$X\\backslash T' \\subseteq E\\backslash\\left( T\\cup B_0 \\right)$ consists of sources in $D'$. Therefore $(D',(T\\cap X) \\cup B_0, X)$ is a standard\n\trepresentation, and we give an indirect argument that $(D',(T\\cap X) \\cup B_0, X)$ represents $M| X$.\n\tClearly, $(D',(T\\cap X) \\cup B_0 \\cup \\left( Y\\backslash Y_0 \\right), X)$ is a representation of $M| X$.\n\tSince we assume that $(D',(T\\cap X) \\cup B_0, X)$ does not represent $M| X$, there must be a set $X_0\\subseteq X$\n\tsuch that there is a routing $Q_0\\colon X_0\\double{\\rightarrow} (T\\cap X) \\cup B_0 \\cup \\left( Y\\backslash Y_0 \\right)$ and such that\n\tthere is no routing $X_0\\double{\\rightarrow} (T\\cap X) \\cup B_0$. Thus there is a path $q\\in Q_0$ with $q_{-1} \\in Y\\backslash Y_0$\n\tand $q_{1}\\in X$. Consequently we have a routing $Q'_1 = \\SET{q}\\cup\\SET{b\\in {\\mathbf{P}}(D') ~\\middle|~ b\\in B_0}$ in $D'$.\n\tThis implies that there is a routing $B_0\\cup\\SET{q_{1}} \\double{\\rightarrow} Y$ in $D$, a contradiction to the maximal cardinality of the choice of $B_0$\n\tabove. Thus our assumption is wrong and $(D',(T\\cap X) \\cup B_0, X)$ is a standard representation of $M| X$, so\n\t$\\mathrm{C}_A(M| X) \\leq \\mathrm{C}_A(M)$ holds. \n\t%\n\n\tFinally, let $(D,T,E)$ be a standard representation of $M$ with $D=(V,A)$ such that $\\left| V \\right| = \\mathrm{C}_V(M)$. By an analogue argument \n\twe obtain that $\\mathrm{C}_V(M| X) \\leq \\mathrm{C}_V(M)$ holds.\n\tThe previous results combined with Lemma~\\ref{lem:restrictcontractdual} yield that the dual inequalities \\linebreak $\\mathrm{C}_A(M|' X) \\leq \\mathrm{C}_A(M)$ and $\\mathrm{C}_V(M|' X) \\leq \\mathrm{C}_V(M)$ hold, too.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:kVgeqE}\\PRFR{Mar 27th}\n\tLet $M=(E,{\\mathcal{I}})$ be a gammoid.\n\tThen $\\mathrm{C}_V(M) \\geq \\left| E \\right|$.\n\\end{lemma}\n\\begin{proof}\\PRFR{Mar 27th}\n\tClear, since $E\\subseteq V$ for every representation $(D,T,E)$ with $D=(V,A)$.\n\\end{proof}\n\n\n\\begin{remark}\\label{rem:vKeasy}\\PRFR{Mar 27th}\n\tLet $k\\in \\mathbb{N}$. Clearly, the class of gammoids $M$ with $\\mathrm{C}_V(M)\\leq k$ is closed under duality and arbitrary minors,\n\tbut Lemma~\\ref{lem:kVgeqE} shows that this class has only a finite number of pair-wise non-isomorphic matroids. Thus such a class of gammoids is trivially characterized\n\tby a finite number of excluded minors,\n\tbecause there are only finitely many non-isomorphic matroids with $k+1$ elements.\n\\end{remark}\n\n\n\\begin{lemma}\\label{lem:arcCSubAdditive}\\PRFR{Mar 27th}\n\tLet $M=(E,{\\mathcal{I}})$ and $N =(E',{\\mathcal{I}}')$ be gammoids with $E\\cap E' = \\emptyset$.\n\tThen $M\\oplus N$ is a gammoid,\n\t\\[ \\mathrm{C}_A(M\\oplus N) \\leq \\mathrm{C}_A(M) + \\mathrm{C}_A(N), {\\mathop{\\text{~and~}}}\n\t \\mathrm{C}_V(M\\oplus N) \\leq \\mathrm{C}_V(M) + \\mathrm{C}_V(N). \\]\n\\end{lemma}\n\\begin{proof}\\PRFR{Mar 27th}\n\tLet $(D,T,E)$ and $(D',T',E')$ be standard representations of $M$ and $N$, respectively, such that $D =(V,A)$ and $D'=(V',A')$\n\twith $\\left| A \\right| = \\mathrm{C}_A(M)$ and $\\left| A' \\right| = \\mathrm{C}_A(N)$, and\n\tsuch that $V\\cap V'=\\emptyset$.\n\tLet $D_\\oplus = (V\\mathbin{\\dot{\\cup}} V', A\\mathbin{\\dot{\\cup}} A')$. Then \\mbox{$M\\oplus N = \\Gamma(D_\\oplus, T\\mathbin{\\dot{\\cup}} T', E\\mathbin{\\dot{\\cup}} E')$}\n\tbecause there are no arcs in $D_\\oplus$ connecting vertices from $V$ with $V'$ or vice versa.\n\tThus every routing $R_\\oplus\\colon X_\\oplus \\double{\\rightarrow} T\\mathbin{\\dot{\\cup}} T'$ in $D_\\oplus$ is the disjoint union of the routings\n\t\\linebreak $R = \\SET{p\\in R_\\oplus ~\\middle|~ \\left| p \\right|\\subseteq V}$ \n\tand $R' = \\SET{p'\\in R_\\oplus ~\\middle|~ \\vphantom{A^A}\\left| p' \\right|\\subseteq V'}$, and conversely every pair of routings\n\t$R\\colon X\\double{\\rightarrow} T$ and $R'\\colon X'\\double{\\rightarrow} T'$ yields a routing $R_\\oplus = R\\mathbin{\\dot{\\cup}} R'$ since $V\\cap V'=\\emptyset$.\n\tThus a set $X\\subseteq E\\mathbin{\\dot{\\cup}} E'$ is independent in $\\Gamma(D_\\oplus, T\\mathbin{\\dot{\\cup}} T', E\\mathbin{\\dot{\\cup}} E')$ if and only if \n\t$X\\cap E$ is independent in $M$ and $X\\cap E'$ is independent in $N$. \\mbox{$(D_\\oplus, T\\mathbin{\\dot{\\cup}} T', E\\mathbin{\\dot{\\cup}} E')$} is a \n\tstandard representation of $M\\oplus N$ with $\\mathrm{C}_A(M) + \\mathrm{C}_A(N)$ arcs, therefore\n\t $$\\mathrm{C}_A(M\\oplus N) \\leq \\mathrm{C}_A(M) + \\mathrm{C}_A(N)$$ holds.\n\tThe same construction applied to representations $(D,T,E)$ and $(D',T',E')$ with $D=(V,A)$, $D'=(V',A')$,\n\t$V\\cap V'=\\emptyset$, $\\left| V \\right| = \\mathrm{C}_V(M)$, and $\\left| V' \\right| = \\mathrm{C}_V(N)$ yields that \\(\\mathrm{C}_V(M\\oplus N) \\leq \\mathrm{C}_V(M) + \\mathrm{C}_V(N)\\)\n\tholds, too.\n\\end{proof}\n\n\\begin{corollary}\\label{cor:extWithLoopCoLoop}\\PRFR{Mar 27th}\n\tLet $M=(E,{\\mathcal{I}})$ be a gammoid, $F$ and $L$ finite sets such that $F\\cap L = E\\cap F = E\\cap L = \\emptyset$.\n\tThen $M\\oplus (F,2^F) \\oplus (L,\\SET{\\emptyset})$ is a gammoid and\n\t\\[ \\mathrm{C}_A\\left( M\\oplus (F,2^F) \\oplus (L,\\SET{\\emptyset}) \\right) = \\mathrm{C}_A(M) \\]\n\\end{corollary}\n\\begin{proof}\\PRFR{Mar 27th}\nThis is a direct consequence of Lemma~\\ref{lem:arcCSubAdditive}\nand the fact that \\[\\Gamma((F,\\emptyset),F,F) = (F,2^F) \\,\\,{\\mathop{\\text{~and~}}}\\,\\,\\Gamma((L,\\emptyset),\\emptyset,L) = (L,\\SET{\\emptyset}).\n\\qedhere \\]\n\\end{proof}\n\n\n\n\\begin{lemma}\\label{lem:disunionArcCZero}\\PRFR{Mar 27th}\n\tLet $M=(E,{\\mathcal{I}})$ be a gammoid with $\\mathrm{C}_A(M) = 0$.\n\tThen there is a subset $X\\subseteq E$ such that\n\t\t\\[ M = (X,2^X) \\oplus (E\\backslash X, \\SET{\\emptyset}).\\]\n\\end{lemma}\n\\begin{proof}\\PRFR{Mar 27th}\n\tSince there is a representation $(D,T,E)$ of $M$ with $D = (V,\\emptyset)$, we obtain that the sets $X\\subseteq E$ that\n\tare linked to $T$ are precisely the subsets of $T$. An element of $E\\backslash T$ can never be linked to $T$ since ${\\mathbf{P}}(D)$ only consists of\n\ttrivial paths. Thus ${\\mathcal{I}} = 2^T$ and obviously $M = (T,2^T) \\oplus (E\\backslash T, \\SET{\\emptyset})$.\n\\end{proof}\n\n\\begin{theorem}\\PRFR{Mar 27th}\n\tLet ${\\mathcal{G}}_0$ be the class of gammoids $M$ with $\\mathrm{C}_A(M) = 0$. Then ${\\mathcal{G}}_0$ is closed under duality, minors, and direct sums;\n\tand ${\\mathcal{G}}_0$ is characterized by the excluded minor $U=(E,2^E\\BSET{E})$ with $E=\\dSET{a,b}$.\n\\end{theorem}\n\\begin{proof}\\PRFR{Mar 27th}\n\tLemma~\\ref{lem:arcCkVDualityAndMinors} yields that ${\\mathcal{G}}_0$ is closed under duality and minors.\n\tLet $M_1,M_2\\in {\\mathcal{G}}_0$ with disjoint ground sets.\n\tBy Lemma~\\ref{lem:arcCSubAdditive} we have $$\\mathrm{C}_A(M_1\\oplus M_2) \\leq \\mathrm{C}_A(M_1) + \\mathrm{C}_A(M_2) = 0,$$\n\tso $\\mathrm{C}_A(M_1\\oplus M_2) = 0$, thus ${\\mathcal{G}}_0$ is closed under direct sums.\n\n\n\n\n\n\n\n\n\n\n\t\\noindent\n\tNow let $X\\subsetneq E$, then $U|' X = (X,\\SET{\\emptyset})$ and $U| X = (X,\\SET{X,\\emptyset})$. \n\tClearly, $\\mathrm{C}_A(U|' X) = 0$ and $\\mathrm{C}_A(U| X) = 0$. Thus every proper minor of $U$ is in ${\\mathcal{G}}_0$.\n\tNow let $M\\in {\\mathcal{G}}_0$, then $M=(F,2^F)\\oplus(L,\\SET{\\emptyset})$ for some finite sets $F$ and $L$.\n\tTherefore ${\\mathcal{C}}(M) = \\SET{\\SET{l} ~\\middle|~ l\\in L}$, so every circuit of a matroid $M\\in {\\mathcal{G}}_0$ has cardinality $1$.\n\tBut ${\\mathcal{C}}(U) = \\SET{\\SET{a,b}}$, thus $U\\notin {\\mathcal{G}}_0$.\n\tNow let $M=(Q,{\\mathcal{I}})$ be any matroid. If there is some $C\\in{\\mathcal{C}}(M)$ with $\\left| C \\right| > 1$, then $M\\notin{\\mathcal{G}}_0$\n\tand\t$M| C = (C,2^C\\BSET{C})$ is a uniform matroid. Now, let $c_1,c_2\\in C$ with $c_1\\not= c_2$,\n\tthen $(M| C)|' \\SET{c_1,c_2} = (\\SET{c_1,c_2},\\SET{\\emptyset,\\SET{c_1},\\SET{c_2}})$ is a rank-$1$ uniform matroid on a\n\t $2$-elementary ground set. Therefore $(M| C)|' \\SET{c_1,c_2}$ is a minor of $M$ that is isomorphic to $U$. \n\tIf there is no $C\\in {\\mathcal{C}}(M)$ with $\\left| C \\right| > 1$,\n\t then let $L_Q = \\SET{q\\in Q ~\\middle|~ \\SET{q}\\in{\\mathcal{C}}(M)}$ and $F_Q = Q\\backslash L$.\n\tClearly, $M = (F_Q,2^{F_Q})\\oplus (L_Q,\\SET{\\emptyset})$ and it is easy to see that $\\mathrm{C}_A(M) = 0$, thus $M\\in{\\mathcal{G}}_0$.\n\tTherefore the class ${\\mathcal{G}}_0$ is characterized by the single excluded minor $U$.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:oplusReprOfGk}\\PRFR{Mar 27th}\n\tLet $k\\in \\mathbb{N}$ and $M=(E,{\\mathcal{I}})$ be a gammoid with $\\mathrm{C}_A(M) = k$.\n\tThen there is a partition $E_1\\mathbin{\\dot{\\cup}} E_2\\mathbin{\\dot{\\cup}} E_3$ of $E$ such that\n\t\\[ M = \\left( M| E_1 \\right) \\oplus (E_2,2^{E_2}) \\oplus (E_3,\\SET{\\emptyset}),\\]\n\tand such that $\\mathrm{C}_A(M| E_1) = k$.\n\tFurthermore, $\\left| E_1 \\right| \\leq 2k$, $\\mathrm{rk}_M(E_1) \\leq k$,\n\t and there is a set $X_0\\subseteq E_1$ with cardinality at most $\\mathrm{rk}_M(E_1)$\n\tsuch that for every $X\\subsetneq E_1$ with $X_0\\subseteq X$\n\t\\[ \\mathrm{C}_A(M| X) < k .\\]\n\\end{lemma}\n\\begin{proof}\\PRFR{Mar 27th}\n\tLet $(D,T,E)$ be a standard representation of $M$ with $D=(V,A)$ and $\\left| A \\right| = k$.\n\tWe may partition $V$ into $V_1 = \\SET{v\\in V ~\\middle|~ \\exists u\\in V\\colon\\,(u,v)\\in A {\\mathop{\\text{~or~}}} (v,u)\\in A}$,\n\tthe set of vertices incident with an arc,\n\tand $V_2 = V\\backslash V_1$, the set of isolated vertices. \n\tThere are no arcs in the induced digraph $D'=(V_2,A\\cap \\left( V_2\\times V_2 \\right))=(V_2,\\emptyset)$,\n\tthus we obtain that \n\t\\[ M' = M| \\left( E\\cap V_2 \\right) = \\Gamma(D,T,E\\cap V_2) = \\Gamma(D',T\\cap V_2, E\\cap V_2) \\] and consequently\n\twe have $\\mathrm{C}_A(M| V_2) = 0$. Therefore there are disjoint $F,L\\subseteq V_2$ such that\n\t$M| V_2 = (F,2^F)\\oplus (L,\\SET{\\emptyset})$ (Lemma~\\ref{lem:disunionArcCZero}).\n\tNow let $X\\subseteq V_1 \\cap E$ with $X\\in{\\mathcal{I}}$.\n\tThen there is a routing $R\\colon X\\double{\\rightarrow} T$ in $D$,\n\tand since no arc of $D$ is incident with $v\\in V_2$, we obtain that $\\left| p \\right|\\subseteq V_1$ for all $p\\in R$.\n\tTherefore we may conclude that $X$ is independent in $M'' = \\Gamma(D'',T\\cap V_1, E\\cap V_1)$ for\n\t$D'' = (V_1,A)$, thus $M'' = M| \\left( E\\cap V_1 \\right)$. Therefore, for all $X\\subseteq E$ with $X\\in{\\mathcal{I}}$, \n\twe have that $X\\cap V_1$ is independent in $M''$,\n\tand that $X\\cap V_2$ is independent in $M'$. Thus\n\t\\[ M = M''\\oplus M' = \\left(\\vphantom{2^{T\\cap V_2}} M| \\left( E\\cap V_1 \\right) \\right)\n\t\t\\oplus \\left( T\\cap V_2, 2^{T\\cap V_2} \\right) \\oplus (\\vphantom{2^{T\\cap V_2}}V_2\\backslash T, \\SET{\\emptyset}).\\]\n\tAssume that $\\mathrm{C}_A(M| \\left( E \\cap V_1 \\right)) < \\mathrm{C}_A(M)$, then we could take a standard representation of\n\t$M| \\left( E\\cap V_1 \\right)$ and augment it with isolated vertices in order to obtain a standard representation of $M$ \n\twith fewer than $\\mathrm{C}_A(M)$ arcs --- yielding a contradiction. Therefore $\\mathrm{C}_A(M| \\left( E \\cap V_1 \\right)) = \\mathrm{C}_A(M)$.\n\tSince every element of $E_1=E\\cap V_1$ must be incident with at least one arc in $D$, and every arc is incident with two vertices,\n\tand since $\\left| A \\right|=k$, we obtain that $\\left| E_1 \\right| \\leq \\left| V_1 \\right| \\leq 2k$. Furthermore,\n\tevery arc in $D$ is incident with at most one source and at most one sink, thus $\\left| T\\cap V_1 \\right| \\leq k$,\n\tand therefore $\\mathrm{rk}_M(E_1) \\leq k$.\n\n\t\\noindent\n\tNow we show that $\\mathrm{C}_A(M| X) < \\mathrm{C}_A(M)$ holds for every $X\\subsetneq E_1$ with $T\\cap V_1 \\subseteq X$ \n\tby constructing a smaller representation.\n\tLet $X\\subsetneq E_1$ such that $\\mathrm{C}_A(M| X) = \\mathrm{C}_A(M)$ and $T\\cap V_1 \\subseteq X$, \n\tand let $x\\in E_1\\backslash X$.\n\n\tSince $x\\notin T\\cap V_1$ we have\n\t $x\\notin T$. Let\n\t $$D_x=\\left(\\vphantom{A^A}V_1\\BSET{x},A\\cap\\left(\\left( V_1\\BSET{x} \\right)\\times \\left( V_1\\BSET{x} \\right)\\right)\\right)$$\n\tbe the digraph induced from $D$ by removing the source $x$.\n\tClearly, $D_x$ has fewer arcs than $D$ because at least one arc in $D$ is incident with $x$.\n\tBut then the contraction of $M$ to $X$ satisfies the equation $M| X = \\Gamma(D_x,T,X)$, \n\twhich implies $\\mathrm{C}_A(M| X) < \\mathrm{C}_A(M)$.\n\\end{proof}\n\n\n\n\\begin{theorem}\\label{thm:arcCquiteEasy}\\PRFR{Mar 27th}\n\tLet $k\\in \\mathbb{N}$, $k \\geq 1$, and let ${\\mathcal{G}}_k$ be the class of gammoids $M$ with $\\mathrm{C}_A(M) \\leq k$. \n\tThen ${\\mathcal{G}}_k$ is closed under duality and minors, but not under direct sums;\n\tand ${\\mathcal{G}}_k$ is characterized by finitely many excluded minors.\n\\end{theorem}\n\\begin{proof}\\PRFR{Mar 27th}\n\tLet $k\\in \\mathbb{N}$ be arbitrarily fixed from now on.\n\tLemma~\\ref{lem:arcCkVDualityAndMinors} yields that ${\\mathcal{G}}_k$ is closed under duality and minors.\n\tNow let $M_i = (\\dSET{a_i,b_i},\\SET{\\emptyset,\\SET{a_i},\\SET{b_i}})$ for\n\t\\linebreak\n\t $i\\in \\SET{1,2,\\ldots,k+1}$,\n\tsuch that $\\dSET{a_i,b_i}\\cap \\dSET{a_j,b_j}= \\emptyset$ for all $i,j\\in \\SET{1,2,\\ldots,k+1}$ with $i\\not= j$.\n\tThen $\\mathrm{C}_A(M_i) = 1$, because $M_i$ is neither free nor does $M_i$ consist of loops, and it can be represented\n\tby $((\\SET{a_i,b_i},\\SET{(a_i,b_i)}),\\SET{b_i},\\SET{a_i,b_i})$. \n\tNow let $N = \\bigoplus_{i=1}^{k+1} M_i$, and let $(D,T,E)$ be a standard representation of $N$ with $D=(V,A)$. \n\tThen $\\left| T \\right| = \\mathrm{rk}_N(E) = k+1$ and $\\left| E \\right| = 2k +2$. Now assume that $\\left| A \\right| \\leq k$,\n\ti.e. that $N\\in{\\mathcal{G}}_k$. There is some $e\\in E\\backslash T$ such that $e$ is not incident with an arc from $A$, thus $\\SET{e}$ cannot\n\tbe linked to $T$ in $D$.\n\tBut then $\\mathrm{rk}_N(\\SET{e}) = 0$ follows, which is a contradiction to the fact that $\\mathrm{rk}_{M_i}(\\SET{e})= 1$ for the appropriate\n\tindex $i$. Thus $N\\notin {\\mathcal{G}}_k$, and consequently, ${\\mathcal{G}}_k$ is not closed under direct sums.\n\n\t\\noindent\n\tNow let $M =(E,{\\mathcal{I}})$ be a matroid.\n\tIf $M\\in{\\mathcal{G}}_k$, then Lemma~\\ref{lem:oplusReprOfGk} yields that there is a partition $E_1\\mathbin{\\dot{\\cup}} E_2\\mathbin{\\dot{\\cup}} E_3=E$\n\twith\\[ M = \\left( M| E_1 \\right) \\oplus (E_2,2^{E_2}) \\oplus (E_3,\\SET{\\emptyset}) \\]\n\tand $\\left| E_1 \\right| \\leq 2k$ such that $\\mathrm{C}_A(M| E_1) \\leq k$.\n\n\n\tNow let $M=(E,{\\mathcal{I}})$ be an excluded minor for ${\\mathcal{G}}_k$. Then for all $e\\in E$ the restriction\n\t$M|\\left( E\\BSET{e} \\right)\\in {\\mathcal{G}}_k$. Thus Lemma~\\ref{lem:arcCSubAdditive}\n\t yields that for all $e\\in E$\n\t\\[ M|\\left( E\\BSET{e} \\right)\\oplus(\\SET{e},\\SET{\\emptyset,\\SET{e}}) \n\t\\not= M \\not= M|\\left( E\\BSET{e} \\right)\\oplus(\\SET{e},\\SET{\\emptyset}), \\]\n\ti.e. $M$ has neither a loop nor a coloop. In this case, \n\tLemma~\\ref{lem:oplusReprOfGk} implies that $\\left| E\\BSET{e} \\right| \\leq 2k$,\n\tso $\\left| E \\right| \\leq 2k+ 1$, thus every excluded minor for ${\\mathcal{G}}_k$ has at most $2k+1$ elements. But up to isomorphism, \n\tthere are only finitely many\n\tmatroids on ground sets with at most $2k+1$ elements, so ${\\mathcal{G}}_k$ is characterized by finitely many excluded minors.\n\\end{proof}\n\n\\noindent We have seen that subclasses of gammoids, that are defined by limiting the number of arcs or the number of vertices \navailable in a standard representation, merely consist of a finite number of matroids which may be extended with an arbitrary amount of\nloops and coloops. Moreover, except for ${\\mathcal{G}}_0$, those classes are not closed under direct sums.\n\n\\needspace{6\\baselineskip}\n\\begin{definition}\\label{def:arcWf}\\PRFR{Mar 27th}\n\tLet $f\\colon \\mathbb{N}\\longrightarrow \\mathbb{N}\\BSET{0}$ be a non-decreasing function, and let\n\t\\linebreak\n\t $M=(E,{\\mathcal{I}})$ be a gammoid. The \\deftext[width of a gammoid]{$\\bm f$-width of $\\bm M$}\n\tshall be \\label{n:arcWfM}\n\t\\[ \\mathrm{W}_f(M) = \\max\\SET{\\frac{\\mathrm{C}_A\\left( \\left( M|' Y \\right)| X \\right))}{f\\left( \\left| X \\right|\n\n\t \\right) }\n\t\t ~\\middle|~ X\\subseteq Y\\subseteq E}. \\]\n\tLet $k\\in \\mathbb{N}$, then the \\deftextX{$\\bm k$-width of $\\bm M$} shall be\n\t\\[ \\mathrm{W}^k(M) = \\mathrm{W}_{f_k}(M) \\] where \n\t\\[ f_k\\colon \\mathbb{N}\\longrightarrow \\mathbb{N}\\BSET{0},\\,n\\mapsto \\max\\SET{1, k\\cdot n} . \\qedhere\\]\n\\end{definition}\n\n\\noindent Clearly $\\mathrm{W}^{0}(M) = \\mathrm{C}_A(M)$ for all gammoids $M$.\n\n\\begin{corollary}\\label{cor:WfClosedUnderMinors}\\PRFR{Mar 27th}\n\tLet $M=(E,{\\mathcal{I}})$ be a gammoid, $X\\subseteq Y\\subseteq E$.\n\tThen \\[ \\mathrm{W}_f(M) = \\mathrm{W}_f(M^\\ast) \\,\\,{\\mathop{\\text{~and~}}}\\,\\, \\mathrm{W}_f\\left( \\left( M|' Y \\right)| X \\right) \\leq \\mathrm{W}_f(M) .\\]\n\\end{corollary}\n\n\\begin{proof}\\PRFR{Mar 27th}\n\tThe second inequality is a direct consequence of the Definition~\\ref{def:arcWf}.\n\t\tLet $M=(E,{\\mathcal{I}})$ be a gammoid and $X\\subseteq Y\\subseteq E$,\n\t then \n\t \\[ (M^\\ast|' Y)| X = \\left( (M| Y)|' X \\right)^\\ast = \\left( (M|' E\\backslash\\left( Y\\backslash X \\right))| X \\right)^\\ast \\]\n\t holds due to Lemmas~\\ref{lem:restrictcontractdual} and \\ref{lem:contractrestrictcommutes}, \n\t and Remark~\\ref{rem:contractRestrictCommutingFormula}.\n\t Since $N$ and $N^\\ast$ share the same ground set and\n\t $\\mathrm{C}_A(N) = \\mathrm{C}_A(N^\\ast)$ for all gammoids $N$ (Lemma~\\ref{lem:arcCkVDualityAndMinors}), we obtain that\n\t \\[ \\mathrm{W}_f(M) = \\mathrm{W}_f(M^\\ast). \\qedhere\\]\n\\end{proof}\n\n\n\\begin{definition}\\PRFR{Mar 27th}\n\tLet $f\\colon \\mathbb{N}\\longrightarrow \\mathbb{N}\\BSET{0}$ be a non-decreasing function. We say that $f$ is \\deftext{super-additive},\n\tif for all $n,m\\in \\mathbb{N}\\BSET{0}$\n\t\\[ f(n+m) \\geq f(n) + f(m) \\]\n\tholds.\n\\end{definition}\n\n\\needspace{3\\baselineskip}\n\\begin{lemma}\\PRFR{Mar 27th}\n\tLet $f\\colon \\mathbb{N}\\longrightarrow \\mathbb{N}\\BSET{0}$ be a non-decreasing and super-additive function, let $k\\in \\mathbb{N}$,\n\tand let ${\\mathcal{W}}_{f,k}$ denote the class of gammoids $M$ with $\\mathrm{W}_f(M) \\leq k$.\n\tThen ${\\mathcal{W}}_{f,k}$ is closed under duality, minors, and direct sums.\n\\end{lemma}\n\n\\begin{proof}\\PRFR{Mar 27th}\n\tIt is clear from Corollary~\\ref{cor:WfClosedUnderMinors} that ${\\mathcal{W}}_{f,k}$ is closed under minors and duality.\n\t \t\n\t Now, let $M=(E,{\\mathcal{I}})$ and $N=(E',{\\mathcal{I}}')$ with $E\\cap E' =\\emptyset$ and $M,N\\in {\\mathcal{W}}_{f,k}$.\n\t Furthermore, let $X\\subseteq Y\\subseteq E\\cup E'$. Then, by Lemmas~\\ref{lem:directSumAndRestrictionCommute} and \\ref{lem:directSumAndContraction}, we have that\n\t \\begin{align*}\n\t \t \\left( (M\\oplus N)|' Y \\right)| X & \\,\\,\\,=\\,\\,\\,\n\t \t \\left(\\left( M|' Y\\cap E \\right) \\oplus ( N|' Y'\\cap E )\\right)| X\n\t \t \\\\& \\,\\,\\,=\\,\\,\\,\n\t \t \t\\left(\\vphantom{A^A}\\left( M|' Y\\cap E \\right)| X\\cap E\\right) \\oplus \n\t \t \t\\left( ( N|' Y\\cap E' )| X\\cap E'\\right).\n\t \\end{align*}\n\t holds.\n\t With Lemma~\\ref{lem:arcCSubAdditive} we obtain\n\t \\begin{align*}\n\t \t\\mathrm{C}_A\\left(\\vphantom{A^A} \\left( (M\\oplus N)|' Y \\right)| X \\right) & \\,\\,\\,\\leq \\,\\,\\,\n\t \t\\mathrm{C}_A\\left(\\vphantom{A^A} \\left( M|' Y\\cap E \\right)| X\\cap E\\right) \n\t \t+ \\mathrm{C}_A\\left(\\vphantom{A^A} ( N|' Y\\cap E' )| X\\cap E'\\right) .\n\t \\end{align*}\n\t We use the super-additivity of $f$ in order to derive\n\t \\begin{align*}\n\t \t\\frac {\\mathrm{C}_A\\left(\\vphantom{A^A} \\left( (M\\oplus N)|' Y \\right)| X \\right)}{f\\left( \\left| X \\right|\n\t \n\t \t\\right)} &\n\t \t\\,\\,\\, \\leq\\,\\,\\,\n\t \t\\frac { \\mathrm{C}_A\\left(\\vphantom{A^A} \\left( M|' Y\\cap E \\right)| X\\cap E\\right) \n\t \t+ \\mathrm{C}_A\\left(\\vphantom{A^A} ( N|' Y\\cap E' )| X\\cap E'\\right) }{f\\left( \\left| X \\right|\n\t \n\t \t\\right) } \\\\\n\t \t\n\t \t\\,\\,\\, \\leq\\,\\,\\, \\frac{k\\cdot f\\left( \\left| X\\cap E \\right|\n\t \n\t \t\\right) + k\\cdot f\\left( \\left| X\\cap E' \\right|\n\t \n\t \t \\right)}{f\\left( \\left| X \\right|\n\t \n\t \t \\right) } \\\\\n\t\n\t \t\\,\\,\\, =\\,\\,\\, \\frac{k\\cdot \\left(\\vphantom{A^A} f\\left( \\left| X\\cap E \\right|\n\t \n\t \t\\right) + f\\left( \\left| X\\cap E' \\right|\n\t \n\t \t\\right) \\right)}\n\t \t{f\\left( \\left| X \\right|\n\t \n\t \t \\right)} \\\\ &\n\t \n\t \t\\,\\,\\, \\leq\\,\\,\\, k\\cdot \\frac{f\\left( \\left| X \\right|\n\t \n\t \t\\right)}{f\\left( \\left| X \\right|\n\t \n\t \t\\right)} \\,\\,\\, = \\,\\,\\, k\t \t,\n\t \\end{align*}\n\t where the second inequality follows from\n\t the fact that \\\n\t \\frac{\\mathrm{C}_A(G)}{f\\left( \\left| F \\right|\n\t \n\t \\right)} \\leq \\mathrm{W}_f(G) \\leq k\\] holds\n\t for every $G=(F,{\\mathcal{J}})\\in {\\mathcal{W}}_{f,k}$, thus it holds for all minors of $M$ and $N$ \n\t (Corollary~\\ref{cor:WfClosedUnderMinors}).\n\t As a consequence, $\\mathrm{W}_f(M\\oplus N) \\leq k$, and therefore\n\t $M\\oplus N\\in {\\mathcal{W}}_{f,k}$ holds.\n\\end{proof}\n\n\\PRFR{Apr 1st}\n\\noindent We may consider a class of matroids, that is closed under direct sums, and that contains a matroid, that is\nneither trivial nor free, to be truly infinite, as opposed to a class that consists of matroids, that are \ndirect sums of free matroids, trivial matroids, and one matroid that is isomorphic to a member of a finite family of\nmatroids.\n\n\\begin{theorem}\\label{ref:infiniteChainOfSubclasses}\\PRFR{Apr 1st}\n\tLet $\\left(M_k\\right)_{k\\in \\mathbb{N}}$ with $M_k = (E_k,{\\mathcal{I}}_k)$ be a sequence of gammoids \n\twith \\[ \\mathrm{C}_A(M_k) \\geq k\\cdot \\left| E_k \\right|.\\]\n\tThen there is an infinite chain of strictly bigger classes of gammoids that are closed under\n\tduality, minors, and direct sums in the family of classes\n\t\\[ {\\mathcal{W}}^\\mathbb{N} = \\SET{ {\\mathcal{W}}^k ~\\middle|~ {\\mathcal{W}}^k \\text{~is the class of all gammoids $M$ with~} \\mathrm{W}^k(M) \\leq 1,\\,k\\in\\mathbb{N} }. \\]\n\\end{theorem}\n\\begin{proof}\\PRFR{Apr 1st}\n\tClearly, we have that $\\mathrm{W}^k(M) > \\mathrm{W}^{k'}(M)$ and\n\t $\\mathrm{W}^k(M_{k'}) \\geq \\frac{k'}{k} > 1$ for all $k,k'\\in \\mathbb{N}$ with $k' > k$,\n\tso every class ${\\mathcal{W}}^k$ contains at most $k$ elements of the matroid sequence $\\left( M_k \\right)_{k\\in \\mathbb{N}}$,\n\tand every class ${\\mathcal{W}}^{k'}$ contains the class ${\\mathcal{W}}^k$ if $k' > k$.\n\tFurthermore, $M_{k'}$ is contained in ${\\mathcal{W}}^{\\mathrm{C}_A(M_{k'})}$, therefore every matroid of the sequence is\n\teventually contained in some ${\\mathcal{W}}^k$. Consequently, ${\\mathcal{W}}^\\mathbb{N}$ must contain a countable chain of strictly\n\tbigger subclasses of gammoids.\n\\end{proof}\n\n\\noindent Conjecture~\\ref{conj:uniformArcs} would imply that there is a strict chain of truly \ninfinite subclasses of gammoids that are closed under\nminors and duality, and that ${\\mathcal{W}}^i$ is a proper subclass of ${\\mathcal{W}}^{i+1}$ for all $i\\in \\mathbb{N}$.\n\n\n\n\n\n\\begin{lemma}\\label{lem:uniformArcs}\\PRFR{Apr 1st}\n\tLet $E$ be a finite set, $r\\in \\mathbb{N}$ with $r \\leq \\left| E \\right|$, and let $$U = \\left( E, \\SET{X\\subseteq E ~\\middle|~\n\t\\vphantom{A^A} \\left| X \\right| \\leq r} \\right)$$\n\tbe the uniform matroid of rank $r$ on $E$.\n\tThen \\[ \\mathrm{C}_V(U) = \\left| E \\right| \\,\\,\\,{\\mathop{\\text{~and~}}}\\,\\,\\, \\mathrm{C}_A(U) \\leq r\\cdot \\left( \\left| E \\right| - r \\right) .\\]\n\\end{lemma}\n\\begin{proof}\\PRFR{Apr 1st}\n\tLet $T \\subseteq E$ with $\\left| T \\right| = r$ and let $D=(E,A)$ be the digraph on the vertex set $E$\n\t where $A = \\SET{(e,t)~\\middle|~ e\\in E\\backslash T,\\, t\\in T}$.\n\tClearly, $(D,T,E)$ is a standard representation with $U = \\Gamma(D,T,E)$. \n\tTherefore $\\mathrm{C}_V(U) \\leq \\left| E \\right|$ and $\\mathrm{C}_A(U) \\leq r\\cdot \\left( \\left| E \\right| - r \\right)$.\n\tObviously, the vertex complexity is bounded from below by the size of the ground set, thus $\\mathrm{C}_V(U) = \\left| E \\right|$.\n\\end{proof}\n\n\n\n\n\n\n\\PRFR{Apr 1st}\n\\noindent The following kind of matroids is usually defined as matroids, whose ground sets consist of edges of undirected \ngraphs, such that subsets of these edges are independent, if they contain no subgraph that consists of {\\em (i)}\ntwo cycles with a single common vertex ($\\infty$-graph), {\\em (ii)} two cycles which\nshare a common line segment ($\\Theta$-graph),\nor {\\em (iii)} two cycles each of which has a special vertex and those special vertices are connected by a line (hand-cuffs graph).\nWe use L.R.~Matthews's characterization in order to define bicircular matroids.\n\\begin{definition}[\\cite{Ma77}, Corollary~3.3 and Theorem~3.5]\\PRFR{Apr 1st}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid. Then $M$ is a \\deftext{bicircular matroid}, if there is a family ${\\mathcal{A}}=(A_i)_{i=1}^{\\mathrm{rk}_M(E)}$ of subsets of $E$ with the property that $\\left| \\SET{i\\in I~\\middle|~e\\in A_i}\\right| \\in \\SET{1,2}$ holds\n\t for all $e\\in E$, and such that $M = M({\\mathcal{A}})$.\n\\end{definition}\n\n\\noindent It is clear that bicircular matroids are special gammoids.\n\n\n\\begin{lemma}\\PRFR{Apr 1st}\n\tLet $M=(E,{\\mathcal{I}})$ be a bicircular matroid. Then\n\t\\[ \\mathrm{C}_A(M) \\leq 2\\cdot \\left| E \\right| \\quad{\\mathop{\\text{~and~}}}\\quad \\mathrm{C}_V(M) \\leq \\left| E \\right| + \\mathrm{rk}_M(E) .\\]\n\\end{lemma}\n\\begin{proof}\\PRFR{Apr 1st}\n\tLet $I$ be a set with $\\left| I \\right| = \\mathrm{rk}_M(E)$ and ${\\mathcal{A}} =(A_i)_{i\\in I}$ be a family of subsets of $E$\n\tsuch that $M= M({\\mathcal{A}})$ and such that $\\left| \\SET{i\\in I~\\middle|~e\\in A_i}\\right| \\in \\SET{1,2}$ for all $e\\in E$.\n\tFor technical reasons, let us further assume that $I\\cap E = \\emptyset$. Let $D_0 = (V,A)$ with $V=E\\mathbin{\\dot{\\cup}} I$ and\n\t$A = \\SET{(e,i)~\\middle|~e\\in E,\\,i\\in I,\\,e\\in A_i}$.\n\t Then $M = \\Gamma(D_0,I,E)$ and $\\left| A \\right| \\leq 2\\cdot \\left| E \\right|$.\n\tWe obtain a standard representation of $M$ by pivoting in an \n\tarbitrary base $T\\in {\\mathcal{B}}(M)$ as it is done in the proof of Theorem~\\ref{thm:gammoidRepresentationWithBaseTerminals}.\n\tThis operation does not introduce any new arcs or vertices,\n\t therefore $\\mathrm{C}_A(M) \\leq \\left| A \\right| \\leq 2\\cdot \\left| E \\right|$ \n\t and $\\mathrm{C}_V(M) \\leq \\left| E \\right| + \\left| I \\right| = \\left| E \\right| + \\mathrm{rk}_M(E)$ holds.\n\\end{proof}\n\n\n\n\\subsection{Essential Arcs and Vertices}\n\n\\PRFR{Apr 1st}\n\\noindent Let $(D,T,E)$ be a representation of a gammoid, and let $D=(V,A)$. In this section, we are concerned with the\nquestion when an arc $a\\in A$ or a vertex $v\\in V$ is\nessential for the representation of $\\Gamma(D,T,E)$. It turns out that this kind of question may be answered by\ninspection of the family of independent sets of\na derived gammoid.\n\n\\begin{definition}\\label{def:ACD}\\PRFR{Apr 1st}\n\tLet $(D,T,E)$ with $D=(V,A)$ be a representation of the gammoid $\\Gamma(D,T,E)=(E,{\\mathcal{I}})$,\n\tand let $a\\in A$ be an arc of $D$.\n\tThe arc $a$ shall be called \\deftext[essential arc of DTE@essential arc of $(D,T,E)$]{essential arc of $\\bm(\\bm D\\bm, \\bm T\\bm, \\bm E\\bm)$},\n\tif there is some $X\\in {\\mathcal{I}}$ such that $X$ is not independent with respect to $\\Gamma(D_a,T,E)$ where\n\t$D_a = (V,A\\BSET{a})$.\n\\end{definition}\n\n\\begin{remark}\\PRFR{Apr 1st}\n If $(D,T,E)$ with $D=(V,A)$ is a representation of $M= \\Gamma(D,T,E)$ such that $\\left| A \\right| = \\mathrm{C}_A(M)$,\n then every arc $a\\in A$ is essential. Also, the converse is not true: Let $(D,T,E)$\n be a representation of a gammoid\n such that every arc of $D=(V,A)$ is essential. If we subdivide an arc of $D$ \n with a newly introduced auxiliary vertex, \n then the resulting digraph $D'$ still consists only of essential arcs\n with respect to $(D',T,E)$ --- but $(D',T,E)$ can no longer have an arc set of minimal cardinality.\n \\end{remark}\n\n\\begin{lemma}\\label{lem:essentialArcsC}\\PRFR{Apr 1st}\n\tLet $M=(E,{\\mathcal{I}})$ be a gammoid, and let $(D,T,E)$ be a representation of $M$ with $D=(V,A)$.\n\tLet $(u,v)\\in A$ be an essential arc of $(D,T,E)$,\n\tand let \n\t\\linebreak\n\t $N = \\Gamma(D,T,V)$ and $N' = \\Gamma(D',T,V)$ where\n\t$D'=(V,A\\BSET{(u,v)})$.\n\tThere is a circuit $C\\in {\\mathcal{C}}(N')$ with $u\\in C$ such that $C$ is independent in $N$.\n\\end{lemma}\n\\begin{proof}\\PRFR{Apr 1st}\n\tClearly, if $N = N'$, then $(u,v)$ is not an essential arc of $(D,T,E)$.\n\tTherefore there is a subset $X\\subseteq E \\subseteq V$ that is independent in $N$ yet dependent in $N'$.\n\tSince every routing in $D$ is a routing in $D'$ unless it traverses the arc $(u,v)$,\n\twe observe that every routing $R\\colon X\\double{\\rightarrow} T$ in $D$ must traverse the arc $(u,v)$.\n\tSince $X$ is dependent in $N'$, there is an minimum-cardinality $X$-$T$-separator\n\t $S'$ in $D'$ with $\\left| S' \\right| < \\left| X \\right|$.\n\tWith the previous observation we obtain that $S = S'\\cup\\SET{u}$ is\n\tan $X$-$T$-separator in $D$ with $\\left| S \\right| = \\left| X \\right|$.\n\tFurthermore, we see that $u\\notin S'$, because otherwise $S'$ would be an $X$-$T$-separator in $D$,\n\twhich would lead us to the contradiction $\\mathrm{rk}_{N}(X) \\leq \\left| S' \\right| < \\left| X \\right| = \\mathrm{rk}_N(X)$ ---\n\tas $X$ is an independent set of $N$.\n\tCorollary~\\ref{cor:Menger} yields that we may cut off the initial parts of the paths of a maximal $X$-$T$-connector in $D$\n\tand thereby obtain a routing from $S$ to $T$ in $D$, so $S$ is independent in $N$.\n\tWe give an indirect argument that $S$ is dependent in $N'$.\n\t Assume that $S$ is independent in $N'$. $S'$ is a minimal cardinality $X$-$T$-separator in $D'$,\n\t thus $S'\\subseteq \\mathrm{cl}_{N'}(X)$ (Corollary~\\ref{cor:Menger}).\n\t If there is a path $p\\in {\\mathbf{P}}(D)$ with\n\t $p_1\\in X\\backslash S'$ and $p_{-1}=u$\n\t that does not visit a vertex $s\\in S'$, then $S'\\cup\\SET{p_{1}}$ is independent,\n\t and so we obtain \n\t \\[\\mathrm{rk}_{N'}(X) = \\mathrm{rk}_{N'}\\left( \\mathrm{cl}_{N'}(X) \\right) \n\t \\geq \\mathrm{rk}_{N'}(S'\\cup\\SET{p_{1}}) = \\left| S' \\right| + 1 = \\left| X \\right|.\\]\n\t Thus $X$ would be independent in $N'$ --- a contradiction.\n\t To avoid this contradiction, every path $p\\in {\\mathbf{P}}(D)$ with $p_1 \\in X$ and $p_{-1}=u$\n\t must visit a vertex $s\\in S'$. But then $S'$ is an $X$-$S$-separator in $D$,\n\t and since $S$ is an $X$-$T$-separator in $D$,\n\t we have that $S'$ is an \\linebreak $X$-$T$-separator in $D$.\n\t Again, this yields the contradiction\n\t $\\mathrm{rk}_N(X) \\leq \\left| S' \\right| < \\left| X \\right| = \\mathrm{rk}_N(X)$.\n\t Therefore\n\t we may dismiss our assumption and we conclude that $S$ is dependent in $N'$.\n\t Remember that $S'$ is independent in $N'$ because it is a minimal-cardinality $X$-$T$-separator in $D'$,\n\t thus there is a circuit $C\\in{\\mathcal{C}}(N')$ with $C\\subseteq S$ and $C\\not\\subseteq S'$, so $u\\in C$;\n\t and since $S$ is independent in $N$, we obtain that $C$ is independent in $N$, too.\n\\end{proof}\n\n\n\n\\begin{definition}\\PRFR{Apr 1st}\n\tLet $D=(V,A)$ be a digraph\n\twith $V\\cap \\left( \\left( V\\times V \\right)\\times \\SET{1,2} \\right) = \\emptyset$.\n\tThe \\deftext[arc-cut digraph]{arc-cut digraph for $\\bm D$} shall be the \\label{n:arcCutDigraph}\n\tdigraph $\\mathrm{AC}(D) = (V_{D},A_{D})$ where\n\t\\begin{align*}\n\t V_D \\,\\,=\\,\\,\\hphantom{\\cup\\,\\,} & V\\mathbin{\\dot{\\cup}} \\left( \\SET{(u,v)\\in V\\times V~\\middle|~ u\\not= v}\\times \\SET{1,2} \\right)\n\t \\quad{\\mathop{\\text{~and~}}}\\\\\n\t A_D \\,\\,=\\,\\, \\hphantom{\\cup\\,\\,}& \\SET{\\left(u, \\left( (u,v),1 \\right) \\right),\\left( \\left( (u,v),1 \\right), v \\right)\n\t ~\\middle|~ (u,v)\\in A,\\,u\\not=v} \n\t\\\\ \\cup\\,\\, &\\SET{\\left( \\left( (u,v),1 \\right),\\left( (u,v),2 \\right) \\right) ~\\middle|~ u,v\\in V,\\,u\\not= v}.\n\t\\end{align*}\n\tIn other words, for all $u,v\\in V$ with $u\\not=v$ we do the following in order to obtain $\\mathrm{AC}(D)$ from $D$:\n\t If there is an arc $(u,v)$ in $D$, \n\twe add two new vertices and turn it into a top-left-to-bottom-right-oriented $\\top$-shaped-junction. If there is no arc $(u,v)$ in $D$,\n\twe add two new vertices and connect one with the other.\n\\end{definition}\n\n\n\\needspace{6\\baselineskip}\n\\vspace*{-\\baselineskip}\n\\begin{wrapfigure}{l}{8.3cm}\n\\vspace{\\baselineskip}\n\\begin{centering}~~\n\\includegraphics{ACD}\n\\end{centering}%\n\\vspace*{-1\\baselineskip}\n\\end{wrapfigure}\n~\n \n\\begin{example}\\PRFR{Apr 1st}\n\tConsider the digraph $D = \\left( \\SET{a,b},\\SET{(a,b)} \\right)$.\n\tThen $\\mathrm{AC}(D)=(V_D,A_D)$\n\t is the digraph where $V_D = \\{a,$ $b,$ $\\left( (a,b),1 \\right),$ $\\left( (a,b),2 \\right),$ $\\left( (b,a),1 \\right),$\n\t$ \\left( (b,a),2 \\right)\\}$ and where $A_D =$ $ \\{ \\left( a, \\left( (a,b),1 \\right) \\right),$ \n\t $\\left( \\left( (a,b),1 \\right), b \\right),$ \\hfill{~} \\linebreak\n\t $\\hphantom{\\{} \\left( \\left( (a,b),1 \\right), \\left( (a,b),2 \\right) \\right),$\n\t \\linebreak\n\t $\\hphantom{\\{}\\left( \\left( (b,a),1 \\right), \\left( (b,a),2 \\right) \\right)\\}$.\n\\end{example}\n\n\\needspace{6\\baselineskip}\n\\begin{definition}\\label{def:ACDTE}\\PRFR{Apr 1st}\n\tLet $(D,T,E)$ be a representation of a gammoid where $D=(V,A)$, and such that\n\t $V\\cap \\left( \\left( V\\times V \\right) \\times \\SET{1,2} \\right) = \\emptyset$. \\label{n:ACDTE}\n\t The \\deftext[arc-cut matroid]{arc-cut matroid for $\\bm(\\bm D\\bm, \\bm T\\bm, \\bm E\\bm)$}\n\t shall be the matroid $\\mathrm{AC}(D,T,E) = \\Gamma(\\mathrm{AC}(D),T',E')$\n\t where $$E' = E \\cup \\SET{\\left( (u,v), i \\right) ~\\middle|~ u,v\\in V,\\,u\\not= v,\\,i\\in\\SET{1,2}}$$\n\t and where \n\t \\[ T' = T\\cup \\SET{\\left( (u,v), 2 \\right) ~\\middle|~ u,v\\in V,\\,u\\not= v}. \\qedhere\\]\n\\end{definition}\n\n\\needspace{5\\baselineskip}\n\\begin{lemma}\\label{lem:indepACDTE}\\PRFR{Apr 1st}\n\tLet $(D,T,E)$ be a representation of a gammoid where $D=(V,A)$, and such that\n\t $V\\cap \\left( \\left( V\\times V \\right) \\times \\SET{1,2} \\right) = \\emptyset$.\n\t Then $X\\subseteq E$ is independent with respect to $\\Gamma(D,T,E)$, if and only if\n\t \\(X' = X\\cup\\SET{\\left( (u,v),2 \\right)~\\middle|~ u,v\\in V,\\,u\\not=v} \\)\n\t is independent with respect to $\\mathrm{AC}(D,T,E)$.\n\\end{lemma}\n\\begin{proof}\\PRFR{Apr 1st}\n\tLet $X$ be independent with respect to $M = \\Gamma(D,T,E)$. There is a routing $R\\colon X\\double{\\rightarrow} T$ in $D$.\n\tThus we have a routing $R' = \\SET{p'~\\middle|~ p\\in R} \\cup \\SET{\\left( (u,v), 2 \\right) ~\\middle|~ u,v\\in V,\\,u\\not= v}$ in $D'$\n\twhere\n\t $$p' = p_1\\left( (p_1,p_2),1 \\right)p_2\\left( (p_2,p_3),1 \\right) \\ldots\n\tp_{n-1}\\left( (p_{n-1},p_{n}),1 \\right)p_n$$ denotes the path in $\\mathrm{AC}(D)$ that is obtained from $p=(p_i)_{i=1}^n$ by\n\tsubdividing every arc $(u,v)$ traversed by $p$ with $\\left( (u,v),1 \\right)$.\n\tConsequently, the derived set $X'$ is independent in $N = \\mathrm{AC}(D,T,E)$.\n\tNow let $X$ be dependent in $M$, therefore there is no routing from $X$ to $T$ in $D$. Now assume that there is a routing $R'$\n\tfrom the derived set $X'$ to $T'=T\\cup\\SET{\\left( (u,v),2 \\right)~\\middle|~ u,v\\in V,\\,u\\not=v} $ in $\\mathrm{AC}(D)$, i.e. that\n\t$X'$ is independent with respect to $N$. Then $R'$ routes every $x\\in X$ to some element $t_x\\in T'\\backslash \\left( X' \\backslash X \\right) = T$\n\tin $\\mathrm{AC}(D)$. By omitting the subdivision vertices in the corresponding paths $p'\\in R$, we obtain a routing from $X$ to\n\t$T$ in $D$ --- a contradiction. Therefore $X'$ is dependent in $N$ if $X$ is dependent in $M$.\n\\end{proof}\n\n\\begin{lemma}\\PRFR{Apr 1st}\n\tLet $(D,T,E)$ be a representation of a gammoid where $D=(V,A)$, and such that\n\t $V\\cap \\left( \\left( V\\times V \\right) \\times \\SET{1,2} \\right) = \\emptyset$.\n\t Furthermore, let $a\\in A$. The arc $a$ is an essential arc of $(D,T,E)$\n\tif and only if there is a circuit $C\\in {\\mathcal{C}}\\left( \\mathrm{AC}(D,T,E) \\right)$ with \n\t\\[ \\left( a,1 \\right) \\in C \\subseteq E \\cup \\SET{\\left( a,1 \\right)} \\cup \\SET{\\left( (u,v),2 \\right)~\\middle|~ u,v\\in V,\\,u\\not= v,\\,a\\not=(u,v)} .\\]\n\\end{lemma}\n\\begin{proof}\\PRFR{Apr 1st}\nFirst, let us assume that $a$ is an essential arc of $(D,T,E)$.\n\tLet $X\\subseteq E$ be independent with respect to $\\Gamma(D,T,E)$,\n\t such that every routing $R\\colon X\\double{\\rightarrow} T$ in $D$ traverses the arc $a$.\n\tThen every routing from $X$ to $T$ in $\\mathrm{AC}(D)$ visits the vertex $(a,1)$.\n\tTherefore every routing from $X' = X\\cup\\SET{\\left( (u,v),2 \\right)~\\middle|~ u,v\\in V,\\,u\\not= v,\\,a\\not=(u,v)}$\n\tto $T'=T\\cup\\SET{\\left( (u,v),2 \\right)~\\middle|~ u,v\\in V,\\,u\\not=v} $ in $\\mathrm{AC}(D)$ also\n\thas to visit the vertex $(a,1)$. This implies that $X'\\cup\\SET{(a,1)}$ must be dependent. From Lemma~\\ref{lem:indepACDTE} we obtain that\n\t $X'$ is independent \n\tin $\\mathrm{AC}(D,T,E)$, and consequently there is a circuit $C\\subseteq X'\\cup\\SET{(a,1)}$ such that $\\left( a,1 \\right)\\in C$.\n\t%\n\tNow assume that $a$ is not an essential arc of $(D,T,E)$.\n\tLet $X\\subseteq E$ be independent with respect to $\\Gamma(D,T,E)$, then there is a routing $R\\colon X\\double{\\rightarrow} T$ in $D$\n\tsuch that the arc $a$ is not traversed by $R$.\n\tThus there is a routing $R'$ from $X' = X\\cup\\SET{\\left( (u,v),2 \\right)~\\middle|~ u,v\\in V,\\,u\\not= v,\\,a\\not=(u,v)}$ \n\tto $T'$ in $\\mathrm{AC}(D)$\n\tthat does not visit the vertex $(a,1)$. It is clear from Definition~\\ref{def:ACD} that such a routing $R'$ cannot visit $(a,2)$ either.\n\tTherefore $R'\\cup\\SET{(a,1)(a,2)}$ is a routing in $\\mathrm{AC}(D)$ and $X'\\cup\\SET{(a,1)}$ is independent with\n\trespect to $\\mathrm{AC}(D,T,E)$. Consequently, if $ C \\subseteq E \\cup \\SET{\\left( a,1 \\right)} \\cup \\SET{\\left( (u,v),2 \\right)~\\middle|~ u,v\\in V,\\,u\\not= v,\\,a\\not=(u,v)} $ is a circuit of $\\mathrm{AC}(D,T,E)$, then $C\\cap E$ is dependent,\n\ttherefore $\\left( a,1 \\right) \\notin C$.\n\\end{proof}\n\n\\noindent A.W.~Ingleton and M.J.~Piff showed the following nice theorem about representations of strict gammoids\nwhere every arc is essential, which they call {\\em minimal presentation of $\\Gamma(D,T,V)$}.\n\n\\begin{theorem}[\\cite{IP73}, Theorem~3.12]\\label{thm:IPEssentialStars}\\PRFR{Apr 1st}\n\tLet $(D,T,V)$ be a representation of a gammoid where $D=(V,A)$ and where all $a\\in A$ are essential arcs of $(D,T,V)$,\n\tand let $u\\in V\\backslash T$.\n\tThen \n\t\\[ S_u = \\SET{v\\in V~\\middle|~ (u,v)\\in A} \\cup \\SET{u} \\in {\\mathcal{C}}(\\Gamma(D,T,V)).\\]\n\\end{theorem}\n\n\\noindent For a proof, see \\cite{IP73} p.60.\n\n\\begin{corollary}\\label{cor:arcEstimatesDTE}\n\tLet $D=(V,A)$ be a digraph, $T\\subseteq V$, and $E\\subseteq V$.\n\tFurthermore, let $M=\\Gamma(D,T,E)$ and $N=\\Gamma(D,T,V)$.\n\tThen \\begin{align*}\n\t \t \\mathrm{C}_A(M) \\leq \\mathrm{C}_A(N)& \\leq\n\t\t \\left| V\\backslash T \\right| + \\sum_{u\\in V\\backslash T} \\mathrm{rk}_N\\left(\\vphantom{A^A} \\SET{v\\in V~\\middle|~(u,v)\\in V} \\right)\n\t\t\\\\& \\leq \\left( \\left| V \\right| - \\mathrm{rk}_N(V) \\right)\\cdot \\left( \\mathrm{rk}_N(V) + 1 \\right).\n\t\\end{align*}\n\\end{corollary}\n\\begin{proof}\n\tSince $M$ is a minor of $N$, we have $\\mathrm{C}_A(M) \\leq \\mathrm{C}_A(N)$ (Lemma~\\ref{lem:arcCkVDualityAndMinors}).\n\tThe last inequality follows from Lemma~\\ref{lem:rankMonotone}.\n\tLet $D'=(V,A')$ be a digraph obtained from $D$ by successively removing one non-essential arc of $(D',T,V)$\n\tafter another from $A'$ until every remaining arc $a\\in A'$ is an essential arc of $(D',T,V)$. Let $u\\in V\\backslash T$, then\n\tTheorem~\\ref{thm:IPEssentialStars} yields that $S_u = \\SET{v\\in V~\\middle|~ (u,v)\\in A} \\cup \\SET{u} \\in {\\mathcal{C}}(N)$,\n\tthus the process of removing non-essential arcs stops no sooner than\n\twhen $O_u = \\SET{v\\in V~\\middle|~ (u,v)\\in A'}$ is independent \n\tin $N$ for all $u\\in V\\backslash T$. Clearly, no arc leaving a vertex $t\\in T$ is essential for $(D',T,V)$. Thus\n\t$$\\left| A' \\right| = \\sum_{u\\in V\\backslash T} \\mathrm{rk}_N\\left(\\vphantom{A^A} \\SET{v\\in V~\\middle|~(u,v)\\in V} \\right)$$\n\tholds. We may obtain a standard representation of $N$ from $(D',T,V)$ by first renaming all $v\\in V\\backslash T$\n\tto $v'$ and then adding a new source $v\\in V\\backslash T$ and a new arc $(v,v')$ to $D'$. \n\tConsequently, \n\t\\[ \\mathrm{C}_A(N) \\leq\n\t\t \\left| V\\backslash T \\right| + \\sum_{u\\in V\\backslash T} \\mathrm{rk}_N\\left(\\vphantom{A^A} \\SET{v\\in V~\\middle|~(u,v)\\in V} \\right)\n\t\t \\leq \\left| V\\backslash T \\right| + \\sum_{u\\in V\\backslash T} \\mathrm{rk}_N(V). \\qedhere\\]\n\\end{proof}\n\n\\noindent Corollary~\\ref{cor:arcEstimatesDTE} together with Remark~\\ref{rem:upperBoundForV} implies that\nevery gammoid $M=(E,{\\mathcal{I}})$ may be represented on a digraph with at most $k = \\mathrm{rk}_M(E)^2 \\cdot \\left| E \\right| + \\mathrm{rk}_M(E) + \\left| E \\right|$ vertices and with at most $(k-\\mathrm{rk}_M(E))\\cdot \\left( 1+\\mathrm{rk}_M(E) \\right)$ arcs. \n\n\\begin{lemma}\\label{lem:uniformStrictStdRep}\\PRFR{Apr 1st}\n\tLet $r\\in \\mathbb{N}$, $U=(E,{\\mathcal{I}})$ be a uniform matroid with $r \\leq \\left| E \\right|$,\\linebreak i.e.\n\t${\\mathcal{I}} = \\SET{\\vphantom{A^A}X\\subseteq E~\\middle|~ \\left| X \\right| \\leq r}$, and\n\tlet $(D,T,E)$ with $D=(E,A)$ be a strict representation of $U$.\n\tThen $$\\left| A \\right| \\geq r\\cdot\\left( \\left| E \\right| - r \\right).$$\n\\end{lemma}\n\\begin{proof}\\PRFR{Apr 1st}\n\tWithout loss of generality we may assume that no digraph occurring in this proof contains a loop arc $(v,v)$.\n\tLet $(D,T,E)$ with $D=(E,A)$ be a strict representation of $U$ \n\twith a minimal number of arcs among all such representations.\n\tDue to that minimality, every $t\\in T$ is a sink of $D$, and every arc $a\\in A$ is an essential arc.\n\tObserve that $\\left| C \\right| = r+1$ for all $C\\in{\\mathcal{C}}(U)$.\n\tThus we obtain from Theorem~\\ref{thm:IPEssentialStars}\n\tthat \n\t\\begin{align*}\n\t \\left| A \\right| & = \\left| \\bigcup_{u\\in E\\backslash T} \\SET{(x,v)\\in A~\\middle|~ x=u} \\right| \n\t \\\\& = \\sum_{u\\in E\\backslash T}\n\t\\left| \\SET{(x,v) \\in A~\\middle|~ x=u} \\right| \\\\ \n\t\t& = \\sum_{u\\in E\\backslash T} \\left| \\SET{v\\in V~\\middle|~ (u,v)\\in A} \\right| \\\\\n\t\t& \\geq \\left| E\\backslash T \\right|\\cdot \\min \\SET{\\vphantom{A^A} \\left| C \\right| - 1 ~\\middle|~ C\\in{\\mathcal{C}}(U)} = \\left( \\left| E \\right| - r \\right) \\cdot r.\n\t\t\\end{align*}\n\tThus, every strict representation of $U$ has at least $r\\cdot \\left( \\left| E \\right| - r \\right)$ arcs.\n\tThe strict standard representation constructed in Lemma~\\ref{lem:uniformArcs} yields that this bound is attained.\n\\end{proof}\n\\noindent\n\tIn general, strict gammoids that are not transversal matroids exist and such matroids cannot\n\thave a standard representation\n\tthat is also a strict representation, because their duals do not have a strict representation.\n\n\\needspace{4\\baselineskip}\n\\begin{definition}\\PRFR{Apr 1st}\n\tLet $(D,T,E)$ with $D=(V,A)$ be a representation of the gammoid $\\Gamma(D,T,E)=(E,{\\mathcal{I}})$,\n\tand let $q\\in V$ be a vertex of $D$.\n\tThen $q$ shall be called \\deftext[essential vertex of DTE@essential vertex of $(D,T,E)$]{essential vertex of $\\bm(\\bm D\\bm, \\bm T\\bm, \\bm E\\bm)$},\n\tif either $q\\in E$ or\n\tif $q\\in V\\backslash E$ and there is some $X\\in {\\mathcal{I}}$ such that $X$ is not independent with respect to $\\Gamma(D_q,T,E)$ where\n\t\\[ D_q = (V\\BSET{q},\\SET{(u,v)\\in A~\\middle|~u\\not= q{\\mathop{\\text{~and~}}} v\\not=q}). \\qedhere \\]\n\\end{definition}\n\n\\needspace{6\\baselineskip}\n\n\\vspace*{-\\baselineskip}\n\\begin{wrapfigure}{r}{3cm}\n\\vspace{\\baselineskip}\n~~~\\includegraphics{essentialVtx}\n\\vspace*{-\\baselineskip}\n\\end{wrapfigure}\n~\n \n\n\n\\begin{remark}\\PRFR{Apr 1st}\n\tClearly, if $(u,v)$ is an essential arc of $(D,T,E)$, then $u$ and $v$ are essential vertices of $(D,T,E)$. \n\tOn the other hand, not every essential vertex $v$ of $(D,T,E)$ is incident with an essential arc of $(D,T,E)$.\n\tFor instance, let\\linebreak $D=(V,A)$ be the digraph where $V=\\dSET{a,b,c,d,e,f,g}$ and\n\t\\linebreak $A = \\SET{(a,b),(a,c),(b,d),(c,d),(d,e),(d,f),(e,g),(f,g)}$.\n\tThen $d$ is an essential vertex of $(D,\\SET{g},\\SET{a})$, but $(D,\\SET{g},\\SET{a})$ has no essential arcs.\n\\end{remark}\n\n\\begin{lemma}\\label{lem:vtxInMinReprs}\\PRFR{Apr 1st}\n\tLet $M=(E,{\\mathcal{I}})$ be a gammoid, and let $(D,T,E)$ be a representation of $M$, and let $D=(V,A)$.\n\n\n\tLet $q\\in V\\backslash E$ be an essential vertex of $(D,T,E)$,\n\tand let $N = \\Gamma(D,T,V\\BSET{q})$ and $N' = \\Gamma(D_q,T,V\\BSET{q})$\n\twhere $$D_q = \\left(\\vphantom{A^A} V\\BSET{q}, \\SET{(u,v)\\in A~\\middle|~u\\not= q {\\mathop{\\text{~and~}}} v\\not=q} \\right).$$\n\tThen there is a circuit $C\\in {\\mathcal{C}}(N')$ with $C\\cap \\SET{u \\in V~\\middle|~ (u,q)\\in A} \\not= \\emptyset$\n\tsuch that \\linebreak $C$ is independent in $N$.\n\\end{lemma}\n\n\\begin{proof}\\PRFR{Apr 1st}\nWithout loss of generality we may assume\n\tthat $(q,q)\\notin A$, and we do induction on the number of arcs entering $q$ in $D$. \n\tThere has to be at least one arc $(u,q)\\in A$ with $u\\in V$,\n\tbecause there is a subset $X\\subseteq E$, such that every routing $R\\colon X\\double{\\rightarrow} T$\n\tvisits the vertex $q\\notin E$. So $q$ cannot be a source in $D$.\n\tTherefore, the base case of the induction is the case where precisely one arc $(u,q)$ enters $q$ in $D$.\n\tThis arc is essential\n\twith respect to $(D,T,E)$,\n\tsince it is traversed by every routing from $X$ to $T$ in $D$.\n\tLemma~\\ref{lem:essentialArcsC} yields a desired circuit $C$ with $u\\in C$.\n\tIf there is a non-essential arc entering $q$, then $\\Gamma(D,T,E) = \\Gamma(D',T,E)$\n\twhere $D'=(V,A\\BSET{(u,q)})$ for an arbitrarily chosen non-essential arc $(u,q)\\in A$,\n\tand then the existence of $C$ follows by induction hypothesis on $(D',T,E)$.\n\tIf all arcs entering $q$ are essential for $(D,T,E)$, then we pick an arbitrary choice $(u,q)\\in A$,\n\tand throw away all other arcs entering $q$. Let $D'' = (V,A'')$\n\t where $A'' = \\SET{(x,y)\\in A~\\middle|~ y\\not= q}\\cup\\SET{(u,q)}$,\n\t and let $M' = \\Gamma(D'',T,E)$. Clearly, every independent set of $M'$ is also independent in $M$,\n\t and if we delete $q$ and all incident arcs from $D''$ we obtain $D_q$.\n\t Furthermore, there is exactly one arc\n\t entering $q$ in $D''$,\n\t and therefore we obtain a circuit $C\\in{\\mathcal{C}}(N')$ with $u\\in C$ \n\t that is independent in $\\Gamma(D'',T,V\\BSET{q})$ -- and therefore independent in $N$ -- \n\t from the induction hypothesis applied to $(D'',T,E)$.\n\\end{proof}\n\n\\subsection{Digraphs as Black Boxes}\n\n\\PRFR{Mar 7th}\n\\noindent Lemma~\\ref{lem:sourcesinkrepresentation} states that each gammoid $M$ may be represented by\na triple $(D,T,E)$, where $T\\subseteq E$ is a base of $M$ and where $D=(V,A)$ is a digraph, such that\nall $t\\in T$ are sinks and all $e\\in E\\backslash T$ are sources of $D$. Given such a representation, we may\ndisregard the structure of $D$. Instead, we may regard $D$ merely as a function, which assigns\nto each pair $(X,S)$ --- where $X\\subseteq E$ and where $S\\subseteq T$ \nsuch that\n$X\\cap T \\subseteq S$ holds --- the\nminimal cardinality of an $X$-$S$-separator in $D$. Clearly, the value of this function with respect to $D$ equals the rank of $X$\nwith respect to the contraction $M|'\\left( V\\backslash T \\right) \\cup S$, and therefore the function derived from $D$\ndoes not depend on\nthe choice of the representation $(D,T,E)$ of $M$, it is already determined by $M$ alone.\nIn this section, we will elaborate this idea.\n\n\\begin{definition}\\label{def:Mblackbox}\\PRFR{Mar 7th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, $B\\subseteq E$, $\\rho\\colon 2^E \\times 2^B \\longrightarrow \\mathbb{N}$ a map.\n\t The pair $(B,\\rho)$\\label{n:Brho} shall be called \\deftext[M-black box@$M$-black box]{$\\bm M$-black box},\n\tif $B$ is a base of $M$ and if for all $X\\subseteq E$ and all $S\\subseteq B$\n\tthe equation\n\t\\[ \\rho(X,S) = \\mathrm{rk}_{M|'\\left( E\\backslash B \\right)\\cup S} (X\\backslash (B\\backslash S)) \\]\n\tis satisfied. If $B$ is clear from the context, we also denote the $M$-black box $(B,\\rho)$ by $\\rho$ alone.\n\\end{definition}\n\n\\noindent Clearly, for every $B\\in{\\mathcal{B}}(M)$, there is a unique $M$-black box $(B,\\rho)$.\n\n\\begin{definition}\\label{def:digraphBlackbox}\\PRFR{Mar 7th}\n\tLet $D=(V,A)$ be a digraph, and $X,Y\\subseteq V$.\n\tThe \\deftext[black box for $(X,D,Y)$]{black box for $\\bm (\\bm X\\bm,\\bm D\\bm,\\bm Y\\bm)$} shall be the map\n\t\\[ \\lambda_{(X,D,Y)} \\colon 2^X \\times 2^Y \\longrightarrow \\mathbb{N} \\]\n\twhere for all $S\\subseteq X$ and all $T\\subseteq Y$\n\t\\[ \\lambda_{(X,D,Y)}(S,T) = \\min \\SET{\\left| C \\right| ~\\middle|~\\vphantom{A^A} C\\subseteq V \\text{~s.t.~} C \\text{~is an }S-T-\\text{separator in~} D} .\\]\n\tIf $(X,D,Y)$ is clear from the context, we may denote $\\lambda_{(X,D,Y)}$ by $\\lambda$, too.\n\\end{definition}\n\n\\begin{definition}\\PRFR{Mar 7th}\n\tLet $X,Y$ be finite sets with $X\\cap Y = \\emptyset$,\n\tand let\n\t $\\lambda\\colon 2^X\\times 2^Y\\longrightarrow \\mathbb{N}$ be a map.\n\t Then $\\lambda$ shall be called a \\deftext[D-black box@$D$-black box]{$\\bm D$-black box},\n\tif there is a digraph $D=(V,A)$ with $X\\cup Y\\subseteq V$ such that for all $X'\\subseteq X$ and $Y'\\subseteq Y$\n\twith $X'\\cap Y \\subseteq Y'$\n\t\\[\\lambda(X',Y') = \\lambda_{(X,D,Y)}(X',Y').\\] \n\n\tIn this case, we say that $\\lambda$ is a $D$-black box represented by $(X,D,Y)$.\n\\end{definition}\n\n\n\\begin{corollary}\\PRFR{Mar 7th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, $B\\in{\\mathcal{B}}(M)$ a base of $M$, and let $(B,\\rho)$ be the corresponding $M$-black box.\n\t Then $M$ is a gammoid if and only if $\\rho$ is a $D$-black box.\n\\end{corollary}\n\\begin{proof}\\PRFR{Mar 7th}\n\tThere is a standard representation $(D,T,E)$ of the gammoid $M$ such that $D=(V,A)$ and $T=B$ (Remark~\\ref{rem:standardRepresentation}).\n\tThen $(E, D, T)$ \n\trepresents $\\rho$ due to Menger's Theorem~\\ref{thm:MengerGoering}, Definition~\\ref{def:gammoid}, and the fact that \n\tfor all $T'\\subseteq T$, we have the equality $\\Gamma(D,T,E)|'(E\\backslash T') = \\Gamma(D',T\\backslash T',E\\backslash T')$ \n\twhere $D'=(V\\backslash T', A\\cap\\left( (V\\backslash T')\\times (V\\backslash T')\\right)$ --- this is a special case of the construction used in the proof of\n\tLemma~\\ref{lem:contractionStrictGammoid}. Therefore every $M$-black box is a $D$-black box.\n\tConversely, if the $M$-black box $\\rho$ is represented by $(X,D,Y)$,\n\tthen $M = \\Gamma(D,Y,X)$ and so $M$ is a gammoid.\n\\end{proof}\n\n\\begin{definition}\\label{def:cascadeDigraph}\\PRFR{Mar 7th}\n\tLet $D=(V,A)$ be a digraph. Then $D$ shall be called \\deftext{cascade digraph},\n\tif there is a partition $V_1\\mathbin{\\dot{\\cup}} V_2 \\mathbin{\\dot{\\cup}} \\cdots \\mathbin{\\dot{\\cup}} V_k = V$\n\tsuch that $A\\subseteq \\bigcup_{i=1}^{k-1}\\left( V_i \\times V_{i+1} \\right)$.\n\\end{definition}\n\n\\begin{definition}[\\cite{M72}]\\PRFR{Mar 7th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid.\n\tThen $M$ is a \\deftext{cascade},\n\tif there is a digraph $D=(V,A)$, such that there is a partition $V_1\\mathbin{\\dot{\\cup}} V_2 \\mathbin{\\dot{\\cup}} \\cdots \\mathbin{\\dot{\\cup}} V_k = V$\n\twith $A\\subseteq \\bigcup_{i=1}^{k-1} \\left( V_i \\times V_{i+1} \\right)$, and such that\n\t\\[ M = \\Gamma(D,V_k,V_1). \\qedhere\\]\n\\end{definition}\n\n\n\\begin{proposition}[\\cite{M72}, \\cite{Ma70thesis}\\textsuperscript{\\ref{ftn:Ma70}}]\\label{prop:cascadesNonDual}\\PRFR{Mar 7th}\nLet ${\\mathcal{C}}{\\mathcal{M}}$ be the class of all cascades. Then\n\\[ \\SET{M^\\ast ~\\middle|~ M\\in {\\mathcal{C}}{\\mathcal{M}}} \\not\\subseteq {\\mathcal{C}}{\\mathcal{M}} \\quad{\\mathop{\\text{~and~}}}\\quad\n \\SET{M|' E' ~\\middle|~ M=(E,{\\mathcal{I}})\\in {\\mathcal{C}}{\\mathcal{M}},\\,E'\\subseteq E} \\not\\subseteq {\\mathcal{C}}{\\mathcal{M}}.\\]\nIn other words, the class of cascades is neither closed under taking duals nor under contraction.\n\\end{proposition}\n\\stepcounter{footnote}\n\\footnotetext[\\thefootnote]{\\label{ftn:Ma70}Unfortunately, we were not able to acquire a copy of J.H.~Mason's thesis from within Europe before the printing deadline. The thesis contains the proof that the dual of a certain cascade is not a cascade itself, which is only cited in \\cite{M72}. It appears to be available at the Memorial Library,\nUW Madison Theses Basement North\nAWB M411 J655.}\n\n\\PRFR{Mar 7th}\n\\noindent Clearly, every cascade digraph is acyclic, and it is also clear that\n the \\deftext{transitive triple digraph} $\\left(\\SET{x,y,z},\\SET{(x,y),(y,z),(x,z)}\\right)$ is an acyclic digraph but not a cascade digraph. But regarding $D$-black boxes, the class of cascade digraphs and the class of acyclic digraphs have the same expressiveness.\n\n\n\\needspace{6\\baselineskip}\n\\begin{lemma}\\label{lem:DblackBoxArcSubdivision}\\PRFR{Mar 7th}\n\tLet $D=(V,A)$ be a digraph and let $X,Y\\subseteq V$, and $a=(u,w) \\in A$. Furthermore, let $v\\notin V$ be a new element.\n\tThen\n\t$$\\lambda_{(X,D,Y)} = \\lambda_{(X,D',Y)}$$ where\n\t$$ D' = (V\\mathbin{\\dot{\\cup}}\\SET{v}, A\\BSET{a} \\cup \\SET{(u,v),(v,w)}) $$\n\tdenotes the digraph obtained from $D$ by subdividing the arc $a$ with the new vertex $v$.\n\\end{lemma}\n\\begin{proof}\\PRFR{Mar 7th} Clearly, $v\\notin X\\cup Y \\subseteq V$.\n\tThe statement of the lemma follows from the fact that for all $x\\in X$ and $y\\in Y$ there is an obvious bijection\n\t\\[ \\phi \\colon {\\mathbf{P}}(D;x,y) \\longrightarrow {\\mathbf{P}}(D';x,y), \\, p\\mapsto \\begin{cases}[r]\n\t\t\t\tp & \\quad \\text{if~}a\\notin \\left| p \\right|_A,\\\\\n\t\t\t\tqvr & \\quad \\text{otherwise,}\n\t\t\t\\end{cases}.\\]\n\twhere $q = (p_1,p_2,\\ldots, p_j)$ and $r = (p_j,p_{j+1},\\ldots, p_n)$ for\n\t$p = (p_i)_{i=1}^n$ and $j\\in \\SET{1,2,\\ldots,n}$ such that $p_j = u$, and consequently, $p_{j+1} = w$.\n\tLet $X'\\subseteq X$ and $Y'\\subseteq Y$. The map $\\phi$ yields that\n\t every $X'$-$Y'$-separator in $D$ is also an $X'$-$Y'$-separator in $D'$, as well as\n\tevery $X'$-$Y'$-separator $S$ in $D'$ with $v\\notin S$ is an $X'$-$Y'$-separator in $D$.\n\tFurthermore, if $S$ is an $X'$-$Y'$-separator in $D'$ with $v\\in S$, then $S\\BSET{v}\\cup\\SET{u}$\n\tis an $X'$-$Y'$-separator in $D$ of the same or less cardinality. Therefore $\\lambda_{(X,D,Y)} = \\lambda_{(X,D',Y)}$,\n\tsince the values of those maps only depend on the cardinality of their respective minimal separators.\n\\end{proof}\n\n\n\n\\begin{lemma}\\label{lem:acyclicToCascade}\\PRFR{Mar 7th}\n\tLet $D=(V,A)$ be an acyclic digraph, and let $X,Y\\subseteq V$.\n\tThen there is a cascade digraph $D'$ such that\n\t\\[ \\lambda_{(X,D,Y)} = \\lambda_{(X,D',Y)}.\\]\n\\end{lemma}\n\\begin{proof}\\PRFR{Mar 7th}\n\tWithout loss of generality we may assume that $X\\cap Y =\\emptyset$, since otherwise we could introduce a copy $v'$\n\tfor every vertex $v\\in X\\cap V$ and add a single arc leaving $v$ and entering $v'$ to $D$, and then\n\tcontinue with $Y' = Y\\backslash X \\cup \\SET{v'~\\middle|~v\\in X\\cap Y}$. Using similar constructions, \n\twe may also assume without loss of generality, that $X$ consists\n\tof sources of $D$ and $Y$ consists of sinks of $D$, as well as that $D$ has no sources and no sinks in $V\\backslash\\left( X\\cup Y \\right)$.\n\tPossibly renaming elements from $V$, we may further assume that $V\\cap \\left( A\\times \\mathbb{N} \\right) = \\emptyset$.\n\tSince $D$ is acyclic, there is a strict linear order\\footnote{That is a binary relation, which is irreflexive, \n\t\tantisymmetric, and transitive.} $\\prec$ on $V$ such that $u \\prec v$ holds for all $(u,v)\\in A$.\n\tLet $D' = (V', A')$ be the digraph where\n\t \\begin{align*} V' = & V \\mathbin{\\dot{\\cup}} \\SET{((u,v),i) \\in A\\times \\mathbb{N} ~\\middle|~ \\vphantom{A^A}1 \\leq i \\leq \\left| \\SET{x\\in V~\\middle|~ u \\prec x \\prec v} \\right| } \\text{~and}\\\\\n\t A' =& \\hphantom{\\cup}\\,\\,\\SET{(u,v)\\in A ~\\middle|~\\vphantom{A^A} \\SET{x\\in V~\\middle|~ u \\prec x \\prec v} = \\emptyset} \\\\\n\t & \\cup \\SET{(u,((u,v),1)) ~\\middle|~\\vphantom{A^A}(u,v)\\in A,\\, \\SET{x\\in V~\\middle|~ u \\prec x \\prec v} \\not= \\emptyset}\\\\\n\t & \\cup \\SET{(((u,v),k),v) ~\\middle|~\\vphantom{A^A}(u,v)\\in A,\\, k = \\left| \\SET{x\\in V~\\middle|~ u \\prec x \\prec v} \\right| \\not= 0}\\\\\n\t & \\cup \\SET{(((u,v),k),((u,v),k+1)) ~\\middle|~\\vphantom{A^A}(u,v)\\in A, k\\in \\mathbb{N},\\, 1 \\leq k < \\left| \\SET{x\\in V~\\middle|~ u \\prec x \\prec v} \\right|}.\n\t \\end{align*}\n\tIn words, every arc $(u,v)\\in A$ is subdivided \n\tby $k$ new vertices, where $k$ equals the number of vertices that the arc $(u,v)$ skips with\n\trespect to the strict linear order $\\prec$ on $V$. For instance, if $(u,v)\\in A$ and $u \\prec x \\prec y \\prec v$ is a maximal chain\n\tconnecting $u$ with $v$ in $\\prec$, then the arc $(u,v)$ is subdivided by the new vertices $((u,v),1)$ and $((u,v),2)$.\n Repeated application of Lemma~\\ref{lem:DblackBoxArcSubdivision} yields that\n\t\\[ \\lambda_{(X,D,Y)} = \\lambda_{(X,D',Y)} .\\]\n\tWe define the map \n\t\\begin{align*} \\phi\\colon V' \\longrightarrow &\\SET{\\vphantom{A^A}1,2,\\ldots,\\left| V\\backslash\\left( X\\cup Y \\right) \\right|+1},\n\t\\\\v'\\mapsto &\\begin{cases}[r]\n\t\t\t\t\t1 & \\quad \\text{if~}v'\\in X,\\\\\n\t\t\t\t\t1 + \\left| \\SET{u\\in V~\\middle|~ u \\prec v'} \\right| & \\quad \\text{if~}v'\\in V\\backslash X,\\\\\n\t\t\t\t\t1 + i + \\left| \\SET{x\\in V~\\middle|~ x \\prec u} \\right| & \\quad \\text{if~}v'=((u,v),i)\\in A\\times \\mathbb{N}.\\\\\n\t\t\t\t\\end{cases}\n\t\\end{align*}\n\tLet $k = \\left| V\\backslash\\left( X\\cup Y \\right) \\right|+1$, then there is a partition $V'_1,V'_2,\\ldots,V'_k$ of $V'$\n\twith \\[ V'_i = \\SET{v'\\in V'~\\middle|~ \\phi(v') = i} \\] for all $i\\in\\SET{1,2,\\ldots,k}$ that has the property that\n\t\\[ A' \\subseteq \\bigcup_{i=1}^{k-1}\\left( V'_i \\times V'_{i+1} \\right).\\]\n\tTherefore, $D'$ is a cascade digraph where the set $X = V_1$ and the set $Y = V_k$.\n\\end{proof}\n\n\\needspace{6\\baselineskip}\n\\begin{example}\\PRFR{Mar 7th}\n\tThe construction from Lemma~\\ref{lem:acyclicToCascade} applied to\n\t\\begin{center}\n\t\\includegraphics{Figs\/acyclicDigraph}\n\t\\end{center}\n\tyields\n\t\\begin{center}\n\t\\includegraphics{Figs\/acyclicDigraphB},\n\t\\end{center}\n\twhere the vertices and arcs that do not belong to the original digraph are depicted red.\n\\end{example}\n\n\n\\needspace{4\\baselineskip}\n\\begin{corollary}\\label{cor:acyclicDigraphIsCascade}\\PRFR{Mar 7th} Let $D=(V,A)$ be an acyclic digraph, $E,T\\subseteq V$. Then there is a cascade digraph $D'=(V',A')$\nwith a partition $V'_1 \\mathbin{\\dot{\\cup}} V'_2\\mathbin{\\dot{\\cup}}\\cdots\\mathbin{\\dot{\\cup}} V'_k = V'$ such that $A'\\subseteq \\bigcup_{i=1}^{k-1}\\left( V'_i\\times V'_{i+1} \\right)$\nand such that\n\t\\[\\Gamma(D,T,E) = \\Gamma(D',V'_k, V'_1).\\]\nEvery gammoid that can be represented using an acyclic digraph is a cascade.\n\\end{corollary}\n\\begin{proof}\n\tDirect consequence of the proof of Lemma~\\ref{lem:acyclicToCascade}.\n\\end{proof}\n\n\\begin{remark}\\label{rem:weNeedCycles}\\PRFR{Mar 7th}\n\tCorollary~\\ref{cor:acyclicDigraphIsCascade}, together with Proposition~\\ref{prop:cascadesNonDual} stating that cascades are not closed under duality, implies that cycle walks are inevitable in representations of some gammoids, since the class of gammoids is closed under duality (Lemma~\\ref{lem:dualityrespectingrepresentation}).\n\\end{remark}\n\n\n\n\\section{Strict Gammoids}\\label{sec:StrictGammoids}\n\n\\begin{definition}\\PRFR{Jan 22nd}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid. $M$ is a \\deftext{strict gammoid} if there is a digraph $D=(V,A)$ and a set $T\\subseteq V$ such that\\label{n:GDTV} \\( M = \\Gamma(D,T,V) \\).\\footnote{Note that this implies that $V=E$, so $D=(E,A)$ is a digraph where the ground set of $M$ is the vertex set of $D$.}\n\\end{definition}\n\n\\PRFR{Jan 22nd}\n\\noindent It is clear from this definition, that every gammoid is a deletion-minor of a strict gammoid, as $\\Gamma(D,T,E)$ for $D=(V,A)$ is a deletion-minor of $\\Gamma(D,T,V)$.\nGiven a strict gammoid representation $(D,T,V)$, then $T\\subseteq V$ is a base of $M$ and $\\mathrm{rk}_M(V) = \\left| T \\right|$.\nThe following characterization of the rank function of a strict gammoid was given by C.~McDiarmid in \\cite{McD72}, where it is used in order to proof that gammoids are indeed matroids.\n\n\\begin{theorem}\\PRFR{Jan 22nd}\n\tLet $D=(V,A)$ and $T\\subseteq V$. Let $M=\\Gamma(D,T,V)$ be the strict gammoid represented by $(D,T,V)$. Then for $X\\subseteq V$,\n\t\\[ \\mathrm{rk}(X) = \\min_{U\\subseteq V\\backslash T} \\left( \\left| X\\backslash \\Dcl{U} \\right| + \\left| \\partial U \\right| \\right). \\]\n\\end{theorem}\n\n\\begin{proof}\\PRFR{Jan 22nd}\n\tLet $X\\subseteq V$ and $U\\subseteq V\\backslash T$.\n\tTrivially, $\\partial U$ separates $X\\cap \\Dcl{U}$ from $T$\n\t in $D$.\n\tLet $S = \\partial U \\cup \\left( X\\backslash \\Dcl{U} \\right)$, then $S$ is an\n\t$X$-$T$-separator in $D$: let $x\\in X \\backslash S$, then $x\\in \\Dcl{U}$. Since $U\\cap T=\\emptyset$, every path $p\\in{\\mathbf{P}}(D)$ with $p_1 = x$ and $p_{-1}\\in T$ must leave the\n\t set $U$ at some point. Consequently, it must visit a vertex from $\\partial U$, \n\t so $\\left| p \\right|\\cap S\\not=\\emptyset$. It follows that\n\t \\[ \\mathrm{rk}(X) \\leq \\left| S \\right| = \\left| \\partial U \\cup \\left( X\\backslash \\Dcl{U} \\right) \\right|\n\t \\leq \\left| X\\backslash \\Dcl{U} \\right| + \\left| \\partial U \\right| ,\\]\n\t i.e. that the right-hand side of the equation in the lemma is an upper bound for the left-hand side of that equation.\n\n\t \\needspace{3\\baselineskip}\n \n\t \\noindent\n\t Now let $S\\subseteq V$ be an $X$-$T$-separator in $D$ \n\t with $\\left| S \\right| = \\mathrm{rk}(X)$, its existence is guaranteed by \n\t Menger's Theorem~\\ref{thm:MengerGoering}. \n\t \\input{Text\/Fig\/McDiarmid-Fig.tex}\t \n\t Let\n\t \\[ U = \\SET{p_{-1} \\in V ~\\middle|~\\vphantom{A^A} p\\in {\\mathbf{P}}(D)\\colon\\, p_1\\in X {\\mathop{\\text{~and~}}} \\left| p \\right|\\cap S=\\emptyset}\\]\n\t denote the set of vertices, that can be reached from any vertex $x\\in X$ by a path not visiting $S$. \n\t Clearly, $U\\subseteq V\\backslash T$ holds because $S$ is an $X$-$T$-separator in $D$, and the outer margin\n\t $\\partial U$ is a subset of $S$ whereas $S\\cap U = \\emptyset$ by construction. Let $S' = S\\cap \\partial U$ and let\n\t $X' = S\\backslash \\Dcl{U}$. $X'$ is indeed a subset of $X$: Let $s\\in S\\backslash X$,\n\t since $S$ is a minimal $X$-$T$-separator, every maximal $X$-$T$-connector $R$ has\n\t a path $p\\in R$ with $S\\cap \\left| p \\right|=\\SET{s}$.\n\t We obtain that $s = p_k$ for some $k\\in\\mathbb{N}$,\n\t and since $s\\notin X$ but $p_1\\in X$, we have $k > 1$. Therefore $p_{k-1}\\in U$ and\n\t then $s\\in \\partial U \\subseteq \\Dcl{U}$, so $s\\notin X'$ --- this establishes $X'\\subseteq X$. This yields $X' = X\\cap X' = X\\cap \\left( S \\backslash \\Dcl{U} \\right) = \\left( X\\cap S \\right)\\backslash \\Dcl{U}$, and since $X\\backslash S\\subseteq U \\subseteq \\Dcl{U}$, we have $X' = X\\backslash \\Dcl{U}$.\n\t Since $S\\cap \\Dcl{U} = S\\cap\\left( U\\cup \\partial U \\right) = \\left( S\\cap U \\right)\\cup \\left( S\\cap\\partial U \\right) = S'$, we have\n\t \\( \\left| S \\right| = \\left| X' \\right| + \\left| S' \\right|\n\t \t \\).\n\t Now assume that $S'\\subsetneq \\partial U$, then there is a vertex $u\\in\\partial U$\n\t with $u\\notin S$, such that there is a path $p\\in {\\mathbf{P}}(D)$ with $p_1\\in X$, $\\left| p \\right|\\cap S =\\emptyset$, and there is an arc $(p_{-1},u)\\in A$. But then we obtain that $u\\in U$, and therefore $u\\notin \\partial U$: \n\t Since $pu\\in {\\mathbf{P}}(D)$ is a path with $\\left| pu \\right| = \\left| p \\right|\\cup\\SET{u}$,\n\t and since $u\\notin S$, we have $\\left| pu \\right|\\cap S = \\emptyset$, and so $u$ qualifies as a member of $U$.\n\t Therefore $u\\notin S$ cannot be the case and so $S' = \\partial U$ holds.\n\t Thus we obtain\n\t \\[ \\mathrm{rk}(X) = \\left| S \\right| = \\left| X' \\right| + \\left| S' \\right| = \\left| X\\backslash \\Dcl{U} \\right| + \\left| \\partial U \\right| \\]\n\t and therefore, on the right-hand side of the equation in the lemma, the minimum expression ranges over an upper bound that is equal to\n\t $\\mathrm{rk}(X)$, \n\t therefore both sides of the equation must be equal.\n\\end{proof}\n\n\\PRFR{Jan 22nd}\n\\noindent\nJ.H.~Mason gives a necessary and sufficient condition for when a matroid $M$ is a strict\ngammoid in \\cite{M72}. In order to present the proof, we need the Lemma~2.1 \nfrom \\cite{M72}. Here, we give a slightly more detailed version of J.H.~Mason's proof.\nBut first, we want to introduce the following notion of a special $X$-$T$-separator.\n\n\\begin{definition}\\label{def:rightmost-separator}\\PRFR{Jan 22nd}\n\tLet $D=(V,A)$ be a digraph, and let $X\\subseteq V$ and $T\\subseteq V$ be sets of vertices.\n\tThe \\deftext[barrier between X and T in D@barrier between $X$ and $T$ in $D$]{barrier between $\\bm X$ and $\\bm T$ in $\\bm D$}\n\tis defined to be the set\\label{n:barrier}\n\t\\[ \\delta_D(X,T) = \\SET{x\\in X~\\middle|~\\vphantom{A^A} \\left( \\partial_D \\SET{x}\\right) \\cap \\left( V\\backslash X \\right) \\not=\\emptyset} \\,\\cup\\, \\left( X\\cap T \\right). \\qedhere\\]\n\\end{definition}\n\n\\begin{lemma}\\label{lem:barrier}\\PRFR{Jan 22nd}\n\tLet $D=(V,A)$ be a digraph, and let $X\\subseteq V$ and $T\\subseteq V$ be sets of vertices.\n\tThen the barrier $\\delta_D(X,T)$ is an $X$-$T$-separator in $D$.\n\\end{lemma}\n\\begin{proof}\\PRFR{Jan 22nd}\n \tLet $R$ be an $X$-$T$-connector, and let \n\t$p=(p_i)_{i=1}^k \\in R$ be a path that does not end in a vertex from $X\\cap T$. Then \n\tthere is a maximal integer $1\\leq j < k$ such that $p_j\\in X$. Then $p_{j+1}\\notin X$, \n\tyet $(p_j,p_{j+1})\\in A$, thus $p_{j+1}\\in \\partial\\SET{p_j}$ and so $p_j\\in \\delta_D(X,T)$.\n\t If otherwise $p\\in R$ is a path that ends in $X\\cap T$ we clearly have $p_{-1}\\in \\delta_D(X,T)$ --- thus $\\delta_D(X,T)$\n\t is an $X$-$T$-separator.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:minFTSeparatorInFStrictGammoid}\\PRFR{Jan 22nd}\n\tLet $D=(V,A)$ be a digraph, $T\\subseteq V$, $F\\in {\\mathcal{F}}(\\Gamma(D,T,V))$, and let $S\\subseteq V$ be an $F$-$T$-separator in $D$\n\twith minimal cardinality. Then $S\\subseteq F$.\n\\end{lemma}\n\\begin{proof}\\PRFR{Jan 22nd}\n\tLet $M= \\Gamma(D,T,V)$,\n\tand let $R\\colon B_F\\double{\\rightarrow} T$ be a maximal $F$-$T$-connector in $D$ where\n\t$B_F$ is a base of $F$ in $M$.\n\tThen every $s\\in S$ is visited by a path $p\\in R$ (Corollary~\\ref{cor:Menger}), thus any path\n\t$q\\in {\\mathbf{P}}(D)\\backslash R$ with $q_1\\in S$ has the property that $\\SET{s}\\subseteq \\left| q \\right|\\cap \\left| p \\right|$,\n\tso $R\\cup\\SET{q}$ can never be a routing in $D$.\n\t Therefore $R$ cannot be extended by a path starting in $s$, so $R$ is also a maximal $F\\cup S$-$T$-connector in $D$.\n\t Therefore $\\mathrm{rk}(F) = \\mathrm{rk}(F\\cup S)$, so $S\\subseteq \\mathrm{cl}(F) = F$.\n\\end{proof}\n\n\\needspace{6\\baselineskip}\n\n\\begin{lemma}\\label{lem:flatofstrictgammoidisstrictgammoid}\\PRFR{Jan 22nd}\n\tLet $M=(E,{\\mathcal{I}})$ be a strict gammoid, and $F\\in{\\mathcal{F}}(M)$ be a flat of $M$.\n\tThen the restriction $M| F$ is a strict gammoid.\n\tFurthermore, if $D=(E,A)$ and $T\\subseteq E$ with $M=\\Gamma(D,T,E)$, then\n\tthe barrier $\\delta_D(F,T)$ is an $F$-$T$-separator of minimal cardinality in $D$.\n\\end{lemma}\n\n\n\\begin{proof}\\PRFR{Jan 22nd}\n\tLet $M=\\Gamma(D,T,E)$ for suitable $D=(E,A)$ and $T\\subseteq E$.\n\t\\begin{figure}[t]\n\t\\begin{center}\n\t\\includegraphics[scale=1.2]{lemmaflatsstrictgammoid}\n\n\t\\end{center}\n\t\\caption{Situation of the ``right-most'' $F$-$T$-separator $B$ in $D$.\\label{fig:FTBD}}\n\t\\end{figure}\n\tNow let\n\t\\[ B = \\delta_D(F,T) = \\SET{f\\in F~\\middle|~ \\vphantom{A^A}\\left( \\partial \\SET{f} \\right) \\cap \\left(E \\backslash F\\right) \\not= \\emptyset} \\cup \\left( F\\cap T \\right) \\]\n\tbe the barrier between $F$ and $T$ in $D$ (Fig.~\\ref{fig:FTBD}), i.e. the set\n\twhich consists of those $f\\in F$, that are either targets of the representation $(D,T,E)$, or that\n\thave an out-arc which leaves the flat $F$. Clearly, $B$ is an $F$-$T$-separator in $D$ (Lemma~\\ref{lem:barrier}).\n\tWe give an indirect argument that $B$ is a\n\tminimal $F$-$T$-separator in $D$:\n\tAssume that $B$ is not a minimal $F$-$T$-separator, then there is a set $S\\subseteq F$, which is\n\ta minimal $F$-$T$-separator (Lemma~\\ref{lem:minFTSeparatorInFStrictGammoid}), and there is an element $b\\in B\\backslash S$, since $\\left| B \\right| > \\left| S \\right|$.\n\t Clearly, $b\\notin T$ since $F\\cap T$ is a subset of every $F$-$T$-separator.\n\t Further, there is an element $e\\in E\\backslash F$ such that $(b,e)\\in A$ according to the definition of $B$, \n\t and since $e\\notin F=\\mathrm{cl}_M(F)$, \n\t there is a maximal $F$-$T$-connector $R$ and a path $p\\in {\\mathbf{P}}(D)$ with $p_1 = e$ \n\t and $p_{-1}\\in T$, such that $R\\cup\\SET{p}$ is a routing in $D$. \n\t Thus the path $p$ does not visit any vertex that belongs to a minimal $F$-$T$-separator in $D$, \n\t and therefore the path $bp$ does not visit any \n\t vertex of $S$, too --- which contradicts the assumption that $S$ is an $F$-$T$-separator, \n\t thus $B$ must be a minimal $F$-$T$-separator.\n\n\t \\par \\vspace*{.6\\baselineskip}\\noindent \\mbox{---\\hspace*{2.5mm}} \\PRFR{Jan 22nd}\n\n\tLet $D'=\\left( F,A\\cap\\left( F\\times F \\right) \\right)$ be the restriction of $D$ to $F$, and let $M' = \\Gamma(D',B,F)$ be the strict gammoid presented by the \n\trestriction of $D$ and the target set $B$.\n\tLet $R$ be a routing from $X_0\\subseteq F$ to $T$ in $D$,\n\tthen every path $p=(p_i)_{i=1}^k \\in R$ has a smallest integer $1\\leq j(p) \\leq k$,\n\tsuch that $p_{j(p)} \\in B$. By construction of $B$, we have that $\\SET{p_1,p_2,\\ldots,p_{j(p)}} \\subseteq F$. Thus $R$ induces a routing $R'=\\SET{p_1 p_2 \\ldots p_{j(p)} ~\\middle|~p\\in R}$ in $D'$ which routes $X_0$ to $B$. So $\\mathrm{rk}_{M'}(X_0) \\geq \\mathrm{rk}_M(X_0)$.\n\tWe give an indirect argument that the inequality $\\mathrm{rk}_{M'}(X_0) \\leq \\mathrm{rk}_M(X_0)$ holds, too. \n\tLet $S\\subseteq E$ be a minimal $X_0$-$T$-separator in $D$, then we have $S\\subseteq \\mathrm{cl}_M(X_0) \\subseteq F$ (Lemma~\\ref{lem:minFTSeparatorInFStrictGammoid}). \n\tAssume that there is a routing $R'$ from $X_0$ to $B$ in $D'$, such that $\\left| R' \\right| > \\left| S \\right|$. \n\tThen there must be some $p\\in R'$ such that $\\left| p \\right|\\cap S = \\emptyset$ and $p_{-1}\\in B$. If $p_{-1}\\in T$, then $p$ \n\tis a contradiction to $S$ being an $X_0$-$T$-separator in $D$. If otherwise $p_{-1}\\in B\\backslash T$, we have again the situation \n\twhere there is some $e\\in E\\backslash F$ with $(p_{-1},e)\\in A$, such that there is a path $q\\in {\\mathbf{P}}(D)$ with $q_1=e$ and\n\t $q_{-1}\\in T$, that avoids every $F$-$T$-separator. So, consequently, $\\left| q \\right|\\cap F=\\emptyset$.\n\tThe path $pq\\in {\\mathbf{P}}(D)$ contradicts $S$ being an $X_0$-$T$-separator in $D$. \n\tTherefore $\\left| R' \\right| \\leq \\left| S \\right|$, and\n\twe just proved, that for any $X\\subseteq F$ the equation $\\mathrm{rk}_M(X) = \\mathrm{rk}_{M'}(X)$ holds. \n\tThus $M| F = \\Gamma(D',B,F)$ is a strict gammoid.\n\\end{proof}\n\n\n\\needspace{4\\baselineskip}\n\n\\begin{corollary}\\label{cor:gammoidrestrictionsamerank}\\PRFR{Jan 22nd}\n\tLet $M=(E,{\\mathcal{I}})$ be a gammoid. Then there is a strict gammoid $M'=(V,{\\mathcal{I}}')$ such that\n\t$\\mathrm{rk}_M(E) = \\mathrm{rk}_{M'}(V)$ and $M = M'| E$.\n\\end{corollary}\n\\begin{proof} \\PRFR{Jan 22nd}\n\tLet $(D,T,E)$ be a representation of $M$, where $D=(V,A)$. Let \\linebreak $M_0=\\Gamma(D,T,V)$ be the strict gammoid arising naturally from the representation of $M$, and let $F=\\mathrm{cl}_{M_0}(E)$ be the smallest flat in $M_0$ that contains $E$. Then $\\mathrm{rk}_{M_0}(F) = \\mathrm{rk}_{M_0}(E) = \\mathrm{rk}_{M}(E)$ since $M= M_0| E$. Now, let $M'= M_0| F$. \n\t$M'$ is a strict gammoid (Lemma~\\ref{lem:flatofstrictgammoidisstrictgammoid}), and since $E\\subseteq F$, we have $M = M_0 | E = M'| E$, thus $M$ is the\n\trestriction of a strict gammoid of the same rank.\n\\end{proof}\n\n\n\\begin{lemma}\\label{lem:contractionStrictGammoid}\\PRFR{Jan 26th}\n\tLet $M=(E,{\\mathcal{I}})$ be a strict gammoid, $C\\subseteq E$.\n\t Then $M |' C$ is a strict gammoid.\n\\end{lemma}\n\\begin{proof}\\PRFR{Jan 26th}\n\tLet $B_0$ be a base of $E\\backslash C$ in $M$, and let $B$ be a base of $M$ with $B_0\\subseteq B$ \n\t(Lemma~\\ref{lem:augmentation}). Let further $D=(E,A)$ be a digraph, such that $M=\\Gamma(D,B,E)$\n\tand such that $B$ consists only of sinks in $D$\n\t (Theorem~\\ref{thm:gammoidRepresentationWithBaseTerminals}). We denote the family of independent sets of $M|' C$ by ${\\mathcal{I}}'$.\n\tThen for every $X\\subseteq C$, we have $X\\in {\\mathcal{I}}'$ if and only if \\linebreak $X\\mathbin{\\dot{\\cup}} B_0 \\in {\\mathcal{I}}$ \n\t(Lemma~\\ref{lem:contractionBchoice}). But $X\\mathbin{\\dot{\\cup}} B_0\\in {\\mathcal{I}}$\n\t if and only if there is a routing \\linebreak\n\t$R\\colon X\\mathbin{\\dot{\\cup}} B_0 \\double{\\rightarrow} B$ in $D$. Since $B_0\\subseteq B$ consists of sinks in $D$,\n\tfor every $b_0\\in B_0$, the trivial path $b_0\\in{\\mathbf{P}}(D)$ is a member of $R$.\n\tWe give an indirect argument, that for every $e\\in \\left( E\\backslash C \\right)\\backslash B_0$ \n\tand every $p\\in R$, $e\\notin \\left| p \\right|$ holds: If there would be such a path $p=(p_i)_{i=1}^n\\in R$, \n\tthen for some $j\\in \\mathbb{N}$, $p_j = e$. But then the path $q = p_j p_{j+1}\\ldots p_n \\in {\\mathbf{P}}(D)$ yields \n\ta routing $\\SET{q}\\cup\\SET{b_0\\in {\\mathbf{P}}(D)\\mid b_0\\in B_0}$ which implies that\n\t$\\SET{e}\\cup B_0\\in {\\mathcal{I}}$ -- a contradiction to the maximality of the \n\tbase $B_0$ of $E\\backslash C$ in $M$. Thus the routing $R'=\\SET{p\\in R\\mid p_1\\notin B_0}$ routes $X$ to $B\\backslash B_0$ \n\tin $D'=(C,A\\cap \\left( C\\times C \\right))$, the sub-digraph of $D$ induced by $C$.\n\tConversely, every routing $S\\colon Y\\double{\\rightarrow} B\\backslash B_0$ in $D'$ induces the\n\trouting $S\\cup\\SET{b_0\\in {\\mathbf{P}}(D)\\mid b_0\\in B_0}$ from $Y\\mathbin{\\dot{\\cup}} B_0$ to $B$ in $D$,\n\tso $M|' C = \\Gamma(D',B\\backslash B_0,C)$: the contraction is again a strict gammoid.\n\\end{proof}\n\n\n\\subsection{Mason's $\\alpha$-Criterion}\n\n\\PRFR{Jan 30th}\n\\noindent In the proof of Lemma~\\ref{lem:flatofstrictgammoidisstrictgammoid}\nwe have seen that the elements of a flat $F$ of a strict gammoid $M=\\Gamma(D,T,V)$ fall into two disjoint categories: for some $f\\in F$, we have $\\partial \\SET{f} \\subseteq F$,\nand for an independent subset $I\\subseteq F$, we have $\\partial \\SET{i} \\not\\subseteq F$\nfor all $i\\in I$ -- more precisely, there is a base $B$ of $F$ such that $I=B\\backslash T$. Before we present Mason's criterion, we need one last definition.\n\n\\begin{notation}\\PRFR{Feb 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid and $X\\subseteq E$. The family of those flats of $M$, which are proper subsets of $X$, shall\n\tbe denoted by\\label{n:FcalMX}\n\t\\[ {\\mathcal{F}}(M,X) = \\SET{F\\in{\\mathcal{F}}(M)~\\middle|~ F\\subsetneq X}. \\qedhere\\]\n\\end{notation}\n\n\\needspace{6\\baselineskip}\n\\begin{definition}\\label{def:alphaM}\\PRFR{Jan 30th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid. The \\deftext[a-invariant of M@$\\alpha$-invariant of $M$]{$\\bm \\alpha$-invariant of $\\bm M$} shall be the map\\label{n:alphaM}\n\t\\[ \\alpha_M\\colon 2^E \\longrightarrow \\mathbb{Z} \\]\n\tthat is uniquely characterized by the recurrence relation\n\t\\[ \\alpha_M(X) = \\left| X \\right| - \\mathrm{rk}_M(X) - \\sum_{F\\in {\\mathcal{F}}(M,X)} \\alpha_M(F).\\]\n\tIf the matroid $M$ is clear from the context, we also write $\\alpha(X)$ for $\\alpha_M(X)$.\n\\end{definition}\n\n\n\n\\begin{remark}\\label{ref:AlphaIndependent}\\PRFR{Jan 30th}\nClearly, $\\alpha(\\emptyset) = 0$ for any matroid $M$. Furthermore,\nthe value $\\alpha(X)$ for $X\\subseteq E$ may be calculated from the values $\\alpha(X')$\ncorresponding to proper subsets $X'\\subsetneq X$ and the rank of $X$, so $\\alpha$ is well-defined.\n\\end{remark}\n\n\n\\PRFR{Jan 30th}\n\\noindent Just like it is the case for the rank function, the family of bases, and the family of circuits of a matroid,\n we can use $\\alpha_M$ to reconstruct the matroid $M$; thus $M$ is already uniquely determined by $\\alpha_M$.\n\n\\begin{definition}\\label{def:FcalAlpha}\\PRFR{Jan 30th}\n\tLet $E$ be a finite set and let $\\alpha\\colon 2^E \\longrightarrow \\mathbb{Z}$ be a map. \n\tThe \\deftext[zero-family of a@zero-family of $\\alpha$]{zero-family of $\\bm \\alpha$} \\label{n:Ialpha}\n\tshall be\n\t\\[ {\\mathcal{I}}_\\alpha = \\SET{X\\subseteq E ~\\middle|~\\vphantom{A^A} \\forall Y\\subseteq X\\colon\\, \\alpha(Y) = 0}.\\]\n\tThe family of \\deftext[a-flats@$\\alpha$-flats]{$\\bm \\alpha$-flats}\n\tshall be defined as\\label{n:FcalAlpha}\n\t \\[ {\\mathcal{F}}(\\alpha) = \\SET{F\\subseteq E ~\\middle|~\\vphantom{A^A} \\forall e\\in E\\backslash F,\\,X\\subseteq F\\colon\\,\\,\\, X\\in {\\mathcal{I}}_\\alpha {\\mathop{\\text{~and~}}} \\SET{e}\\in{\\mathcal{I}}_\\alpha \\Rightarrow X\\cup\\SET{e}\\in{\\mathcal{I}}_\\alpha}. \\]\n\tFurthermore, we define the pair \\label{n:MAlpha}\n\t\\[ M(\\alpha) = (E,{\\mathcal{I}}_\\alpha). \\qedhere\\]\n\n\n\n\\end{definition}\n\n\\begin{lemma}\\label{lem:alphaIndependent}\\PRFR{Jan 30th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid and let $\\alpha=\\alpha_M$ be its $\\alpha$-invariant.\n\tThen ${\\mathcal{I}}={\\mathcal{I}}_\\alpha$, $\\alpha(X) = 0$ for all $X\\in {\\mathcal{I}}$, and $\\alpha(C) = 1$ for all $C\\in{\\mathcal{C}}(M)$.\n\\end{lemma}\n\n\n\\begin{proof}\\PRFR{Jan 30th}\n\tLet $X\\in{\\mathcal{I}}$, we show $\\alpha(X) = 0$ by induction on $\\left| X \\right|$.\n\tIn the base case, we have $$\\alpha(\\emptyset) = \\left| \\emptyset \\right| - \\mathrm{rk}(\\emptyset) = 0 - 0 = 0.$$\n\tFor the induction step, we may assume by induction hypothesis that for all $Y\\subsetneq X$ the equality $\\alpha(Y) = 0$ holds.\n\tThus\n\t$$ \\alpha(X) = \\left| X \\right| - \\mathrm{rk}(X) - \\sum_{F\\in{\\mathcal{F}}(M,X)} \\alpha(F) = \\left| X \\right| - \\left| X \\right| - \\sum_{F\\in{\\mathcal{F}}(M,X)} 0 = 0.$$\n\tTherefore we obtain ${\\mathcal{I}} \\subseteq {\\mathcal{I}}_\\alpha$.\n\tNow let $X\\subseteq E$ with $X\\notin {\\mathcal{I}}$. Then there is a circuit $C\\in{\\mathcal{C}}(M)$ such that $C\\subseteq X$.\n\tFor all $D\\subsetneq C$, we have $D\\in {\\mathcal{I}}$, therefore $\\alpha(D) = 0$.\n\tSo clearly\n\t$$ \\alpha(C) = \\left| C \\right| - \\mathrm{rk}(C) - \\sum_{F\\in{\\mathcal{F}}(M,C)} \\alpha(F) = \\left| C \\right| - \\left( \\left| C \\right| - 1 \\right) - \\sum_{F\\in{\\mathcal{F}}(M,C)} 0 = 1,$$\n\twhich implies $X\\notin {\\mathcal{I}}_\\alpha$, and we obtain that ${\\mathcal{I}} = {\\mathcal{I}}_\\alpha$.\n\\end{proof}\n\n\n\\begin{corollary}\\PRFR{Jan 30th}\n\tLet $M=(E,{\\mathcal{I}})$ and $N=(E,{\\mathcal{I}}')$ be two matroids defined on the same ground set $E$.\n\tThen $M = N$ if and only if $\\alpha_M = \\alpha_N$.\n\\end{corollary}\n\n\n\\begin{remark}\\PRFR{Jan 30th}\n\tWe may express the rank of a matroid $M=(E,{\\mathcal{I}})$ in terms of the $\\alpha$-invariant of $M$ in different ways.\n\tLet $X\\subseteq E$, then\n\t\\begin{align*}\n\t\t \\mathrm{rk}(X) & = \\left| X \\right| - \\sum_{F\\in{\\mathcal{F}}(M),\\,F\\subseteq X} \\alpha(F) \\\\\n\t\t \t\t& = \\max \\SET{\\left| I \\right|\\vphantom{A^A} ~\\middle|~ I\\subseteq X,\\,\\forall J\\subseteq I\\colon\\, \\alpha(J) = 0} \\\\\n\t\t \t\t& = \\max \\SET{\\left| I \\right|\\vphantom{A^A} ~\\middle|~ I\\subseteq X,\\,I\\in {\\mathcal{I}}_\\alpha }.\n\t\\end{align*}\n\tWe may use this equation in order to give an axiomatization of matroids in terms of its $\\alpha$-invariant,\n\twhich admittedly appears to be not very helpful.\n\t\\begin{enumerate}\n\t\t\\item[\\em (A1)] $\\alpha(\\emptyset) = 0$.\n\t\t\\item[\\em (A2)] For all $X,Y\\subseteq E$ with $\\left| X \\right| < \\left| Y \\right|$ and for which\n\t\t\tthe restrictions\n\t\t\t $\\alpha|_{2^X}$ and $\\alpha|_{2^Y}$ are constantly zero, there is an element\n\t\t\t $y\\in Y\\backslash X$, such that $\\alpha|_{2^{X'}}$ is constantly zero, where $X' = X\\cup\\SET{y}$.\n\t\t\\item[\\em (A3)] For all $X\\subseteq E$ \n\t\t\t\\[ \\alpha(X) = \\left| X \\right| - \\max \\SET{\\left| I \\right| \\vphantom{A^A}~\\middle|~ I\\subseteq X,\\,I\\in{\\mathcal{I}}_\\alpha} - \\sum_{F\\in {\\mathcal{F}}(\\alpha),\\,F\\subsetneq X} \\alpha(F) .\\]\n\t\\end{enumerate}\n\tClearly, {\\em (A2)} resembles the augmentation axiom {\\em (I3)} for ${\\mathcal{I}}_\\alpha$ and {\\em (A1)} \n\tguarantees that $\\emptyset\\in{\\mathcal{I}}_\\alpha$. {\\em (I2)} trivially holds for ${\\mathcal{I}}_\\alpha$ by construction,\n\tand so $M(\\alpha) = (E,{\\mathcal{I}}_\\alpha)$ is a matroid. Then {\\em (A3)} guarantees that $\\alpha = \\alpha_{M(\\alpha)}$,\n\ti.e. that $\\alpha$ behaves like the $\\alpha$-invariant for $M(\\alpha)$ on the dependent sets.\n\\end{remark}\n\n\n\\needspace{6\\baselineskip}\n\\begin{theorem}[\\cite{M72}, Theorem~2.2]\\label{thm:AlphaCriterion}\\PRFR{Jan 30th}\n\tLet $D=(V,A)$, $T\\subseteq V$, and $M=\\Gamma(D,T,V)$ be a strict gammoid.\n\tThen for all $X\\subseteq V$, we have $\\alpha(X) \\geq 0$.\n\t%\n\tFurthermore, if $F\\in{\\mathcal{F}}(M)$\n\tthen $\\alpha(F)$ is the number of elements of $F\\backslash T$\n\twith the property,\n\tthat $\\partial \\SET{f}$ is a subset of $F$ but not of any proper sub-flat $F'\\subsetneq F$.\n\\end{theorem}\n\n\n\n \\noindent We present a slightly polished version of the proof in \\cite{M72}.\n\n\\begin{proof}\\PRFR{Jan 30th}\n\tFor every $X\\subseteq E$ we define the subsets\n\t\\begin{align*}\n\t X_1 & = \\SET{x\\in X\\backslash T~\\middle|~\\vphantom{A^A}\\partial \\SET{x} \\not\\subseteq X} \\cup \\left( X\\cap T \\right) = \\delta_D(X,T),\\\\\n\t X_2 & = \\SET{x\\in X\\backslash T~\\middle|~\\vphantom{A^A} \\partial \\SET{x} \\subseteq X{\\mathop{\\text{~and~}}} \\forall F\\in{\\mathcal{F}}(M)\\colon\\,F\\subsetneq X \\Rightarrow \\partial \\SET{x} \\not \\subseteq F},\\text{ and} \\\\\n\t X_3 & = \\SET{x\\in X\\backslash T~\\middle|~\\vphantom{A^A} \\exists F\\in{\\mathcal{F}}(M)\\colon\\,F \\subsetneq X {\\mathop{\\text{~and~}}} \\partial\\SET{x}\\subseteq F}.\n\t \\end{align*}\n\tThen $X= X_1\\mathbin{\\dot{\\cup}} X_2 \\mathbin{\\dot{\\cup}} X_3$ is the disjoint union of $X_1$, $X_2$, and $X_3$.\n\tFurthermore $X_3$ is the disjoint union of the sets $F_2$, where $F$ ranges over all flats in $M$ that are proper subsets of $X$, \n\tbecause if $\\partial\\SET{x}\\subseteq F\\in {\\mathcal{F}}(M)$ and $x\\notin T$, then every path from $x$ to some $t\\in T$ must visit a vertex \n\tof $F$. Therefore every $F$-$T$-separator in $D$ is also an $\\left( F\\cup\\SET{x} \\right)$-$T$-separator in $D$, \n\tso $\\mathrm{rk}(F) = \\mathrm{rk}(F\\cup\\SET{x})$, thus $x\\in F$, so $x\\in F_2$ for some flat $F\\subsetneq X$.\n\tNow assume that $x\\in F_2\\cap G_2$ for some $F,G\\in{\\mathcal{F}}(M)$. \n\tThen $F\\cap G\\in {\\mathcal{F}}(M)$ (Lemma~\\ref{lem:flatsintersectinflat}) and $x\\in F\\cap G$. But $x\\notin F_3$, \n\tso $F\\cap G$ is not a proper subset of $F$, thus $F=F\\cap G$. Analogously $G=F\\cap G$, thus\n\t$F = G$ whenever $x\\in F_2\\cap G_2$. Therefore $F_2\\cap G_2 = \\emptyset$ for every $F,G\\in {\\mathcal{F}}(M)$ with $F\\not= G$.\n\n\t\\par \\vspace*{.6\\baselineskip}\\noindent \\mbox{---\\hspace*{2.5mm}} \n\tFirst, we prove the second claim by induction on the rank of the flat. Let $O=\\mathrm{cl}(\\emptyset)$\n\tbe the unique rank $0$ flat of $M$. Then\n\t$\\alpha(O) = \\left| O \\right| - \\mathrm{rk}(O) = \\left| O \\right|$. We have\n\t$$O=\\SET{v\\in E~\\middle|~\\vphantom{A^A}\\nexists p\\in {\\mathbf{P}}(D)\\colon\\,p_1=v{\\mathop{\\text{~and~}}} p_{-1}\\in T},$$ because\n\t $O$ must consist precisely of those vertices of $D$, which cannot reach any target $t\\in T$.\n\t Therefore $\\partial O = \\emptyset$, which implies that for every $o\\in O$,\n\t $\\partial\\SET{o} \\subseteq O$. Consequently, $O = O_2$ as defined above, \n\t so $\\alpha(O) = \\left| O_2 \\right|$ follows and the induction base is established.\n\t\n\t\\par \\vspace*{.6\\baselineskip}\\noindent \\mbox{---\\hspace*{2.5mm}} \n\tNow let $F\\in{\\mathcal{F}}(M)$ be a flat, and by induction hypothesis we may assume that $\\alpha(F') = \\left| F'_2 \\right|$ for all $F'\\in{\\mathcal{F}}(M)$ with $F'\\subsetneq F$.\n\tThus we may assume the equation\n\t $$\\left| F_3 \\right| = \\sum_{F'\\in{\\mathcal{F}}(M,F)} \\alpha(F').$$\n\tFurthermore, $F_1 = \\delta_D(F,T)$ is a minimal $F$-$T$-separator in $D$ (Lemma~\\ref{lem:flatofstrictgammoidisstrictgammoid}), therefore $F_1$ is a base of $F$, and so\n\t $\\left| F_1 \\right| = \\mathrm{rk}(F)$. We obtain\n\t \\begin{align*}\n\t \t\\left| F \\right| & = \\left| F_1 \\right| + \\left| F_2 \\right| + \\left| F_3 \\right| \\\\\n\t \t& = \\mathrm{rk}(F) + \\left| F_2 \\right| + \\sum_{F'\\in{\\mathcal{F}}(M,F)} \\alpha(F'), \\text{ and so}\\\\\n\t \t\\left| F_2 \\right| & = \\left| F \\right| - \\mathrm{rk}(F) - \\sum_{F'\\in{\\mathcal{F}}(M,F)} \\alpha(F') = \\alpha(F).\n\t \\end{align*}\n\n\t \\par \\vspace*{.6\\baselineskip}\\noindent \\mbox{---\\hspace*{2.5mm}} \n\t Now let $X$ be a subset of $E$ that is not necessarily a flat of $M$. Then $X_1 = \\delta_D(X,T)$ is still an $X$-$T$-separator in $D$\n\t (Lemma~\\ref{lem:barrier}), albeit not necessarily minimal.\n\t Therefore $\\left| X_1 \\right| \\geq \\mathrm{rk}(X)$. Thus we obtain\n\t \\begin{align*}\n\t \t\\alpha(X) & = \\left| X \\right| - \\mathrm{rk}(X) - \\sum_{F\\in{\\mathcal{F}}(M,X)} \\alpha(F) \\\\\n\t \t\t\t & \\geq \\left| X \\right| - \\left| X_1 \\right| - \\left| X_3 \\right| \\\\\n\t \t\t\t & = \\left| X_2 \\right| \\geq 0. \\qedhere\n\t \\end{align*}\n\\end{proof}\n\n\\input{Text\/Ex\/31_Nonstrict}\n\n\\begin{definition}\\label{def:alphaSystem}\\PRFR{Jan 30th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid.\n\tThe \\deftext[a-system of M@$\\alpha$-system of $M$]{$\\bm \\alpha$-system of $\\bm M$} is defined to be\n\tthe family\\label{n:alphaSystem} ${\\mathcal{A}}_M = (A_i)_{i\\in I} \\subseteq E$, where \n\t\\[ I = \\SET{(F,n)\\in {\\mathcal{F}}(M)\\times \\mathbb{N} ~\\middle|~\\vphantom{A^A} 1\\leq n \\leq \\alpha_M(F)} \\]\n\tand $A_{(F,n)} = F$ for all $(F,n)\\in I$.\n\\end{definition}\n\n\n\\noindent J.H.~Mason also proved that the condition $\\alpha_M \\geq 0$ is sufficient for $M$ to be a strict gammoid. First, we need a sufficient condition that allows us to recognize that a \ntriple $(D,T,V)$ satisfies the equality $M=\\Gamma(D,T,V)$.\n\n\\needspace{8\\baselineskip}\n\n\\begin{lemma}[\\cite{M72}, Lemma~2.3]\\label{lem:presentsStrictGammoid}\\PRFR{Jan 30th}\n\tLet $M=(V,{\\mathcal{I}})$ be a matroid, $D=(V,A)$ be a digraph and $T\\subseteq V$.\n\tIf for all $X\\subseteq V$, the barriers $\\delta_D(X,T)$ have the property\n\t\t\\[ \\mathrm{rk}_M\\left( \\delta_D(X,T) \\right) = \\mathrm{rk}_M(X) \\] and\n\tif the barriers of flats are independent, i.e. for all $F\\in {\\mathcal{F}}(M)$\n\t\\[ \\left| \\delta_D(F,T) \\right| = \\mathrm{rk}_M(F), \\]\n\tthen $M = \\Gamma(D,T,V)$.\n\\end{lemma}\n\n\\begin{proof}\\PRFR{Jan 30th}\n\tLet $N = \\Gamma(D,T,V)$ throughout this proof.\n\tLet $B\\subseteq V$ be a base of $M$, and assume that $B$ is not independent in $N$, i.e. that \n\tthere is a $B$-$T$-separator $S$ in $D$,\n\tsuch that $\\left| S \\right| < \\left| B \\right|$. Let\n\t $$ X = \\SET{p_{-1}\\in V~\\middle|~\\vphantom{A^A}\n\t\t p\\in {\\mathbf{P}}(D)\\colon\\,p_1\\in B {\\mathop{\\text{~and~}}} \\left| p \\right| \\cap S = \\emptyset} \\cup S.$$\n\tBy construction, $X$ consists of $S$ together with all vertices, that can be reached from some $b\\in B\\backslash S$ \n\twithout traversing an element from $S$.\n\tSince $S$ separates $B$ from $T$ in $D$, $S$ is\n\tan $X$-$T$-separator in $D$, too. By construction of $S$, we have $\\partial \\left( X\\backslash S\\right) \\subseteq S$ \n\tand $X\\cap T = S\\cap T$. \n\tTherefore, for all $x\\in X$ we have that $\\partial\\SET{x}\\not\\subseteq X$ implies that $x\\in S$.\n\tTogether with the fact that $S$ is a minimal $B$-$T$-separator and $B\\subseteq X$, we arrive at\n\t $\\delta_D\\left( X,T \\right) = S$. Using the premise of the lemma we arrive at the contradiction to {\\em (R1)},\n\t $$\n\t\\mathrm{rk}_M\\left(\\delta_D(X,T)\\right) = \\mathrm{rk}_M(X) \\geq \\mathrm{rk}_M(B) = \\left| B \\right| > \\left| S \\right| = \\left| \\delta_D(X,T) \\right|.$$\n\tIn other words, if we assume that the base $B$ of $M$ is dependent in $N$, we can construct a set $X$ that spans $M$,\n\tbut that has a barrier in $D$ which is smaller than $\\left| B \\right|$, and therefore the barrier property\n\t$\\mathrm{rk}_M\\left( \\delta_D(X,T) \\right) = \\mathrm{rk}_M(X)$ cannot hold. Consequently,\n\t $B$ must be independent in $N$ and we have $\\mathrm{rk}_N(X) \\geq \\mathrm{rk}_M(X)$.\n\n\t \\noindent\n\t Now let $X\\subseteq V$, then let $F = \\mathrm{cl}_M(X)$ be the smallest flat containing $X$ in $M$.\n\t By Lemma~\\ref{lem:barrier}, $\\delta_D(F,T)$ is an $F$-$T$-separator in $D$, therefore\n\t $$\\mathrm{rk}_N(X) \\leq \\mathrm{rk}_N(F) \\leq \\left| \\delta_D(F,T) \\right| = \\mathrm{rk}_M(F) = \\mathrm{rk}_M(X).$$\n\t Consequently, $\\mathrm{rk}_M = \\mathrm{rk}_N$, and so $M = N = \\Gamma(D,T,V)$.\n\\end{proof}\n\n\n\n\\begin{theorem}[\\cite{M72}, Theorem~2.4]\\label{thm:AlphaCriterion2}\\PRFR{Jan 30th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid.\n\tIf $\\alpha(X) \\geq 0$ holds for all $X\\subseteq E$, then $M$ is a strict gammoid.\n\\end{theorem}\n\n\\PRFR{Jan 30th}\n\\noindent J.H.~Mason's proof \\cite{M72} uses the following line of arguments: \nFirst, observe that $\\alpha \\geq 0$ is a sufficient \ncondition for the $\\alpha$-system of $M$ to have a transversal\n $T_0$. Let $T_0$ be such a transversal, and the map $\\sigma' \\colon T_0 \\longrightarrow {\\mathcal{F}}(M)$ shall be the projection on the first coordinate of the bijection \n $\\sigma\\colon T_0\\longrightarrow I$ witnessing the transversal property of $T_0$ with respect to ${\\mathcal{A}}_M$. Then\nlet $T=E\\backslash T_0$ be the target set, and let $D=(E,A)$ be the digraph where $(u,v)\\in A$ if and only if\n$u\\in T_0$ and\n$v \\in \\sigma'(u)$. We have $M= \\Gamma(D,T,E)$. Now, let us see this proof in detail.\n\n\\begin{proof}\\PRFR{Jan 30th}\n\tLet $I$ and ${\\mathcal{A}}_M$ be as in Definition~\\ref{def:alphaSystem}.\nIt follows from Hall's Theorem (Corollary~\\ref{cor:Hall}) that ${\\mathcal{A}}_M$\n\thas a transversal $T_0$ if and only if for all $J\\subseteq I$ the inequality\n\t\\( \\left| \\bigcup_{i\\in J} A_i \\right| \\geq \\left| J \\right| \\) holds. For ${\\mathcal{A}}_M$, this is the case if and only if for all ${\\mathcal{G}} \\subseteq {\\mathcal{F}}(M)$, the inequality\n\t\\begin{align}\n\t\\left| \\bigcup_{F\\in {\\mathcal{G}}} F \\right| \\geq \\sum_{F\\in {\\mathcal{G}}} \\alpha(F)\\label{ineq:Thm24}\n\t\\end{align}\n\tholds. \n\t For every $X\\subseteq E$, the recurrence relation of $\\alpha$ (Definition~\\ref{def:alphaM}) can be written as the\n\tequation \\[ \\alpha(X) + \\sum_{F'\\in {\\mathcal{F}}(M,F)} \\alpha(F') = \\left| X \\right| - \\mathrm{rk}(X).\\]\n\tConsequently, we obtain the inequality\n\t\\[ \\left| X \\right| - \\mathrm{rk}(X) =\n\t\\alpha(X) + \\sum_{F'\\in {\\mathcal{F}}(M,X)} \\alpha(F') \\geq \n\t \\sum_{F'\\in {\\mathcal{F}}(M),\\,F'\\subseteq X} \\alpha(F'),\\]\n\t where equality holds whenever $X\\in{\\mathcal{F}}(M)$ or $\\alpha(X)=0$.\n\t Now let ${\\mathcal{G}}\\subseteq {\\mathcal{F}}(M)$, then we can use the last inequality\n\t together with the property that $\\alpha \\geq 0$, and we obtain\n\t \\begin{align*}\n\t \t\\left| \\bigcup_{G\\in{\\mathcal{G}}} G \\right| \\,\\,\\geq\\,\\, \\mathrm{rk}\\left( \\bigcup_{G\\in{\\mathcal{G}}} G \\right) \n\t \t+ \\sum_{F'\\in {\\mathcal{F}}(M),\\, F'\\subseteq\\,\\bigcup{\\mathcal{G}}} \\alpha(F') \\,\\,\\geq\\,\\, \\sum_{G\\in{\\mathcal{G}}} \\alpha(G),\n\t \\end{align*}\n\t therefore the inequality~\\ref{ineq:Thm24} holds for ${\\mathcal{G}}$. Consequently, ${\\mathcal{A}}_M$ has a transversal. \n\\par \\vspace*{.6\\baselineskip}\\noindent \\mbox{---\\hspace*{2.5mm}} \n\tLet $T_0$ be a transversal of ${\\mathcal{A}}_M$, and let $\\sigma\\colon T_0 \\longrightarrow I$ be a bijective map\n\twith the property that for all $t\\in T_0$, $t\\in A_{\\sigma(t)}$. We set $T= E\\backslash T_0$ and define the map\n\t \\[ \\sigma' \\colon T_0 \\longrightarrow {\\mathcal{F}}(M),\\, t\\mapsto F_t \\]\n\t where $F_t\\in {\\mathcal{F}}(M)$ such that there is some $i_t\\in \\mathbb{N}$ with $\\sigma(t) = (F_t,i_t)$.\n\tNow, define $D=(E,A)$ to be the digraph on $E$ where\n\t$(u,v)\\in A$ if and only if $u\\in T_0$ and $v\\in \\sigma'(u)$.\n\tLet $N= \\Gamma(D,T,E)$ be the strict gammoid represented by $(D,T,E)$. We want to use Lemma~\\ref{lem:presentsStrictGammoid} in order to show that $M=N$.\n\tLet $X\\subseteq E$ be a subset of $E$. We have to show that the set\n\t\\[ B_X = \\delta_D(X,T) = \\SET{x\\in X\\backslash T ~\\middle|~\\vphantom{A^A} \\sigma'(x) \\not\\subseteq X} \\cup \\left( X\\cap T \\right) \\]\n\tcontains a base of $X$ with respect to $M$, i.e. that $\\mathrm{rk}_M(B_X) = \\mathrm{rk}_M(X)$; and further, if $X\\in{\\mathcal{F}}(M)$, we\n\thave to show that $\\mathrm{rk}_M(B_X) = \\left| B_X \\right|$ holds, too. \n\tAssume for now, that $B_X$ contains a base of $X$, and that $X\\in {\\mathcal{F}}(M)$. Then\n\t $$\\left| X \\right| = \\mathrm{rk}_M(X) + \\sum_{F\\in{\\mathcal{F}}(M),\\,F\\subseteq X} \\alpha_M(F)$$ and the set\n\t $X\\backslash B_X$ consists of all elements $t\\in X \\cap T_0$, such that the flat $\\sigma'(t)$ is a subflat of $X$,\n\t therefore $$\\left| X\\backslash B_X \\right| = \\sum_{F\\in{\\mathcal{F}}(M),\\,F\\subseteq X} \\alpha_M(F),$$\n\t and consequently $\\left| B_X \\right| = \\mathrm{rk}_M(B_X)$.\n\n\t \\par \\vspace*{.6\\baselineskip}\\noindent \\mbox{---\\hspace*{2.5mm}} We give an indirect argument that indeed\n\t $\\mathrm{rk}_M(B_X) = \\mathrm{rk}_M(X)$, so let us assume that \\linebreak $\\mathrm{rk}_M(B_X) < \\mathrm{rk}_M(X)$.\n\t Let $Y\\subseteq \\mathrm{cl}_M(X)$ be a subset that is maximal with respect to set-inclusion among all\n\t subsets of $\\mathrm{cl}_M(X)$ with the property, that $\\mathrm{rk}_M(B_Y) < \\mathrm{rk}_M(Y)$,\n\t where\n\t $B_Y = \\delta_D(Y,T)$.\n\t We show that $\\mathrm{cl}_M(B_Y)\\subseteq Y$ holds for the maximal choice $Y$.\n\t Let $Y' = Y\\cup \\mathrm{cl}_M(B_Y)$, then \n\t \\[ B_{Y'} = \\delta_D(Y',T) =\n\t \\SET{y'\\in Y'\\backslash T~\\middle|~\\vphantom{A^A}\\sigma'(y')\\not\\subseteq Y'}\\cup \n\t \\left( Y'\\cap T \\right) \\subseteq \\mathrm{cl}_M(B_Y),\\]\n\t because $Y'\\cap T = \\left( Y\\cap T \\right) \\cup \\left( \\mathrm{cl}_M(B_Y) \\cap T \\right) \\subseteq \\mathrm{cl}_M(B_Y)$\n\t for the reason that $$Y\\cap T\\subseteq \\delta_D(Y,T) = B_Y;$$\n\t and because\n\t $$ \\SET{y'\\in Y' \\backslash T ~\\middle|~ y'\\notin \\mathrm{cl}_M(B_Y) {\\mathop{\\text{~and~}}} \\sigma'(y')\\not \\subseteq Y'} \\subseteq \\SET{y\\in Y \\backslash T ~\\middle|~ \\sigma'(y)\\not \\subseteq Y} \\subseteq B_Y. $$\n\t This holds since for every $y\\in Y'\\backslash \\mathrm{cl}_M(B_Y) \\subseteq Y$,\n\t the inequality $\\partial \\SET{y} \\cap \\left(T_0\\backslash Y'\\right) \\not= \\emptyset$\n\t implies the inequality $\\partial \\SET{y} \\cap \\left(T_0\\backslash Y\\right) \\not= \\emptyset$ due to $Y\\subseteq Y'$ --- so the left-most\n\t set above is actually empty, because $B_Y\\subseteq \\mathrm{cl}_M(B_Y)$.\n\t %\n\t Since $\\mathrm{cl}_M$ does not change the rank, we obtain $$\\mathrm{rk}_M(B_{Y'}) \\leq \\mathrm{rk}_M(B_Y) < \\mathrm{rk}_M(Y) = \\mathrm{rk}_M(Y'),$$\n\t that means $Y'= Y\\cup\\mathrm{cl}_M(B_Y)$ also satisfies $\\mathrm{rk}_M(B_{Y'}) < \\mathrm{rk}_M(Y')$,\n\t and consequently,\n\t $\\mathrm{cl}_M(B_Y)\\subseteq Y$ for the $\\subseteq$-maximal subset $Y$.\n\n\t \\par \\vspace*{.6\\baselineskip}\\noindent \\mbox{---\\hspace*{2.5mm}} \n\t Now, we want to show that for the $\\subseteq$-maximal choice $Y$, the barrier $B_Y$ is independent in $M$.\n\t We give an indirect argument.\n\t Assume that $B_Y$ is not independent, therefore there is a circuit $C\\subseteq B_Y$. Clearly,\n\t $\\mathrm{cl}_M(C) \\subseteq \\mathrm{cl}_M(B_Y) \\subseteq Y$. Let $F\\in{\\mathcal{F}}(M)$ such that $F\\subseteq \\mathrm{cl}_M(C)$.\n\t From the definition of $B_Y$, it is clear, that any $e\\in E$ with $\\sigma'(e) = F$ has the property, that\n\t $e\\in F\\backslash B_Y \\subseteq Y\\backslash B_Y$. So $\\mathrm{cl}_M(C)$ has at least as many elements as the sum of the $\\alpha$-values of all (not necessarily proper)\n\t subflats of $\\mathrm{cl}_M(C)$ plus the number of elements of $B_Y\\cap \\mathrm{cl}_M(C)$.\n\t Thus we obtain\n\t \\begin{align*}\n\t \t\\left| \\mathrm{cl}_M(C) \\right| \n\t \t & \\,\\,\\geq\\,\\,\n\t \t\\left| \\mathrm{cl}_M(C) \\cap B_Y \\right| + \\sum_{F\\in{\\mathcal{F}}(M), F\\subseteq \\mathrm{cl}_M(C)} \\alpha_M(F) \\\\\n\t \t& \\,\\, \\geq\\,\\, \\hphantom{|} \\mathrm{rk}_M(C) + 1 \\,\\,\\,\\,\\,\\,\\,\\, + \\sum_{F\\in{\\mathcal{F}}(M), F\\subseteq \\mathrm{cl}_M(C)} \\alpha_M(F) \\\\\n\t \t& \\,\\,=\\,\\, \\left| \\mathrm{cl}_M(C) \\right| + 1,\n\t \\end{align*}\n\t and arrive at a contradiction, where the second inequality is due to the fact that $C\\subseteq B_Y$ and $\\mathrm{rk}_M(C) = \\left| C \\right| - 1$. Therefore $B_Y \\in {\\mathcal{I}}$ is independent in $M$.\n\n\t \\par \\vspace*{.6\\baselineskip}\\noindent \\mbox{---\\hspace*{2.5mm}} Now, observe that with $\\alpha_M \\geq 0$, we obtain\n\t \\begin{align*} \\mathrm{rk}_M(Y) > \\mathrm{rk}_M(B_Y) & =\\, \\left| B_Y \\right| \n\t \\\\ & =\\, \\left| Y \\right| - \\sum_{F\\in{\\mathcal{F}}(M),\\,F\\subseteq Y} \\alpha_M(Y)\n\t \\\\ & \\geq\\, \\mathrm{rk}_M(Y) + \\alpha_M(Y) \n\t \\\\ & \\geq\\, \\mathrm{rk}_M(Y). \\end{align*}\n\t \tThis contradiction yields, that the assumption, that there is a maximal subset $Y$ of $\\mathrm{cl}_M(X)$ \n\t with $\\mathrm{rk}_M(B_Y) < \\mathrm{rk}_M(Y)$, is wrong. Consequently, $\\mathrm{rk}_M(B_X) < \\mathrm{rk}_M(X)$ cannot be the case.\n\t Thus $\\mathrm{rk}_M(B_X) = \\mathrm{rk}_M(X)$ and all premises of Lemma~\\ref{lem:presentsStrictGammoid} are met. We just established $M=N=\\Gamma(D,T,E)$, so $M$ is a strict gammoid.\n\\end{proof}\n\n\n\n\\begin{corollary}\\label{cor:MasonAlpha}\\PRFR{Jan 30th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid. Then $M$ is a strict gammoid if and only if for all $X\\subseteq E$\n\tthe inequality $\\alpha(X) \\geq 0$ holds.\n\\end{corollary}\n\\begin{proof}\\PRFR{Jan 30th}\n\tCombine the Theorems \\ref{thm:AlphaCriterion} and \\ref{thm:AlphaCriterion2}.\n\\end{proof}\n\n\\noindent We just saw that we obtain a \nstrict representation of a strict gammoid from every transversal of its $\\alpha$-system.\nThe converse holds, too, in the sense that every representation of a strict gammoid yields a transversal\nof its $\\alpha$-system.\n\n\\needspace{6\\baselineskip}\n\\begin{lemma}\\label{lem:alphaTransversalFromD}\\PRFR{Apr 4th}\n\tLet $D=(V,A)$ be a digraph, $T\\subseteq V$, $M=\\Gamma(D,T,V)$ be a strict gammoid,\n\tand ${\\mathcal{A}}_M=(A_i)_{i\\in I}$ be the $\\alpha$-system of $M$.\n\tLet further $X = V\\backslash T$ and\n\t\\[ \\phi\\colon X \\longrightarrow {\\mathcal{F}}(M),\\,u\\mapsto \\mathrm{cl}\\left(\\vphantom{A^A} \\SET{v\\in V~\\middle|~ (u,v)\\in A} \\right) .\\]\n\tThen $X$ is a transversal of ${\\mathcal{A}}_M$, and there is a bijection $\\psi\\colon X\\longrightarrow I$\n\tsuch that $$x\\in A_{\\psi(x)} = \\phi(x)$$ for all $x\\in X$.\n\\end{lemma}\n\\begin{proof}\\PRFR{Apr 4th} Without loss of generality we may assume that there is no loop arc $(v,v)\\in A$ in $D$.\n\tLet $M=(V,{\\mathcal{I}})$, $x\\in V\\backslash T$, and let $S_x = \\SET{y\\in V~\\middle|~(x,y)\\in A}$.\n\tClearly $S_x$ is an $\\SET{x}\\cup S_x$-$T$-separator in $D$, because every path from $x$ to $t\\in T$\n\tmust use an arc that leaves $x$, and thus this arc visits a vertex from $S_x$. Consequently,\n\t$x\\in \\mathrm{cl}(S_x)$. Since $x\\notin S_x$, we obtain that \n\t$$\\mathrm{rk}(S_x) \\leq \\left| S_x \\right| < \\left| S_x\\cup\\SET{x} \\right| \\leq \\left| \\mathrm{cl}(S_x) \\right|.$$\n\tTherefore $\\phi(x) = \\mathrm{cl}(S_x)$ is a dependent flat of $M$ with $x\\in \\phi(x)$.\n\tLet $F\\in {\\mathcal{F}}(M)$, let $I_F = \\SET{(F',k) \\in I~\\middle|~F' \\subseteq F}$,\n\tand let $X_F = \\SET{x\\in X~\\middle|~ \\phi(x) \\subseteq F}$.\n\tWe show that $X_F$ is a partial transversal of the subfamily ${\\mathcal{A}}_F = (A_i)_{i\\in I_F}$ of ${\\mathcal{A}}_M$,\n\tby induction on the nullity $\\left| F \\right| - \\mathrm{rk}(F)$ of $F$.\n\tIf $F\\in {\\mathcal{I}}$ then $X_F = \\emptyset$, since $\\mathrm{cl}(S_x)$ is dependent for all $x\\in V\\backslash T$.\n\tWe give an indirect argument for the induction step and assume that\n\t$\\left| X_F \\right| > \\left| F \\right| - \\mathrm{rk}(F)$.\n\n\n\n\n\n\n\n\tThere is an $F$-$T$-separator $S_F$ in $D$ with minimal cardinality $\\left| S_F \\right| = \\mathrm{rk}(F)$.\n\tClearly, $S_F\\in {\\mathcal{I}}$ and $S_F\\subseteq \\mathrm{cl}(F) = F$ (Lemma~\\ref{lem:minFTSeparatorInFStrictGammoid}).\n\tSince $\\left| X_F \\right| > \\left| F \\right| - \\mathrm{rk}(F)$ and $X_F \\subseteq F$, we obtain that\n\t$X_F \\cap S_F \\not= \\emptyset$. Let $f\\in X_F\\cap S_F$, \n\tthen $S_f = \\SET{g\\in V~\\middle|~(f,g)\\in A} = \\phi(f) \\subseteq F$. Since $f\\in X_F \\subseteq V\\backslash T$\n\twe have \n\t$f\\notin T$. Now let $f'\\in F$ and $t\\in T$, then every path $p\\in{\\mathbf{P}}(D;f',t)$\n\twith $f\\in \\left| p \\right|$ must also visit\n\tanother element $f''\\in S_f \\subseteq F$ as it continues to $t$. Thus every such $p$ \n\tmust visit an element from $S_F\\BSET{f}$ after visiting $f$\n\t --- a contradiction to the fact that $S_F$ is an $F$-$T$-separator \n\twith minimal cardinality in $D$. Thus $\\left| X_F \\right| \\leq \\left| F \\right| - \\mathrm{rk}(F)$.\n\tConsequently, $X_F$ is a partial transversal of ${\\mathcal{A}}_F$.\n\tObserve that\n\t$$\\left| X \\right| = \\left| V\\backslash T \\right| = \\left| V \\right| - \\mathrm{rk}(V) = \\sum_{F\\in {\\mathcal{F}}(M)} \\alpha_M(F) = \\left| I \\right|.$$\n\tSo $X$ is a transversal of ${\\mathcal{A}}_M$ with the property, that\n\t$\\left| \\SET{x\\in X~\\middle|~\\phi(x) = F} \\right| = \\alpha_M(F)$ holds for all $F\\in {\\mathcal{F}}(M)$,\n\tthus $X$ has the claimed property.\n\\end{proof}\n\n\n\\section{Transversal Matroids}\\label{sec:TransversalMatroids}\n\n\\PRFR{Jan 30th}\nThe notion of transversal matroids has been introduced in section \\ref{sec:shortTransversalMatroids}.\nIn this section, we develop the theory of transversal matroids a little further.\n\n\\begin{lemma}\\label{lem:transersalMatroidsAreGammoids}\\PRFR{Jan 30th}\n\tLet $E$ be a finite set, ${\\mathcal{A}}=(A_i)_{i\\in I}\\subseteq E$ be a finite family of subsets of $E$,\n\tand $M=M({\\mathcal{A}})$ be the transversal matroid presented by ${\\mathcal{A}}$.\n\tThen $M$ is a gammoid.\n\\end{lemma}\n\n\\begin{proof}\\PRFR{Jan 30th}\n\tWithout loss of generality, we may assume that $E\\cap I=\\emptyset$.\n\tLet $D=(V,A)$ be the digraph where $V=E\\mathbin{\\dot{\\cup}} I$ and\n\t$(e,i)\\in A$ if and only if $e\\in E$, $i\\in I$ and $e\\in A_i$.\n\tThen $M({\\mathcal{A}})= \\Gamma(D,I,E)$: The routings $R\\colon X_0\\double{\\rightarrow} I$ in $D$ with $X_0\\subseteq E$\n\tare in correspondence to the injections $\\iota\\colon X_0\\longrightarrow I$ that have the property $x\\in A_{\\iota(x)}$ for\n\tall $x\\in X_0$, where $R(\\iota)=\\SET{x\\iota(x) \\in {\\mathbf{P}}(D)\\mid x\\in X_0}$ is the routing induced by a partial transversal $X_0$ of ${\\mathcal{A}}$ with injective map $\\iota$; and $P(R) = \\SET{p_1\\in E\\mid p\\in R}$ is the partial transversal of ${\\mathcal{A}}$ induced from a routing $R\\colon X_0\\double{\\rightarrow} I$ with $X_0\\subseteq E$ in $D$.\n\\end{proof}\n\n\n\\begin{lemma}\\label{lem:dualOfStrictGammoidIsTransversal}\\PRFR{Jan 30th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid.\n\t If $M$ is a strict gammoid, then $M^\\ast$ is a transversal matroid.\n\\end{lemma}\n\\begin{proof}\\PRFR{Jan 30th}\n\tIt is an immediate consequence from the fact that\n\tthere is a linking from $X$ onto $T$ in $D$ if and only if $V\\backslash X$ is a transversal of\n ${\\mathcal{A}}_{D,T}$ (Lemma~\\ref{lem:linkage}), that the bases of $\\Gamma(D,T,V)$ are precisely those subsets of $V$, for which their complement in $V$ is a base of the transversal matroid $M({\\mathcal{A}}_{D,T})$\ndefined by the linkage system of $D$ to $T$. Thus $M^\\ast = M({\\mathcal{A}}_{D,T})$.\n\\end{proof}\n\n\\noindent The converse statement holds, too.\n\n\\begin{lemma}\\label{lem:dualtransversalstrictgammoid}\\PRFR{Jan 30th}\n\tLet ${\\mathcal{A}}=(A_i)_{i\\in I} \\subseteq E$ be a family of sets, and \n\t$M = M({\\mathcal{A}})$ the transversal matroid presented by ${\\mathcal{A}}$. Then\n\t$M^\\ast$ is a strict gammoid.\n\\end{lemma}\n\n\\begin{proof}\\PRFR{Jan 30th}\n\tWithout loss of generality, we may assume that $E\\cap I = \\emptyset$.\n\tWe define the family $\\hat{\\mathcal{A}} = (\\hat A_i)_{i\\in I} \\subseteq E\\mathbin{\\dot{\\cup}} I$\n\tby setting $\\hat A_i = A_i \\mathbin{\\dot{\\cup}} \\SET{i}$ for all $i\\in I$.\n\tFurther, let $D=(E\\mathbin{\\dot{\\cup}} I, A)$ where \n\t\\[ A = \\SET{(e,i)\\in E\\times I\\mid e\\in A_i}. \\]\n\tIt is easy to see that the linkage system ${\\mathcal{A}}_{D,E}$ of $D$ to $E$ is\n\tprecisely the family $\\hat {\\mathcal{A}}$. Therefore $M(\\hat{\\mathcal{A}})^\\ast = \\Gamma(D,E,E\\mathbin{\\dot{\\cup}} I)$ is\n\ta strict gammoid.\n\tOn the other hand, $M(\\hat {\\mathcal{A}})| E = M({\\mathcal{A}})$ is evident from the construction.\n\tWith Lemma~\\ref{lem:restrictcontractdual} we obtain\n\t\\[\n\t\tM({\\mathcal{A}})^\\ast = \\left( M(\\hat{\\mathcal{A}})| E\\right)^\\ast = \\left( M(\\hat{\\mathcal{A}})^\\ast \\right)|' E,\n\t\\]\n\twhere the last term is the contraction of a strict gammoid, therefore $M^\\ast$ is a strict gammoid (Lemma~\\ref{lem:contractionStrictGammoid}).\n\\end{proof}\n\n\\begin{corollary}\\label{cor:transversalstrictdual}\\PRFR{Jan 30th}\n\tLet $M$ be a matroid. Then $M$ is a transversal matroid if and only if $M^\\ast$ is a strict gammoid.\n\\end{corollary}\n\n\\begin{corollary}\\label{cor:transversalrepresentation}\\PRFR{Jan 30th}\n\tLet $E,I$ be finite sets and ${\\mathcal{A}} = (A_i)_{i\\in I} \\subseteq E$ be a family of subsets.\n\tIf $M=M({\\mathcal{A}})$ is the transversal matroid presented by ${\\mathcal{A}}$ and $r = \\mathrm{rk}_M(E)$,\n\tthen there is a family of subsets ${\\mathcal{A}}' = (A'_i)_{i=1}^r \\subseteq E$, such that\n\t$M=M({\\mathcal{A}}')$.\n\\end{corollary}\n\\begin{proof}\\PRFR{Jan 30th}\n\tBy Lemma~\\ref{lem:dualtransversalstrictgammoid} the dual $M^\\ast$ is a strict gammoid\n\tof rank $\\left| E \\right| - r$. Thus there is a digraph $D=(E,A)$ and a base $T\\subseteq E$ of $M^\\ast$ such that $M^\\ast = \\Gamma(D,T,E)$. Let ${\\mathcal{A}}_{D,T}$ be the linkage system of $D$ to $T$,\n\tthen $M({\\mathcal{A}}_{D,T})^\\ast = M^\\ast$, and ${\\mathcal{A}}_{D,T} = (A^{(D,T)}_i)_{i\\in E\\backslash T}$ consists of $\\left| E \\right| - \\left| T \\right| = r$ sets $A_i$, which may be renumbered by the integers from $1$ through $r$ yielding the desired ${\\mathcal{A}}'$.\n\\end{proof}\n\n\\begin{example}\\label{ex:transversalMatroidDualityRespecting}\\PRFR{Feb 15th}\nIt is particularly easy to obtain a duality respecting representation of a transversal matroid from representations that are in the form of Corollary~\\ref{cor:transversalrepresentation}.\nLet $M=(E,{\\mathcal{I}})$ be a transversal matroid, $r = \\mathrm{rk}(E)$, and ${\\mathcal{A}} = (A_i)_{i=1}^r \\subseteq E$ be a representation of $M$, i.e. $M = M({\\mathcal{A}})$.\nThen there is a base $B\\in{\\mathcal{B}}(M)$ and a bijective map $\\phi \\colon B\\longrightarrow \\SET{1,2,\\ldots,r}$ such that \n$b\\in A_{\\phi(b)}$ holds for all $b\\in B$. Furthermore, if $I=\\dSET{i_1,i_2,\\ldots,i_r}$ is a set with $I\\cap E = \\emptyset$, \nthen the digraph $D=(E\\mathbin{\\dot{\\cup}} I, A)$ with\n\t \\begin{align*} A = \\SET{\\left(i_{\\phi(b)},b\\right) ~\\middle|~ b\\in B} & \\cup \\SET{\\left(i_{\\phi(b)},i_{k}\\right) ~\\middle|~b\\in B,\\, k\\in\\SET{1,2,\\ldots,r}\\BSET{\\phi(b)}\\colon\\, b\\in A_k} \\\\ &\n\t \\cup \\SET{\\left(e,i_k\\right) ~\\middle|~\n\t\t\\vphantom{A^A} k\\in \\SET{1,2,\\ldots,r},\\,e\\in A_k\\backslash B}\n\\end{align*}\nhas the property, that $M = \\Gamma(D,B,E)$, because it is the digraph that arises from the \ndigraph described in Lemma~\\ref{lem:transersalMatroidsAreGammoids} and the construction from the proof \nof Theorem~\\ref{thm:gammoidRepresentationWithBaseTerminals} with respect to the basis $B$.\n Since the premises of Lemma~\\ref{lem:dualityrespectingrepresentation} are satisfied,\n$(D,B,E)$ is a duality respecting representation of $M$.\n\\end{example}\n\n\n\\begin{corollary}\\PRFR{Feb 15th}\n\tLet $M$ be a transversal matroid. There is a representation $(D,T,E)$ where $D=(V,A)$ with $M = \\Gamma(D,T,E)$ and $\\left| V \\right| \\leq \\left| E \\right| + \\mathrm{rk}(E)$. There even is a representation that uses a digraph with $\\left| V \\right| < 2\\cdot \\left| E \\right|$.\n\\end{corollary}\n\\begin{proof}\\PRFR{Feb 15th}\n\tA representation with $\\left| V \\right| \\leq \\left| E \\right| + \\mathrm{rk}(E)$ has been constructed in Example~\\ref{ex:transversalMatroidDualityRespecting}. If $\\mathrm{rk}(E) = \\left| E \\right|$, then $M = (E,2^E)$, i.e. $M$ is the free matroid on $E$, and therefore\n\tthe digraph $D' = \\left( E, \\emptyset \\right)$ yields a representation $M = \\Gamma(D',E,E)$ with strictly fewer than $2\\cdot \\left| E \\right|$ elements.\n\\end{proof}\n\n\n\\section{Constructions within the Class of Gammoids}\n\n\\noindent \\PRFR{Mar 27th}\nIn this section, we explore methods of obtaining new gammoids from old ones. The main application of this section is the following:\nIf we know that a matroid $M$ may be constructed from a matroid $N$ using a construction that does not leave the class of gammoids,\nthen we may conclude that $M$ is a gammoid whenever $N$ is a gammoid.\n\n\\noindent\nLet us start with a well-known result of J.H.~Mason \\cite{M72}.\n\n\\begin{theorem}\\label{thm:GammoidsClosedMinorsDuality}\\PRFR{Mar 27th}\n\tThe class consisting of all gammoids is closed under minors, duality, and direct sums.\n\\end{theorem}\n\\begin{proof}\\PRFR{Mar 27th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid. It is clear from Definition~\\ref{def:gammoid}\n\tthat the representation $(D,T,E)$ of $M$ yields the representation $(D,T,X)$ of\n\t$M| X$ for all $X\\subseteq E$. Thus the class of all gammoids is closed under restriction.\n\tCorollary~\\ref{cor:dualityrespectingrepresentation} yields that if $M$ is a gammoid, then so is\n\tits dual $M^\\ast$. Consequently, the class of all gammoids is closed under duality.\n\tIt follows with Lemma~\\ref{lem:restrictcontractdual} that the class of gammoids is also closed under\n\tcontraction. We showed in Lemma~\\ref{lem:arcCSubAdditive} that the class of gammoids is closed under direct sums.\n\\end{proof}\n\n\\noindent \\PRFR{Mar 27th}\n\tRemember that Corollary~\\ref{cor:extWithLoopCoLoop} established, that\n\tevery extension of a gammoid $M$ by a loop or a coloop is again a gammoid.\n\tTherefore $M$ is a gammoid if and only if $M| X$ is a gammoid, where $X$ consists of all elements of the ground set of $M$,\n\tthat\n\tare neither loops nor coloops.\n\n\\begin{lemma}\\label{lem:principalExtOfStrictGammoidIsStrict}\\PRFR{Mar 27th}\n\tLet $M=(E,{\\mathcal{I}})$ be a gammoid, $e\\notin E$, and let $N\\in {\\mathcal{X}}(M,e)$ such that\n\t\\[ C = \\SET{F\\in {\\mathcal{F}}(M)~\\middle|~e\\in \\mathrm{cl}_N(F)} \\]\n\thas at most one $\\subseteq$-minimal element.\n\tThen $N$ is a gammoid. If $M$ is a strict gammoid, then $N$ is a strict gammoid.\n\\end{lemma}\n\\begin{proof}\\PRFR{Mar 27th}\n\tLet $(D,T,E)$ with $D=(V,A)$ be a strict representation of $M$ if $M$ is a strict gammoid, otherwise let $(D,T,E)$\n\tbe a representation of $M$.\n\tIf $C=\\emptyset$, then $e$ is a coloop in $N$. So $(D',T\\cup\\SET{e},E\\cup\\SET{e})$\n\twith\n\t$D' = (V\\cup\\SET{e},A)$ is a representation of $N$, which is a strict representation if $M$ is a strict gammoid.\n\tOtherwise let $F_0 = \\bigcap C$ be the unique $\\subseteq$-minimal element of $C$.\n\t Then\n\t$D''=(V\\cup\\SET{e},A'')$ with\n\t\\[ A'' = A \\cup \\left( \\SET{e} \\times F_0 \\right)\\]\n\tyields the representation $(D'',T,E\\cup\\SET{e})$ of $N$ -- which is strict if $M$ is a strict gammoid:\n\tLet $N' = \\Gamma(D'',T,E\\cup\\SET{e})$, and let $X\\subseteq E\\cup\\SET{e}$ be independent in $N'$.\n\tIf $X\\subseteq E$, then $X$ is independent in $N$, because by construction, \n\tno path $p\\in {\\mathbf{P}}(D'';x,t)$ for any $x\\in E$ and any $t\\in T$\n\tvisits $e$. Thus every routing $X\\double{\\rightarrow} T$ in $D''$ is also a routing with respect to $D$. If $e\\in X$ for $X$ independent in $N'$,\n\t then the fact that $F_0\\not\\subseteq \\mathrm{cl}_M(X\\BSET{e})$ follows\n\t from the way $D''$ is constructed from $D$: Every path from $e$ to any $t\\in T$ visits some element from $f\\in F_0$.\n\t Thus every routing $X\\double{\\rightarrow} T$ in $D''$ induces a routing $(X\\BSET{e})\\cup\\SET{f}\\double{\\rightarrow} T$ in $D$ for some $f\\in F_0\\backslash X$.\n\t Therefore there is some\n\t$f\\in F_0\\backslash X$ such that $(X\\BSET{e})\\cup\\SET{f}$ is independent in $M$, consequently $f\\notin \\mathrm{cl}_M(X\\BSET{e})$.\n\tWe obtain $\\mathrm{rk}_N(X) = \\mathrm{rk}_M(X\\BSET{e}) + 1$ and so $X$ is independent in $N$.\n\tNow let $X\\subseteq E\\cup\\SET{e}$ be independent in $N$. If $e\\notin X$ holds,\n\t then $X$ is independent in $M$. So $X$ is independent \n\tin $N'$, too, because $A\\subseteq A''$. If $e\\in X$ and $X$ is independent in $N$, then $X' = X\\BSET{e}$ is independent in $M$ \n\tand $F_0\\not\\subseteq \\mathrm{cl}_M(X')$.\n\tBut then there is some $f\\in F_0\\backslash X'$ such that $\\mathrm{rk}_M(X'\\cup\\SET{f}) > \\mathrm{rk}_M(X')$, thus $X'\\cup\\SET{f}$ is independent in $M$, too.\n\tNow let $R\\colon X'\\cup\\SET{f}\\double{\\rightarrow} T$ be a corresponding routing in $D$, and let $p^{(f)}\\in R$ be the path of that routing \n\twhere $p_1^{(f)} = f$. Then $R' = \\left(R\\BSET{p^{(f)}}\\right)\\cup\\SET{ep^{(f)}}$ is a routing from $X'\\cup\\SET{e}$ to $T$ in $D''$. \n\tIt follows that $X$ is independent in $N'$, and consequently $N = N'$.\n\\end{proof}\n\n\\begin{definition}\\label{def:deflateOfM}\\PRFR{Mar 27th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, $X\\subseteq E$. The restriction $N=M| X$ shall be a \\deftext[deflate of a matroid]{deflate of $\\bm M$},\n\tif $E\\backslash X = \\dSET{e_1,e_2,\\ldots,e_m}$ can be ordered naturally,\n\tsuch that for all $i\\in \\SET{1,2,\\ldots,m}$ the modular cut\n\t\\[ C_i = \\SET{F\\in {\\mathcal{F}}\\left( \\vphantom{A^A} M| \\left( X\\cup\\SET{e_1,e_2,\\ldots,e_{i-1}} \\right) \\right)~\\middle|~ e_i \\in \\mathrm{cl}_M(F)} \\]\n\thas precisely one $\\subseteq$-minimal element.\n\\end{definition}\n\n\n\\begin{definition}\\label{def:deflated}\\PRFR{Mar 27th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid. $M$ shall be called \\deftext{deflated}, if the only deflate of $M$ is $M$ itself.\n\\end{definition}\n\n\\needspace{4\\baselineskip}\n\n\\begin{lemma}\\label{lem:excludedMinorsForGammoidsAreDeflated}\\PRFR{Mar 27th}\n\tLet $M$ be an excluded minor for the class of gammoids. Then $M$ is deflated.\n\\end{lemma}\n\\begin{proof}\\PRFR{Mar 27th}\n\tWe give an indirect proof and\n\tassume that $M=(E,{\\mathcal{I}})$ is an excluded minor for the class of gammoids, and $M$ is not deflated.\n\tThen there is an element $e\\in E$ such that \n\t\\[ C = \\SET{F\\in {\\mathcal{F}}\\left( M| (E\\BSET{e}) \\right) \\vphantom{A^A}~\\middle|~ e\\in \\mathrm{cl}_M(F)} \\]\n\thas the property that\n\t\\[ C = \\SET{F\\in {\\mathcal{F}}\\left( M| (E\\BSET{e}) \\right) \\vphantom{A^A}~\\middle|~ F_0\\subseteq F} \\]\n\twhere $F_0 = \\bigcap C$.\n\tSince $M$ is an excluded minor, the restriction $N = M| (E\\BSET{e})$ is a gammoid.\n\tIn this situation, Lemma~\\ref{lem:principalExtOfStrictGammoidIsStrict} yields that $M$ is a gammoid --- a contradiction. Thus every\n\texcluded minor for the class of gammoids is deflated.\n\\end{proof}\n\n\\needspace{4\\baselineskip}\n\\begin{lemma}\\label{lem:deflationLemma}\\PRFR{Mar 27th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, $X\\subseteq E$ and let $N = M| X$ be a deflate of $M$.\n\tThen $M$ is a gammoid if and only if $N$ is a gammoid.\n\\end{lemma}\n\\begin{proof}\\PRFR{Mar 27th}\n\tIf $M$ is a gammoid, then $N$ is a gammoid, too (Theorem~\\ref{thm:GammoidsClosedMinorsDuality}).\n\tNow let $N$ be a gammoid, and let $E\\backslash X = \\dSET{e_1,e_2,\\ldots,e_m}$ be implicitly ordered with the properties required in Definition~\\ref{def:deflateOfM}.\n\tLemma~\\ref{lem:principalExtOfStrictGammoidIsStrict} yields that $M| \\left( X\\cup\\SET{e_1,e_2,\\ldots,e_i} \\right)$ is a gammoid whenever\n\t$M| \\left( X\\cup\\SET{e_1,e_2,\\ldots,e_{i-1}} \\right)$ is a gammoid. \n\tThus, by induction on $m$, we obtain that $M$ is a gammoid whenever $N$ is a gammoid -- a fact that we assumed.\n\\end{proof}\n\n\\noindent The former situation is a special case of the following situation\n\n\\begin{definition}\\PRFR{Mar 27th}\n\tLet $T=(T_0,{\\mathcal{T}})$ be a matroid, $D=(V,A)$ be a digraph with $T_0\\subseteq V$, and let $E\\subseteq V$ be any set.\n\tThe \\deftext[matroid induced by D from R@matroid induced by $D$ from $T$]{matroid on $\\bm E$ induced by $\\bm D$ from $\\bm T$}\n\tshall be the pair \\label{n:IDTE}\n\t\\( I(D,T,E) = (E,{\\mathcal{I}}), \\)\n\twhere $X\\in {\\mathcal{I}}$ if and only if there is a routing $R\\colon X\\double{\\rightarrow} T_0$ in $D$, such that $\\SET{p_{-1}~\\middle|~p\\in R}\\in {\\mathcal{T}}$.\n\tIn other words $X$ is independent in $I(D,T,E)$ if and only if there is a linking from $X$ onto an independent set of $T$ in $D$.\n\\end{definition}\n\n\\noindent It is a result of J.H.~Mason that this generalization of Definition~\\ref{def:gammoid} always produces a matroid.\n\n\\begin{theorem}[\\cite{M72}, Theorem 1.1]\\label{thm:mason11}\\PRFR{Mar 27th}\n\tLet $T=(T_0,{\\mathcal{T}})$ be a matroid, $D=(V,A)$ be a digraph with $T_0\\subseteq V$, and let $E\\subseteq V$ be any set. Then $I(D,T,E)$ is indeed a matroid.\n\\end{theorem}\n\n\\noindent \\PRFR{Mar 27th}\nFor a proof\\footnote{We chose to omit the full proof, because for pure logical reasons,\nwe do not need this theorem for our purposes in this work: the construction in\nLemma~\\ref{lem:digraphInducedGammoidIfTisGammoid} works if we define a triple $(M,D,T)$ to be a digraph induction whenever $M$ and $T$ are matroids, \nsuch that $X$ is independent in $M$ if and only if it can be linked to an independent set of $T$ in $D$. Theorem~\\ref{thm:mason11} states that for every matroid $T$ and every digraph $D$, there is a matroid $M$ on every subset of the vertex set of $D$ such that $(M,D,T)$ is such a digraph induction.}, see \\cite{M72}, p.58; J.H.~Mason constructs the linkage system with respect to two routings from $X$, and $Y$, respectively,\nonto independent subsets of $T_0$\nand then uses Theorem~\\ref{thm:bipartiteInduction} in order to show that if $\\left| X \\right| < \\left| Y \\right|$ the augmentation axiom {\\em (I3)} holds \nfor $I(D,T,V)$. The axioms {\\em (I1)} and {\\em (I2)} follow easily from the definition, thus $I(D,T,V)$ is a matroid, \nand consequently $I(D,T,E) = I(D,T,V)| E$ is a matroid, too. Now let us present the general form of the non-trivial implication of\n Lemma~\\ref{lem:deflationLemma}.\n\n\\begin{lemma}\\label{lem:digraphInducedGammoidIfTisGammoid}\\PRFR{Mar 27th}\n\tLet $T=(T_0,{\\mathcal{T}})$ be a matroid, $D=(V,A)$ be a digraph with $T_0\\subseteq V$, and let $E\\subseteq V$.\n\tIf $T$ is a gammoid, then $I(D,T,E)$ is a gammoid.\n\\end{lemma}\n\\begin{proof}\\PRFR{Mar 27th}\n\n\n\tIf $T$ is a gammoid, then there is a representation $(D',S',T_0)$ with $D'=(V',A')$ \n\tsuch that $T = \\Gamma(D',S,T_0)$. Let $v\\mapsto \\tilde{v}$ denote a renaming scheme such that the renamed vertices are disjoint from $V$,\n\ti.e.\n\t$\\tilde V' \\cap V = \\emptyset$ where $\\tilde V' = \\SET{\\tilde{v} ~\\middle|~ v\\in V'}$. \n\tLet $\\tilde{X} = \\SET{\\tilde{x}~\\middle|~ x\\in X}$ for all $X\\subseteq V'$.\n We define the digraph $D_I = (V\\mathbin{\\dot{\\cup}} \\tilde{V'}, A_I)$ where \n \\[ A_I = A\\mathbin{\\dot{\\cup}} \\SET{\\left(\\tilde u,\\tilde v\\right)~\\middle|~ (u,v)\\in A'} \\mathbin{\\dot{\\cup}} \\SET{\\left(t,\\tilde t\\right)~\\middle|~t\\in T_0}.\n \\]\n We show that $I(D,T,E) = \\Gamma(D_I,\\tilde{S},E)$. First, observe that $T_0$ is a $V$-$\\tilde S$-separator in $D_I$, because every arc between $V$ and\n $\\tilde V'$ leaves $t$ and enters $\\tilde t$ for some $t\\in T_0$. Therefore there is a routing $R_I\\colon X\\double{\\rightarrow} \\tilde S$ in $D_I$ with $X\\subseteq E$\n if and only if there are a linking $R\\colon X\\double{\\rightarrow} T_X$ with $T_X \\subseteq T_0$ in $D$ and a routing $R'\\colon T_X\\double{\\rightarrow} S$ in $D'$ ---\n we may construct $R$ and $R'$ from $R_I$ by splitting every $p_I\\in R_I$ into its $V$- and $\\tilde V'$-components. Conversely, if we have a\n pair of routings $R$ and $R'$ such that $\\SET{p_{-1}~\\middle|~p\\in R} = \\SET{p'_1 ~\\middle|~p'\\in R'}$,\n then we may obtain $R_I$ by joining the corresponding paths in $D_I$.\n The latter routing $R'$ exists if and only if $T_X\\in{\\mathcal{T}}$,\n therefore $X\\subseteq E$ is independent in $\\Gamma(D_I,\\tilde S, E)$ if and only if $X$ may be linked onto an independent subset of $T_0$ with respect to $T$, i.e. if and only if $X$ is independent in $I(D,T,E)$.\n\\end{proof}\n\n\\noindent The following corollary is the corresponding generalization of Lemma~\\ref{lem:deflationLemma}.\n\n\\begin{corollary}\\PRFR{Mar 27th}\n\tLet $T=(T_0,{\\mathcal{T}})$ be a matroid, $D=(V,A)$ be a digraph with $T_0\\subseteq V$, such that every vertex $t\\in T_0$ is a sink in $D$.\n\t Let further $E\\subseteq V$ such that $T_0\\subseteq E$.\n\tThen $T$ is a gammoid if and only if $I(D,T,E)$ is a gammoid.\n\\end{corollary}\n\\begin{proof}\\PRFR{Mar 27th}\n\tIf $T$ is a gammoid, then $I(D,T,E)$ is a gammoid (Lemma~\\ref{lem:digraphInducedGammoidIfTisGammoid}).\n\tSince ${\\mathbf{P}}(D;t,v) =\\emptyset$ for all $t\\in T_0$ and all $v\\in V\\BSET{t}$, we obtain that\n\t$X$ is independent in $I(D,T,E)$ if and only if $X$ is independent in $T$\n\tfor all $X\\subseteq T_0$. Thus $I(D,T,E)| T_0 = T$, and, consequently, if $I(D,T,E)$ is a gammoid, then so is $T$ \n\t(Theorem~\\ref{thm:GammoidsClosedMinorsDuality}).\n\\end{proof}\n\\section{Representation over $\\mathbb{R}$}\n\n\\noindent \n\\PRFR{Feb 15th}\nThere are many ways to arrive at the fact that every gammoid can be represented by a matrix\nover a field ${\\mathbb{K}}$ whenever ${\\mathbb{K}}$ has enough elements. Or, to be more precise, for every field ${\\mathbb{F}}$ and \nevery gammoid $M$ there\nis an extension field ${\\mathbb{K}}$ of ${\\mathbb{F}}$, such that $M$ can be represented by a matrix over ${\\mathbb{K}}$.\nFor the sake of simplicity, we only consider representations of gammoids over the field of the reals $\\mathbb{R}$.\nIn \\cite{Ar06}, F.~Ardila points out that the Lindstr\u00f6m Lemma yields an easy method to construct a matrix $\\mu\\in \\mathbb{R} ^{E\\times B}$ from the digraph $D=(V,A)$ such that $\\Gamma(D,T,E) = M(\\mu)$;\nthe construction is universal in the sense that it works with indeterminates and thus yields a representation over ${\\mathbb{F}}$ whenever these indeterminates can be replaced with elements from ${\\mathbb{F}}$\nwithout zeroing out any nonzero subdeterminants of $\\mu$.\n\n\\begin{definition}\\PRFR{Feb 15th}\n\tLet $D=(V,A)$ be a digraph and $w\\colon A\\longrightarrow \\mathbb{R}$.\n\tThen $w$ shall be called\n\t \\deftext[indeterminate weighting of D@indeterminate weighting of $D$]{indeterminate weighting of $\\bm D$},\n\twhenever the set $\\SET{w(a)\\mid a\\in A}$ is $\\mathbb{Z}$-independent.\n\\end{definition}\n\n\\begin{example}\\PRFR{Feb 15th}\n\tLet $D=(V,A)$ be any digraph, then $\\left| A \\right| < \\infty$. Thus there is a set $X\\subseteq \\mathbb{R}$\n\tthat is $\\mathbb{Z}$-independent with $\\left| X \\right| = \\left| A \\right|$ (Lemma~\\ref{lem:enoughZindependents}). Then\n\tany bijection $\\sigma\\colon A\\longrightarrow X$ induces a indeterminate weighting $w\\colon X\\longrightarrow \\mathbb{R}$\n\twith $w(x) = \\sigma(x)$, thus indeterminate weightings exist for all digraphs.\n\\end{example}\n\n\\begin{notation}\\label{n:prodp}\\PRFR{Feb 15th}\n\tLet $D=(V,A)$ be a digraph and $w\\colon A\\longrightarrow \\mathbb{R}$ be an indeterminate weighting of $D$. Let $q\\in {\\mathbf{W}}(D)$,\n\twe shall write \\[ \\prod q = \\prod_{a\\in \\left| q \\right|_A} w(a). \\]\n\\end{notation}\n\n \\begin{lemma}[Lindstr\u00f6m \\cite{Li73}]\\label{lem:lindstrom}\\PRFR{Feb 15th}\n \tLet $D=(V,A)$ be an acyclic digraph, $n\\in \\mathbb{N}$ a natural number, $S=\\dSET{s_1,s_2,\\ldots,s_n}\\subseteq V$ and $T=\\dSET{t_1,t_2,\\ldots,t_n}\\subseteq V$ be equicardinal subsets of $V$, and let $w\\colon A\\longrightarrow \\mathbb{R}$ be an indeterminate weighting of $D$. Furthermore,\n \t$\\mu\\in \\mathbb{R}^{V\\times V}$ shall be the matrix with\n \t\\[ \\mu(u,v) = \\sum_{p\\in {\\mathbf{P}}(D;u,v)} \\prod p .\\]\n \n \tThen\n \t\\[ \\det \\left( \\mu| S\\times T \\right) = \\sum_{L\\colon S\\double{\\rightarrow} T} \\left( \\mathrm{sgn}(L) \n \t\\prod_{p\\in L} \\left( \\prod p \\right) \\right) \\]\n \twhere $\\mathrm{sgn}(L) = \\mathrm{sgn}(\\sigma)$ for the unique permutation $\\sigma\\in \\mathfrak{S}_n$ with\n \tthe property that for every $i\\in\\SET{1,2,\\ldots,n}$ there is a path $p\\in L$ with $p_1 = s_i$ and \n \t$p_{-1} = t_i$.\n \tFurthermore, \\[ \\det \\left( \\mu| S\\times T \\right) = 0 \\] if and only if\n \tthere is no linking from $S$ to $T$ in $D$.\n \\end{lemma}\n\\noindent As suggested by F.~Ardila, we present the following bijective proof given by I.M.~Gessel and X.G.~Viennot \\cite{GV89p}. \n\\begin{proof}\\PRFR{Feb 15th} The Leibniz formula (Definition~\\ref{def:det}) yields\n\\begin{align*}\n\\det \\left( \\mu| S\\times T \\right) \n& = \\sum_{\\sigma\\in \\mathfrak{S}_n} \\mathrm{sgn}(\\sigma) \\prod_{i=1}^{n} \\mu(s_i,t_{\\sigma(i)}) \\\\\n& = \\sum_{\\sigma\\in \\mathfrak{S}_n} \\mathrm{sgn}(\\sigma) \\prod_{i=1}^{n} \\left( \n\t\t\t\t\\sum_{p\\in {\\mathbf{P}}{\\left(D;s_i,t_{\\sigma(i)}\\right)}} \\prod p \\right) \\\\\n& = \\sum_{\\sigma\\in \\mathfrak{S}_n} \\mathrm{sgn}(\\sigma)\\left( \\sum_{K\\in Q_\\sigma} \\,\\,\n\t\t\t\\prod_{p\\in K} \\left( \\prod p\\right) \\right),\n\\end{align*}\nwhere $$Q_\\sigma = \\SET{\\left. K \\in \\binom{{\\mathbf{P}}(D)}{n} \\,\\,\\right|\\,\\, \\forall i\\in \\SET{1,2,\\ldots,n}\\colon\\,\\exists p\\in K\\colon\\,p_1=s_i {\\mathop{\\text{~and~}}} p_{-1}=t_{\\sigma(i)} }\\,$$ consists of all families of paths connecting $s_i$ with $t_{\\sigma(i)}$ for all $i\\in \\SET{1,2,\\ldots,n}$.\n\n\n Clearly, for $\\sigma,\\tau\\in \\mathfrak{S}_n$ with $\\sigma\\not= \\tau$, the sets $Q_\\sigma\\cap Q_\\tau = \\emptyset$ are disjoint, therefore the following map with the domain $Q=\\bigcup_{\\sigma\\in \\mathfrak{S}_n} Q_\\sigma$ is well defined:\n \\[ \\mathrm{sgn}\\colon Q \\longrightarrow \\SET{-1,1},\\quad K\\mapsto \\mathrm{sgn}(\\sigma) \n \\quad\\text{~where~}\\sigma\\in\\mathfrak{S}_n\\text{~such that~}K\\in Q_\\sigma .\\]\n\n Thus we may write\n \\begin{align*}\n\\det \\left( \\mu| S\\times T \\right) & = \\sum_{K\\in Q} \\mathrm{sgn}(K) \\prod_{p\\in K} \\left( \\prod p \\right).\n\\end{align*}\nFurthermore, if $L\\colon S\\double{\\rightarrow} T$ is a linking from $S$ to $T$ in $D$, then $L\\in Q_\\sigma$ \nwhere $\\sigma\\in\\mathfrak{S}_n$ is the unique permutation mapping the indexes of the initial vertices of the paths in $L$ \nto the indexes of the terminal vertices of the paths in $L$. Let us denote the routings in $Q$ by\n\\[ R = \\SET{L\\in Q\\mid L\\text{~is a routing}}.\\]\nWe prove the first statement of the lemma by showing that there is a bijection\n$\\phi\\colon Q\\backslash R \\longrightarrow Q\\backslash R$, such that for all $K\\in Q\\backslash R$,\n$$\\prod_{p\\in K}\\left( \\prod p \\right) = \\prod_{p\\in \\phi(K)} \\left( \\prod p \\right)$$\nand $\\mathrm{sgn}(K) = -\\mathrm{sgn}(\\phi(K))$. We construct such a map $\\phi$ now.\nLet $$K' = \\SET{p\\in K\\mid \\exists q\\in K\\BSET{p}\\colon\\,\\left| p \\right|\\cap \\left| q \\right|\\not=\\emptyset}$$\nbe the set of paths in $K$ that meet a vertex of another path, clearly $\\left| K' \\right| \\geq 2$\nsince $K$ is not a routing. There is a total order on $K'$:\n let $p,q\\in K'$, then $p\\leq q$ if and only if $i\\leq j$ where $p_1 = s_i$ and $q_1 = s_j$.\nNow let $p=(p_i)_{i=1}^{n(p)}\\in K'$ be chosen \nsuch that $p$ is the minimal element with respect to the above order.\nLet $j(p)\\in \\SET{1,2,\\ldots,n(p)}$ be the smallest index, such that there is some $q\\in K'\\BSET{p}$\nwith $p_{j(p)}\\in \\left| q \\right|$. Now let\n $q=(q_i)_{i=1}^{n(q)}\\in\\SET{k\\in K'\\BSET{p}\\mid p_{j(p)}\\in \\left| q \\right|}$\n be the minimal choice with respect to the above order on $K'$, and let $j(q)\\in \\SET{1,2,\\ldots,n(q)}$\n such that $q_{j(q)} = p_{j(p)}$. Now let\n $p' = p_1 p_2\\ldots p_{j(p)} q_{j(q) + 1} q_{j(q) + 2} \\ldots q_{n(q)}$\n and $q' = q_1 q_2 \\ldots q_{j(q)} p_{j(p)+1} p_{j(p)+2}\\ldots p_{n(p)}$. Since $D$ is acyclic, all walks are paths in $D$, so ${\\mathbf{W}}(D) = {\\mathbf{P}}(D)$.\n Therefore we may set\n $\\phi(K) = \\left( K\\BSET{p,q}\\right)\\cup\\SET{p',q'} \\in Q\\backslash R$.\n Clearly, $\\phi(\\phi(K)) = K$, therefore $\\phi$ is bijective and self-inverse. Furthermore,\n if $K\\in Q_\\sigma$, then $\\phi(K) \\in Q_{\\sigma \\cdot (x y)}$ for a suitable cycle\n $(x y)\\in \\mathfrak{S}_n$. Thus $\\mathrm{sgn}(\\phi(K)) = \\mathrm{sgn}(\\sigma)\\mathrm{sgn}\\left( (x y) \\right) = - \\mathrm{sgn}(\\sigma) = -\\mathrm{sgn}(K)$. Clearly, $K$ and $\\phi(K)$ traverse the same arcs, therefore $\\prod_{p\\in K} (\\prod p) = \\prod{p\\in \\phi(K)} (\\prod p)$.\nThe bijection $\\phi$ implies that the summands $K\\in Q\\backslash R$ add up to zero, thus we have $$\\det \\left( \\mu| S\\times T \\right) = \\sum_{L\\in R} \\mathrm{sgn}(L) \\prod_{p\\in L} \\left( \\prod p \\right).$$\nThe second statement of the lemma follows from the fact that for two routings $L_1,L_2\\in R$,\nwe have $L_1 = L_2$ if and only if $\\bigcup_{p\\in L_1} \\left| p \\right|_A = \\bigcup_{p\\in L_2} \\left| p \\right|_A$. For the non-trivial direction: assume we have a set of arcs $L_A$ that are traversed by the paths of a linking. Then the initial vertices of that linking are the elements of the set\n$S_A = \\SET{u\\in V\\mid \\forall (v,w)\\in L_A\\colon\\,u\\not= w}$. The terminal vertices are the elements of the set\n$T_A = \\SET{w\\in V\\mid \\forall (u,v)\\in L_A\\colon\\,u\\not= w}$, and the paths can be reconstructed from\nthe initial vertices $v\\in S_A$ by following the unique arcs $(v,w),(w,x),\\ldots \\in L_A$ until a vertex $t\\in T_A$ is reached. Clearly, for $L\\in R$, $\\prod_{p\\in L} \\left( \\prod p \\right) \\not= 0$, and since $w$ is an indeterminate weighting, two summands $L,L'\\in R$ can only cancel each other when the corresponding monomials are equal, i.e. $\\prod_{p\\in L} \\left( \\prod p \\right) = \\prod_{p\\in L'} \\left( \\prod p \\right)$; but then $L_A = L'_A$ holds, and so $L = L'$. Thus no summand in the determinant formula\n which belongs to a routing from $R$\ncan be cancelled out by another summand belonging to another routing from $R$. Therefore\n$\\det\\left( \\mu| S\\times T \\right) = 0$ if and only if $R=\\emptyset$, i.e. there is no linking from $S$ to $T$ in $D$.\n\\end{proof}\n\n\\begin{corollary}\\PRFR{Feb 15th}\n\tLet $D=(V,A)$ be an acyclic digraph, $T,E\\subseteq V$, and $w\\colon A\\longrightarrow \\mathbb{R}$ be an indeterminate weighting of $D$. Furthermore,\n\tlet\n\t \t$\\mu\\in \\mathbb{R}^{E\\times T}$ be the matrix with\n \t\\[ \\mu(e,t) = \\sum_{p\\in {\\mathbf{P}}{(D;e,t)}} \\left( \\prod p \\right).\\]\n \n \tThen $\\Gamma(D,T,E) = M(\\mu)$.\n\\end{corollary}\n\\begin{proof}\\PRFR{Feb 15th}\n\tThis is straightforward from the Definition~\\ref{def:Mmu} and the Lindstr\u00f6m Lemma~\\ref{lem:lindstrom}.\n\\end{proof}\n\n\\noindent\n\\PRFR{Feb 15th}\nClearly, for an arbitrary gammoid $M = \\Gamma(D,T,E)$, we cannot assume that $D$ is acyclic (Remark~\\ref{rem:weNeedCycles}). \nThere are several ways to work around this. Either {\\em a)}\\footnote{This is what is implied by the rationale given in \\cite{Ar06}.} \nwe adjust our definition of routing such that routings with non-path walks are allowed, making the class of routings in $D$ \ninfinite whenever there is a cycle in $D$. Then we could use power series to calculate $\\mu$ and the determinant, \nwhere convergence is guaranteed when $\\prod p \\in (0,1)$ for every cycle walk $p\\in {\\mathbf{W}}(D)$. \nA sufficient condition would be to use a weighting $w$ where $0 < w(a) < 1$ for all $a\\in A$.\n The construction of $\\phi$ in the proof of the Lindstr\u00f6m Lemma would still go through, but for the second \n statement we would have to choose the indeterminate weights more carefully, since a cycle walk $q\\in {\\mathbf{W}}(D)$ \n gives rise to the formal power\nseries $\\sum_{i=0}^\\infty \\left( \\prod q \\right)^{i}$ which converges to $\\frac{1}{1-\\prod q}$. \nClearly, a similar cardinality-argument as in Lemma~\\ref{lem:enoughZindependents} \nguarantees that we can find a sufficient number of carefully chosen indeterminates in $\\mathbb{R}$. \nOr {\\em b)} we could try to find a construction that removes cycles from $D$, possibly changing the\ngammoid represented by the resulting digraph $D'$, then use the Lindstr\u00f6m Lemma to obtain a \nmatrix $\\nu$, and then revert the constructions in order to obtain $\\mu$ from $\\nu$; which is what we will do now.\n\n\\begin{definition}\\PRFR{Feb 15th}\n\tLet $D=(V,A)$ be a digraph, $x,t\\notin V$ be distinct new elements, and let $c=(c_i)_{i=1}^n\\in {\\mathbf{W}}(D)$ be a cycle walk.\n\tThe \\deftext[lifting of c in D@lifting of $c$ in $D$]{lifting of $\\bm c$ in $\\bm D$ by $\\bm(\\bm x\\bm,\\bm t\\bm)$} is the digraph\n\t$D^{(c)}_{(x,t)} = (V\\mathbin{\\dot{\\cup}}\\SET{x,t}, A')$ where\n\t\\[ A' = A \\BSET{(c_1,c_2)} \\cup \\SET{(c_1,t),(x,c_2),(x,t)}.\\]\n\\end{definition}\n\n\\noindent Observe that the cycle walk $c\\in{\\mathbf{W}}(D)$ is not a walk in the lifting of $c$ in $D$ anymore.\n\n\\begin{example}\\PRFR{Feb 15th}\n\tConsider $D=(\\SET{c_1,c_2,c_3,c_4},\\SET{(c_1,c_2),(c_2,c_3),(c_3,c_4),(c_4,c_1)})$. Then\n\t$c_1c_2c_3c_4c_1\\in{\\mathbf{W}}(D)$ is a cycle. The lifting of $c$ in $D$ by $(x,t)$ is then defined to be\n\tthe digraph $D'=(\\SET{c_1,c_2,c_3,c_4,x,t},\\SET{(c_1,t),(c_2,c_3),(c_3,c_4),(c_4,c_1),(x,c_2),(x,t)})$.\n\n\t\\begin{center}\n\t\\includegraphics{digraphlifting2}\n\t\\end{center}\n\\end{example}\n\n\\begin{lemma}\\label{lem:liftingNoNewCycles}\\PRFR{Feb 15th}\n\tLet $D=(V,A)$ be a digraph, $x,t\\notin V$, and $c=(c_i)_{i=1}^n\\in {\\mathbf{W}}(D)$ a cycle walk, and let\n\t$D'=D^{(c)}_{(x,t)}$ be the lifting of $c$ in $D$ by $(x,t)$.\n\tIf $c'\\in {\\mathbf{W}}(D')$ is a cycle walk, then\n\t$c'\\in {\\mathbf{W}}(D)$. In other words, the lifting of cycle walks does not introduce new cycle walks.\n\\end{lemma}\n\\begin{proof}\\PRFR{Feb 15th}\n\tLet $D'=(V',A')$.\n\tClearly, $x$ is a source in $D^{(c)}_{(x,t)}$ and $t$ is a sink in $D^{(c)}_{(x,t)}$. Thus $x,t\\notin \\left| c' \\right|$.\n\tBut then $\\left| c' \\right|_A \\subseteq A'\\cap \\left( V\\times V \\right)$ and therefore $c'$ is also a cycle walk in $D$.\n\\end{proof}\n\n\n\\begin{definition}\\label{def:completeLifting}\\PRFR{Feb 15th}\n\tLet $D=(V,A)$ be a digraph. A \\deftext[complete lifting of D@complete lifting of $D$]{complete lifting of $\\bm D$}\n\tis an acyclic digraph $D'=(V',A')$ for which there is a suitable $n\\in \\mathbb{N}$ such that there is a\n\tset $X=\\dSET{x_1,t_1,x_2,t_2,\\ldots,x_n,t_n}$ with $X\\cap V = \\emptyset$,\n\ta family of digraphs $D^{(i)} = (V^{(i)},A^{(i)})$ for $i\\in \\SET{0,1,\\ldots,n}$\n\twhere $D' = D^{(n)}$, $D^{(0)} = D$, and for all $i\\in\\SET{1,2,\\ldots,n}$\n\t $$D^{(i)} = \\left( D^{(i-1)}\\right)^{(c_i)}_{(x_i,t_i)}$$\n\twith respect to a cycle walk $c_i\\in {\\mathbf{W}}\\left( D^{(i-1)}\\right)$.\n\tIn this case, we say that the set $$R = \\SET{(x_i,t_i)\\mid i\\in\\SET{1,2,\\ldots,n}}$$ \\deftextX{realizes} \n\tthe complete lifting $D'$ of $D$.\n\\end{definition}\n\n\\begin{lemma}\\label{lem:completelifting}\\PRFR{Feb 15th}\n\tLet $D=(V,A)$ be a digraph. Then $D$ has a complete lifting.\n\\end{lemma}\n\\begin{proof}\\PRFR{Feb 15th}\n\tBy induction on the number of cycle walks in $D$. If $D$ has no cycle walk, $D$ is a complete lifting of $D$.\n\tNow let $c\\in {\\mathbf{W}}(D)$ be a cycle walk, and let $x,t\\notin V$.\n\tLet $D' = D^{(c)}_{(x,t)}$. By construction $c\\notin {\\mathbf{W}}(D')$,\n\tthus\n\t the number of cycle walks in $D'$ is at least one short of the number of cycle walks in $D$ (Lemma~\\ref{lem:liftingNoNewCycles}), therefore there is a complete lifting $D''$ of $D'$ by induction hypothesis.\n\t Since $D'$ is a lifting of $D$, $D''$ is also a complete lifting of $D$.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:cyclelifting}\\PRFR{Feb 15th}\n\tLet $D=(V,A)$, $E,T\\subseteq V$, $c\\in {\\mathbf{W}}(D)$ a cycle, $x,t\\notin V$, and let $D'=D^{(c)}_{(x,t)}$ be the lifting of $c$ in $D$.\n\tThen $\\Gamma(D,T,E) = \\Gamma(D',T\\cup\\SET{t},E\\cup\\SET{x})|' E$.\n\\end{lemma}\n\\begin{proof}\\PRFR{Feb 15th}\n\tLet $M=\\Gamma(D,T,V)$ be the strict gammoid induced by $(D,T,E)$ and let \\linebreak\n\t$M' = \\Gamma(D',T\\cup\\SET{t},V')$ be the strict gammoid obtained from the lifting of $c$.\n\tThen $M'' = \\left( M' \\right)|'\\left( V\\cup\\SET{t} \\right)$ is a strict gammoid that is represented by $(D'',T,V\\cup\\SET{t})$\n\twhere the digraph\n\t $D'' = (V_0\\BSET{x},A_0\\backslash\\left( V_0\\times\\SET{x} \\right))$\n\tis induced from the $x$-$t$-pivot $D_0$ of $D'$, i.e.\n\t $D_0 = D'_{x\\leftarrow t} = (V_0,A_0)$. This follows from the proof of Lemma~\\ref{lem:contractionStrictGammoid}\n\t along with the single-arc routing $\\SET{xt}\\colon \\SET{x}\\double{\\rightarrow} T\\cup\\SET{t}$ in $D'$.\n\tLet $A''$ denote the arc set of $D''$. \n\tIt is easy to see from the involved constructions (Fig.~\\ref{fig:liftingcycles}), \n\tthat $A'' = \\left( A\\BSET{(c_1,c_2)} \\right) \\cup\\SET{(c_1, t), (t, c_2)}$.\n\t Clearly, a routing $R$ in $D$ can have at most one path $p\\in R$ such that $(c_1,c_2)\\in \\left| p \\right|_A$, \n\t and since $t\\notin V$, we obtain a routing $R'= \\left(R\\BSET{p}\\right)\\cup\\SET{q t r}$ \n\t for $q,r\\in {\\mathbf{P}}(D)$ such that \n\t$p=qr$ with $q_{-1}=c_1$ and $r_1=c_2$.\n\t Clearly, $R'$ routes $X$ to $Y$ in $D''$ whenever $R$ routes $X$ to $Y$ in $D$.\n\tConverserly, let $R'\\colon X'\\double{\\rightarrow} Y'$ a routing in $D''$ with $t\\notin X'$.\n\t Then there is at most one $p\\in R'$ with $t\\in \\left| p \\right|$.\n\t We can invert the construction an let $R''= \\left( R'\\BSET{p} \\right)\\cup\\SET{qr}$\n\t for the appropriate paths $q,r\\in {\\mathbf{P}}(D'')$ with $p=qtr$. \n\tThen $R''$ is a routing from $X'$ to $Y'$ in $D'$. Thus we have shown that $M''| V = M$,\n\tand consequently with $E\\subseteq V$ and Lemma~\\ref{lem:contractrestrictcommutes} follows\n\t\\begin{align*}\n\t\\Gamma(D,T,E) = M| E = \\left( M'' \\right)| E & = \\left( \\Gamma(D',T\\cup\\SET{t},V') |' \\left( V\\cup\\SET{t} \\right) \\right) | E \\\\& = \\Gamma(D',T\\cup\\SET{t},E\\cup\\SET{x})|' E.\n\\end{align*}\n\\end{proof}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics{liftingtrick}\n\\end{center}\n\\caption{\\label{fig:liftingcycles}Constructions involved in Lemma~\\ref{lem:cyclelifting}.}\n\\end{figure}\n\n\\begin{corollary}\\label{cor:acyclicQuasiRepresentationOfGammoids}\\PRFR{Feb 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a gammoid. Then there is an acyclic digraph $D=(V,A)$ and sets $T,E'\\subseteq V$ such that\n\t$M = \\Gamma \\left( D,T,E' \\right)|' E$\n\tand such that $$\\left| T \\right| = \\mathrm{rk}_M(E) + \\left| E'\\backslash E \\right|.$$\n\\end{corollary}\n\\begin{proof}\\PRFR{Feb 15th}\n\tLet $M=\\Gamma(D',T',E)$ with $\\left| T' \\right|=\\mathrm{rk}_M(E)$.\n\tThen let $D$ be a complete lifting of $D'$ (Lemma~\\ref{lem:completelifting}),\n\tand let $D^{(0)},D^{(1)},\\ldots, D^{(n)}$ be the family of digraphs and $c_1,c_2,\\ldots,c_n$ be the cycle walks that correspond to \n\tthe complete lifting $D$ of $D'$ \n\tas required by Definition~\\ref{def:completeLifting},\n\tand let $\\dSET{x_1,t_1,\\ldots,x_n,t_n}$ denote the new elements such that\n\t\\[ D^{(i)} = \\left( D^{(i-1)} \\right)^{(c_i)}_{(x_i,t_i)} \\]\n\tholds for all $i\\in\\SET{1,2,\\ldots,n}$.\n\tInduction on the index $i$ with Lemma~\\ref{lem:cyclelifting} yields that\n\t\\[ \\Gamma(D',T,E) = \\Gamma(D^{(i)},T\\cup\\SET{t_1,t_2,\\ldots,t_i},E\\cup\\SET{x_1,x_2,\\ldots,x_i})|' E\\]\n\tholds for all $i\\in\\SET{1,2,\\ldots,n}$.\n\tClearly, $$\\left| T\\cup\\SET{t_1,t_2,\\ldots,t_n} \\right| = \\left| T \\right| + n = \\mathrm{rk}_M(E) + n = \\mathrm{rk}_M(E) + \\left| \\SET{x_1,x_2,\\ldots,x_n} \\right|.$$\n\\end{proof}\n\n\\begin{theorem}\\label{thm:gammoidOverR}\\PRFR{Feb 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a gammoid, $T=\\dSET{t_1,t_2,\\ldots,t_{\\mathrm{rk}_M(E)}}$. Then there is a matrix $\\mu\\in \\mathbb{R}^{E\\times T}$ such\n\tthat $M= M(\\mu)$.\n\\end{theorem}\n\\begin{proof}\\PRFR{Feb 15th}\n\tBy Corollary~\\ref{cor:acyclicQuasiRepresentationOfGammoids}, there is an acyclic digraph $D=(V,A)$ and\n\tthere are sets $E',T' \\subseteq V$,\n\tsuch that $M = N|' E$ where $N=\\Gamma(D,T',E')$\n\tand $\\left| T' \\right| = \\mathrm{rk}_M(E) + \\left| E' \\backslash E \\right|$.\n\tRemember that $E'\\backslash E$ is independent in $N$,\n\tand every base $B$ of $M$ induces a base $B\\cup \\left( E'\\backslash E \\right)$ of $N$.\n\tThe Lindstr\u00f6m Lemma~\\ref{lem:lindstrom} yields a matrix $\\nu\\in \\mathbb{R}^{E'\\times T'}$ such that \n\t$N = M(\\nu)$. In Lemma~\\ref{lem:contractequalspivot} and Remark~\\ref{rem:contraction} we have seen that we can pivot in the\n\tindependent set $E'\\backslash E$\n\tin $\\nu$, which yields a new matrix $\\nu'\\in \\mathbb{R}^{E'\\times T'}$. Let $T_0 = \\SET{t'\\in T' ~\\middle|~ \\forall e'\\in E'\\backslash E\\colon\\, \\nu'(e',t') = 0 }$ denote the remaining columns of $\\nu'$ that have not been used to pivot in an element of $E'\\backslash E$.\n\tWe set $\\mu = \\nu'| E\\times T_0$. Thus $M(\\mu) = M(\\nu)|' E = N|' E = M$.\n\\end{proof}\n\n\\noindent We want to compare the two methods {\\em a)} and {\\em b)} mentioned above. In our opinion, \nboth methods are connected to aspects of the same underlying phenomenon that cycle paths do not interfere with \nthe existence of linkings between given sets of vertices in a digraph.\n\n\\input{Text\/Ex\/302_repr_circle}\n\n\\noindent\nIn the paper {\\em A parameterized view on matroid optimization problems} \\cite{Marx09}, D.~Marx shows that there is a randomized polynomial time algorithm with respect to the size of the ground set of a gammoid, \nthat constructs a matrix $\\mu$ from $(D,T,E)$ such that $M(\\mu) = \\Gamma(D,T,E)$.\nThe method of D.~Marx starts with the construction of the dual $N^\\ast$ of the underlying strict gammoid $N=\\Gamma(D,T,V)$ \nfor a given representation $(D,T,E)$\nwith $D=(V,A)$ through the linkage system of $D$ to $T$ (Definition~\\ref{def:linkageSystem} and Lemma~\\ref{lem:linkage}). \nThen a matrix $\\nu$ with $M(\\nu) = N^\\ast$ is constructed with a small probability of failure (see Proposition~\\ref{prop:randompolytimeTransversalMatroid} below), \nwhich in turn is converted into a standard representation (Remark~\\ref{rem:stdRep})\nof the form $(I_r\\,\\, A^\\top)^\\top$ using Gaussian Elimination. Then $(-A \\,\\, I_{n-r})^\\top$ is the desired representation of $M$.\nBefore we present the main proposition that leads to this result, we need the following lemma.\n\n\\begin{lemma}[\\cite{Sch80}, Corollary 1]\\label{lem:SchwartzNumberZeros}\n\tLet ${\\mathbb{F}}$ be a field, $X = \\dSET{x_1,x_2,\\ldots,x_n}$,\n\t let $p \\in {\\mathbb{F}}[X]$ be a polynomial\n\twith $p \\not= 0$.\n\tFurthermore, let $F\\subseteq {\\mathbb{F}}$ be a finite subset of elements of the coefficient field with $\\left| F \\right| \\geq c\\cdot \\deg(p)$\n\tfor some $c\\in {\\mathbb{Q}}$ with $c > 0$. Then\n\t \\[\t\\left| \\SET{\\xi \\in F^X ~\\middle|~ p[X=\\xi] = 0} \\right| \\leq \\frac{\\left| F \\right|^n}{c}.\\]\n\\end{lemma}\n\n\\noindent For a formal proof, we refer the reader to J.T.~Schwartz's \n{\\em Fast Probabilistic Algorithms for Verification of Polynomial Identities} \\cite{Sch80}. \nThe proof idea is to do induction on the number of variables involved. The base case is the fact that a polynomial\nin a single variable of degree $d$ can at most have $d$ different zeros. In the induction step, we fix the values of all but one variable,\nif the resulting polynomial in a single variable is the zero polynomial, we may choose any value from $F$ for that variable. Otherwise,\nthere are at most the degree of the resulting polynomial many choices for the last variable such that the polynomial evaluates to zero.\n\n\\begin{lemma}[\\cite{Marx09}, Lemma 1, \\cite{Sch80}, \\cite{Zi79}]\\label{lem:probOfZero}\n\tLet ${\\mathbb{F}}$ be a field, \\linebreak\n\tlet $X = \\dSET{x_1,x_2,\\ldots,x_n}$ be a finite set,\n\t let $p \\in {\\mathbb{F}}[X]$ be a polynomial\n\twith $p \\not= 0$, and let $F\\subseteq {\\mathbb{F}}$ be a finite set.\n\tLet $\\xi$ be a random variable sampled from a uniform distribution on the set $F^X$.\n\tThen the probability that $\\xi$ is a zero of $p$ may be estimated by\n\t\\[ \\Pr\\left(p[X=\\xi] = 0\\right) \\leq \\frac{\\deg(p)}{\\left| F \\right|} .\\]\n\\end{lemma}\n\n\\begin{proof}\\footnote{D.~Marx omits the proof and instead cites \\cite{Sch80} and \\cite{Zi79}.}\n\tIn Lemma~\\ref{lem:SchwartzNumberZeros} we set $c = \\frac{\\left| F \\right|}{\\deg(p)}$ and get\n\t\\[ \\frac{\\left| \\SET{\\xi \\in F^X ~\\middle|~ p[X=\\xi] = 0} \\right|}{ \\left| F^X \\right|} \\leq \\frac{\\left| F \\right|^n}{c\\cdot \\left| F \\right|^n} = \\frac{1}{c} = \\frac{\\deg(p)}{\\left| F \\right|}.\\]\n\\end{proof}\n\n\\begin{proposition}[\\cite{Marx09}, Proposition 3.11]\\label{prop:randompolytimeTransversalMatroid}\n\tLet $E$ be a finite set, $r\\in \\mathbb{N}$, and ${\\mathcal{A}} = (A_i)_{i=1}^r \\subseteq E$ be a family of subsets of $E$.\n\tThen a matrix $\\mu\\in \\mathbb{R}^{E\\times \\SET{1,2,\\ldots,r}}$ with $M(\\mu) = M({\\mathcal{A}})$ can be constructed in randomized polynomial time.\n\\end{proposition}\n\\begin{proof}\n\tFor all $k\\in \\mathbb{N}$ with $k>1$, we write $\\mathrm{unif}(k)$\n\tin order to denote an integer that has been randomly sampled from a uniform distribution on $\\SET{1,2,\\ldots,k}$. Several instances\n\tof $\\mathrm{unif}(k)$ shall denote independently sampled random variables.\n\n\n\tLet $p\\in \\mathbb{N}$ be an arbitrary parameter.\n\tWe define the random matrix $\\mu\\in \\mathbb{R}^{E\\times \\SET{1,2,\\ldots,r}}$ by\\footnote{D.~Marx uses samples \n\tfrom $\\mathrm{unif}\\left( 2^p\\cdot \\left| E \\right|\\cdot 2^{\\left| E \\right|} \\right)$ and uses the argument that there are at most\n\t$2^{\\left| E \\right| }$ independent sets. This line of arguments is valid, \n\tyet it does not use the fact that if $X$ is independent in $M(\\mu)$, then all subsets of $X$ are independent in $M(\\mu)$, too; \n\tconsequently, the probability of failure is overestimated.}\n\t\\[ \\mu(e,i) = \\begin{cases}[r]\n\t\t\t\t\t\\mathrm{unif} \\left( 2^p\\cdot \\left| E \\right|\\cdot Q \\right) & \\quad\\text{if~} e\\in A_i, \\\\\n\t\t\t\t\t0 & \\quad\\text{otherwise,}\n\t\\end{cases} \n\t\\]\n\twhere $$Q = \\binom{\\left| E \\right|}{\\left\\lceil \\frac{\\left| E \\right|}{2}\\right\\rceil}.$$\n\tClearly, $Q \\leq 2^{\\left| E \\right|}$ with equality if $\\left| E \\right| = 1$.\n\tObserve that sampling $\\mathrm{unif}(2^k)$ can be done by sampling $k$ bits from a uniform distribution. Thus $\\mu$ can be obtained\n\tby sampling at most\n\t $\\left| E \\right|\\cdot r \\cdot \\left(\\left| E \\right| + p + \\lceil \\log_2\\left( \\left| E \\right| \\right) \\rceil \\right)$ uniform random bits.\n\tWe show that $\\Pr( M(\\mu) \\not= M({\\mathcal{A}})) \\leq \\frac{1}{2^p}$.\n\tLet $X\\subseteq E$ be independent in $M(\\mu)$. Then ${\\mathrm{idet}~}(M| X\\times \\SET{1,2,\\ldots,r}) = 1$,\n\tso there is an injective map $\\phi\\colon X\\longrightarrow \\SET{1,2,\\ldots,r}$ such that $\\mu(x,\\phi(x)) \\not= 0$\n\tfor all $x\\in X$. By construction of $\\mu$ we obtain that in this case $x\\in A_{\\phi(x)}$. Therefore $X$ is a partial transversal\n\tof ${\\mathcal{A}}$, and so $X$ is indepedent in $M({\\mathcal{A}})$, too.\n\n\t\\noindent\n\tNow let $X\\subseteq E$ be a base of $M({\\mathcal{A}})$. Thus $X$ is a maximal partial transversal of ${\\mathcal{A}}$ and\n\tthere is an injective map $\\phi\\colon X\\longrightarrow \\SET{1,2,\\ldots,r}$ such that $x\\in A_{\\phi(x)}$ for all $x\\in X$.\n\tLet $X=\\dSET{x_1,x_2,\\ldots,x_k}$, then we may define the matrix $\\nu \\in \\mathbb{R}[X]^{X\\times \\phi[X]}$\n\twhere\n\t\t\\[ \\nu(x,i) = \\begin{cases}[r]\n\t\t\t\t\t\tx_i & \\quad\\text{if~} i = \\phi(x),\\\\\n\t\t\t\t\t\t\\mu(x,i) & \\quad\\text{otherwise.}\n\t\t\\end{cases}\\]\n\tThen $\\det(\\nu)$ is a polynomial of degree $\\left| X \\right|$ with leading monomial $x_1x_2\\cdots x_k$ in $\\mathbb{R}[X]$,\n\tand if $\\xi\\in \\mathbb{R}^X$ is the vector\n\twhere $\\xi(x) = \\mu(x,\\phi(x))$ for all $x\\in X$, we have the equality\n\t\\[ \\det(\\mu| X\\times \\phi[X]) = \\left( \\det(\\nu) \\right)[X=\\xi] .\\]\n\tRemember that each value $\\xi(x)$ has been uniformly sampled from a set with cardinality $ 2^p\\cdot \\left| E \\right|\\cdot Q$,\n\tthus Lemma~\\ref{lem:probOfZero} yields\n\t\\[ \\Pr(\\det(\\mu| X\\times\\phi[X]) = 0) \\leq \\frac{\\left| X \\right|}{2^p\\cdot \\left| E \\right|\\cdot 2^{\\left| E \\right|}} \\leq \\frac{1}{2^p\\cdot Q}.\\]\n\tThere are at most $\\binom{\\left| E \\right|}{\\mathrm{rk}_{M({\\mathcal{A}})}(E)}$ different bases in $M({\\mathcal{A}})$,\n\tand the family of all subsets of $E$ with cardinality $\\left\\lceil \\frac{\\left| E \\right|}{2} \\right\\rceil$\n\tis a maximal-cardinality anti-chain in the power set lattice of $E$.\n\tTherefore, there are at most $Q$ different bases\n\tin $M({\\mathcal{A}})$\n\tneeded to detect failure of $M({\\mathcal{A}})=M(\\mu)$.\n\tThus we obtain\n\t\\[ \\Pr(M(\\mu) \\not= M({\\mathcal{A}})) \\leq \\sum_{B\\in {\\mathcal{B}}(M({\\mathcal{A}}))} \\frac{1}{2^p\\cdot Q} \\leq \\frac{1}{2^p} .\\]\n\\end{proof}\n\n\\section{The Recognition Problem}\n\nFirst, we give a formal definition of what we mean when we talk about the problem of {\\em recognizing a gammoid}.\n\n\\begin{definition}\\PRFR{Feb 15th}\n\tLet ${\\mathcal{M}}$ be a class of matroids.\n\tThe \\deftext{gammoid recognition problem for $\\bm {\\mathcal{M}}$} -- or shorter $\\mathrm{Rec}\\Gamma_{\\mathcal{M}}$\\label{n:RecGM} --\n\tis the problem of computing the image of $M\\in {\\mathcal{M}}$ under the class-map\\label{n:GammaM}\n\t\\[ \\Gamma_{\\mathcal{M}} \\colon {\\mathcal{M}} \\longrightarrow \\SET{0,1},\\quad M\\mapsto \\begin{cases}[r] 1 & \\text{if~} M\\text{~is a gammoid,}\\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0 & \\text{otherwise.} \\end{cases} \\]\n\tThe elements $M\\in {\\mathcal{M}}$ are called the \\deftext[instances of RecGM@instances of $\\mathrm{Rec}\\Gamma_{\\mathcal{M}}$]{instances of $\\bm{\\mathbf{Rec}\\Gamma_{\\mathcal{M}}}$}.\n\\end{definition}\n\n\\PRFR{Mar 7th}\n\\noindent In less formal words, the gammoid recognition problem is the problem that given an instance of a matroid $M$,\ndetermine whether $\\Gamma_{\\mathcal{M}}(M) = 1$ or $\\Gamma_{\\mathcal{M}}(M) = 0$ by application of some algorithm. Thus we are interested in\nalgorithms that compute $\\Gamma_{\\mathcal{M}}$ and naturally we are also interested in the run-time complexity of \nthose algorithms as well as lower bounds for the complexity of these algorithms.\nObviously, there is no constant-time algorithm for the computation of $\\Gamma_{\\mathcal{M}}(M)$, therefore we would like to fix\na certain way to encode a matroid $M$ --- up to renaming elements of its ground set,\nyet preserving the implicit linear order of its ground set $E=\\dSET{e_1,e_2,\\ldots, e_n}$.\n\n\\begin{definition}\\PRFR{Mar 7th}\n\tLet $n,r\\in \\mathbb{N}$ with $n\\geq r$. We fix the bijection\\label{n:kth}\n\t\\begin{align*}\n\t\t \\mathrm{kth}(n,r)\\colon \\SET{1,2,\\ldots,\\binom{n}{r}} \\longrightarrow &\\binom{\\SET{1,2,\\ldots,n}}{r}\n\t\\end{align*}\n\twith the defining property that for all $i,j\\in \\mathbb{N}$ with $1\\leq i,j \\leq \\binom{n}{r}$\n\twe have\n\t\\[ \\min\\left(\\vphantom{A^A} \\mathrm{kth}(n,r)(i) \\bigtriangleup \\mathrm{kth}(n,r)(j) \\right) \\,\\,\\in\\,\\, \\mathrm{kth}(n,r)(i) \\quad\\Longleftrightarrow\\quad i < j,\\]\n\twhere $\\bigtriangleup$ denotes the symmetric difference of sets.\n\tIn words, we enumerate all $r$-elementary subsets of $\\SET{1,2,\\ldots,n}$ in ascending order with respect to the linear order that\n\trelates a subset $A$ with a subset $B$ whenever the smallest element in $\\left( A\\cup B \\right) \\backslash \\left( A\\cap B \\right)$\n\tbelongs to $A$.\n\\end{definition}\n\n\\noindent For $n \\geq r+1$ we have $\\mathrm{kth}(n,r)(1) = \\SET{1,2,\\ldots,r}$, $\\mathrm{kth}(n,r)(2) = \\SET{1,2,\\ldots,r-1,r+1}$,\nand $\\mathrm{kth}(n,r)\\left( \\binom{n}{r} \\right) = \\SET{n-r, n-r+1,\\ldots,n}$.\n\n\\needspace{4\\baselineskip}\n\n\\begin{definition}\\label{def:bitsM}\\PRFR{Mar 7th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid and let $E = \\dSET{e_1,e_2,\\ldots,e_n}$ bear an implicit linear order.\n\tWe define the \\deftext[binary encoding of $M$]{binary encoding of $\\bm M$} to \\label{n:bMenc}\n\tbe the vector \\[ \\mathrm{b}(M) = (\\mathrm{b}(M,i))_{i=1}^{N} \\in \\SET{0,1}^{N} \\]\n\twhere $N = \\left| E \\right| + 2 + \\binom{\\left| E \\right|}{\\mathrm{rk}_M(E)}$ is the encoding length of $M$\n\tand where\n\t\\[ \\mathrm{b}(M,i) = \\begin{cases}[r]\n\t\t\t\t\t\t1 & \\quad \\text{if~} i \\leq \\mathrm{rk}_M(E),\\\\\n\t\t\t\t\t\t0 & \\quad \\text{if~} i = \\mathrm{rk}_M(E) + 1, \\\\\n\t\t\t\t\t\t1 & \\quad \\text{if~} \\mathrm{rk}_M(E) + 1 < i \\leq \\left| E \\right| + 1, \\\\\n\t\t\t\t\t\t0 & \\quad \\text{if~} i = \\left| E \\right| + 2, \\\\\n\t\t\t\t\t\t1 & \\quad \\text{if~} i > \\left| E \\right| + 2 {\\mathop{\\text{~and~}}} \\kappa\\left( i - \\left| E \\right| - 2 \\right)\\in {\\mathcal{I}},\\\\\n\t\t\t\t\t\t0 & \\quad \\text{otherwise,}\n\t\t\t\t\t\\end{cases}\\]\n\twhere $\\kappa(k) = \\SET{e_i\\in E ~\\middle|~ i\\in \\mathrm{kth}(\\left| E \\right|, \\mathrm{rk}_M(E))(k)}$.\n\tIn other words, $\\mathrm{b}(M)$ consists of a unary encoding of $\\mathrm{rk}_M(E)$, followed by a unary encoding of $\\left| E \\right| - \\mathrm{rk}_M(E)$,\n\tfollowed by $\\binom{\\left| E \\right|}{\\mathrm{rk}_M(E)}$ bits encoding which of the $\\mathrm{rk}_M(E)$-elementary subsets of $E$ are bases of $M$,\n\tin the ascending order with respect to the implicit linear order on $E$.\n\tFurthermore, the \\deftext[encoding length of $M$]{encoding length of $\\bm M$} shall be denoted by \\label{n:encM}\n\t\\[ {\\mathbf{N}}(M) = \\left| E \\right| + 2 + \\binom{\\left| E \\right|}{\\mathrm{rk}_M(E)}. \\qedhere \\]\n\\end{definition}\n\n\\begin{remark}\\PRFR{Mar 7th}\n\tClearly, we can restore a matroid isomorphic to $M=(E,{\\mathcal{I}})$ from $\\mathrm{b}(M)$.\n\tFurthermore, the laws of the binomial coefficients yield that\n\t$${\\mathbf{N}}(M) = {\\mathbf{N}}(M^\\ast)$$ since $\\binom{n}{k} = \\binom{n}{n-k}$, and for all $X\\subsetneq E$\n\t\\[{\\mathbf{N}}(M| X) = {\\mathbf{N}}(M|' X) < {\\mathbf{N}}(M)\\]\n\tsince $\\binom{n}{k} = \\binom{n-1}{k-1} + \\binom{n-1}{k}$.\n\tWhen $\\left| E \\right|\\geq 4$, a rather rough estimate is \\( {\\mathbf{N}}(M) \\leq 2^{\\left| E \\right|}. \\)\n\\end{remark}\n\n\\needspace{4\\baselineskip}\n\\begin{remark}\\PRFR{Mar 7th}\n\tR.~Pendavingh and J.~van~der~Pol give the following\n\t lower bound for the number of matroids of rank $r$ on an $n$-elementary ground set in \\cite{PP17}, \n\t let $s_{r,n}$ denote this lower bound. Then \n\t \\[ \\log \\left( s_{n,r} \\right) \\geq \\frac{1}{n-r+1}\\cdot \\binom{n}{r}\\cdot \\log\\left( c^{1-r}(n-r+1)(1+o(1)) \\right) \\]\n\t for some constant $c$ independent of $r$ and $n$\n\t (Lemma~9~(3), \\cite{PP17} p.4). Thus, if we write a big list of all matroids with $n$ elements and rank $r$, and then\n\t use the corresponding list index, encoded as a binary number, in order to represent the base vector of the matroid, \n\t we would still have the binomial $\\binom{\\left| E \\right|}{\\mathrm{rk}_M(E)}$ as a factor in\n\t the encoding length. Therefore our encoding $\\mathrm{b}(M)$ from Definition~\\ref{def:bitsM} may be considered not excessively-bloated.\n\\end{remark}\n\n\n\n\n\\PRFR{Mar 7th}\n\\noindent First, we shall examine how easy it is to extract matroid information from an encoded matroid.\nThroughout this work, we assume that checking, whether a set of the correct cardinality is a base of $M$, can be\ndone in $O(1)$ time by reading the corresponding bit from $\\mathrm{b}(M)$.\n\n\\needspace{3\\baselineskip}\n\\begin{algorithm}\\PRFR{Mar 7th}\\label{alg:indep}\\index{algorithm!independence} \\textbf{Check For Independence}\\\\\n\n\\noindent \n\\begin{tabularx}{\\textwidth}{rl}\n\t\\textbf{Input}& {\\em(1)} A matroid $M=(E,{\\mathcal{I}})$ given by $\\mathrm{b}(M)$.\\\\\n\t&{\\em(2)} A subset $X\\subseteq E$, given by a vector of $2^{\\SET{1,2,\\ldots,\\left| E \\right|}}$.\\\\\n\t\\textbf{Output}& $1$ if $X\\in {\\mathcal{I}}$, $0$ otherwise.\\\\\n\t& \\\\\n\t& \\ttfamily for $i = 1 \\ldots \\binom{\\left| E \\right|}{\\mathrm{rk}_M(E)}$ do\\\\*\n\t& \\ttfamily $\\quad $ if $\\mathrm{b}(M,\\left| E \\right|+1+i) = 1 \\text{\\rmfamily ~and~} X \\subseteq \\mathrm{kth}\\left( \\left| E \\right|, \\mathrm{rk}_M(E) \\right)(i)$ \n\t\tthen do \\\\*\n\t& \\ttfamily $\\quad $ $\\quad $ return $1$ and stop\\\\*\n\t& \\ttfamily $\\quad $ end if $\\mathrm{b}(M,\\left| E \\right|+1+i) = 1 \\text{\\rmfamily ~and~} X \\subseteq \\mathrm{kth}\\left( \\left| E \\right|, \\mathrm{rk}_M(E) \\right)(i)$ \\\\\n\t& \\ttfamily $\\quad $ next $i$ \\\\*\n\t& \\ttfamily end for $i$ \\\\*\n\t& \\ttfamily return $0$.\n\\end{tabularx}\n\n\\noindent\nIn order to check for independence we have to test whether $X$ is the subset of a base of $M$.\nTherefore we iterate over at most $O({\\mathbf{N}}(M))$ base candidates $B$ and for each candidate we check whether $B\\in{\\mathcal{B}}(M)$ and\n$B \\subseteq X$, which can be done in $O(\\left| E \\right|)$ bit-comparisons.\nThus the overall run-time is $O(\\left| E \\right|\\cdot {\\mathbf{N}}(M)) = O(n^2)$ \nwhere $n= {\\mathbf{N}}(M) + \\left| E \\right|$ is the total bit-length of the input.\n\\end{algorithm}\n\n\\begin{proof}[Proof of correctness]\\PRFR{Mar 7th}\nLemma~\\ref{lem:augmentation} states that every independent set $X\\in {\\mathcal{I}}$ can be extended to a base $B\\in {\\mathcal{B}}(M)$.\nThe execution invariant at the ``{\\ttfamily next $i$}'' instruction is that\n none of the bases in ${\\mathcal{B}}(M)\\cap \\SET{\\SET{e_k ~\\middle|~ k\\in \\mathrm{kth}(\\left| E \\right|,\\mathrm{rk}_M(E)(j)} ~\\middle|~ j \\leq i}$ contains $X$.\n Thus the invariant at the ``{\\ttfamily end for $i$}''-instruction is\n that there is no $B\\in{\\mathcal{B}}(M)$ with $X\\subseteq B$. And then the return value is $0$.\n Otherwise, if the algorithm returns $1$, then the set $\\SET{e_k ~\\middle|~ k\\in \\mathrm{kth}\\left( \\left| E \\right|, \\mathrm{rk}_M(E) \\right)(i)}$\n is a base of $M$ that proves $X\\in {\\mathcal{I}}$.\n\\end{proof}\n\n\\noindent \\textbf{In order to reduce the technicalities of notation}, we identify the ground set $E=\\dSET{e_1,e_2,\\ldots,e_n}$ with\nthe set $\\dSET{1,2,\\ldots,n}$ through the bijection $i\\mapsto e_i$; therefore we identify $\\SET{e_i~\\middle|~i\\in \\mathrm{kth}(n,r)(j)}$ with $\\mathrm{kth}(n,r)(j)$ for the treatise of algorithms.\n\n\n\\needspace{3\\baselineskip}\n\\begin{algorithm}\\PRFR{Mar 7th}\\index{algorithm!rank} \\textbf{Compute the Rank}\\\\\n\n\\noindent\n\\begin{tabularx}{\\textwidth}{rl}\n\t\\textbf{Input}& {\\em(1)} A matroid $M=(E,{\\mathcal{I}})$ given by $\\mathrm{b}(M)$.\\\\\n\t&{\\em(2)} A subset $X\\subseteq E$, given by a vector $2^{\\SET{1,2,\\ldots,\\left| E \\right|}}$.\\\\\n\t\\textbf{Output}& $\\mathrm{rk}_M(X)$.\\\\\n\t& \\\\\n\t& $r := 0$ \\\\*\n\t& \\ttfamily for $i = 1 \\ldots \\binom{\\left| E \\right|}{\\mathrm{rk}_M(E)}$ do\\\\*\n\t& \\ttfamily $ \\quad $ $r := \\max\\SET{\\vphantom{A^A}r,\\,\\mathrm{b}(M, \\left| E \\right| + 1 + i)\\cdot \n\t\\left| \\mathrm{kth}\\left( \\left| E \\right|, \\mathrm{rk}_M(E) \\right)(i) \\cap X \\right|}$\\\\*\n\t& \\ttfamily $\\quad $ next $i$ \\\\*\n\t& \\ttfamily end for $i$\\\\*\n\t& \\ttfamily return $r$.\n\\end{tabularx}\n\n\\PRFR{Mar 7th}\n\\noindent\nIn order to compute the rank, we have to do $O({\\mathbf{N}}(M))$ iterations of the main loop which consists of one multiplication, \none comparison of values bounded above by $\\left| E \\right|$,\n possibly a copy operation of these values,\n and possibly a calculation of the intersection between \n$\\mathrm{kth}(\\left| E \\right|, \\mathrm{rk}_M(E))(i)$ and $X$, which can be done in $O(\\left| E \\right|)$. \nThus the overall run-time is $O(\\left| E \\right|\\cdot {\\mathbf{N}}(M)) = O(n^2)$ \nwhere $n= {\\mathbf{N}}(M) + \\left| E \\right|$ is the total bit-length of the input.\n\\end{algorithm}\n\n\\begin{proof}[Proof of correctness]\\PRFR{Mar 7th}\nLemma~\\ref{lem:augmentation} and Definition~\\ref{def:rank} yield that the rank of $X\\subseteq E$ equals the maximum cardinality of\nthe intersection of $X$ with a base $B\\in{\\mathcal{B}}(M)$.\nClearly, the invariant for the value of $r$ at the ``{\\ttfamily next $i$}'' instruction is\n $$r = \\max\\SET{\\vphantom{A^A}\\left| X\\cap B \\right| ~\\middle|~ B\\in {\\mathcal{B}}(M) \\cap \\SET{\\mathrm{kth}(\\left| E \\right|,\\mathrm{rk}_M(E)(j) ~\\middle|~ j \\leq i}}.$$\nTherefore the invariant at the ``{\\ttfamily end for $i$}'' instruction is\n$$r = \\max\\SET{\\vphantom{A^A}\\left| X\\cap B \\right| ~\\middle|~ B\\in {\\mathcal{B}}(M)},$$\n thus the returned value $r = \\mathrm{rk}_M(X)$ is correct.\n\\end{proof}\n\n\\needspace{3\\baselineskip}\n\\begin{algorithm}\\PRFR{Mar 7th}\\index{algorithm!closure} \\textbf{Compute the Closure}\\\\\n\n\\noindent\n\\begin{tabularx}{\\textwidth}{rl}\n\t\\textbf{Input}& {\\em(1)} A matroid $M=(E,{\\mathcal{I}})$ given by $\\mathrm{b}(M)$.\\\\\n\t&{\\em(2)} A subset $X\\subseteq E$, given by a vector of $2^{\\SET{1,2,\\ldots,\\left| E \\right|}}$.\\\\\n\t\\textbf{Output}& $\\mathrm{cl}_M(X)$.\\\\\n\t& \\\\\n\t& $C := X$ \\\\*\n\t& $r := \\mathrm{rk}_M(X)$ \\\\*\n\t& \\ttfamily for $e \\in E\\backslash X$ do \\\\*\n\t& \\ttfamily $\\quad $ if $\\mathrm{rk}_M(X\\cup\\SET{e}) = r$ then $C := C \\cup \\SET{e}$ \\\\*\n\t& \\ttfamily $\\quad $ next $e$\\\\*\n\t& \\ttfamily end for $e$\\\\*\n\t& \\ttfamily return $C$.\n\\end{tabularx}\n\n\\PRFR{Mar 7th}\n\\noindent\nIn order to compute the closure, we have to compute the ranks of subsets of $E$ precisely $\\left| E\\backslash X \\right| + 1$ times,\ntherefore we accumulate a running time of $O(\\left| E \\right|\\cdot n^2) = O(n^3)$ where $n= {\\mathbf{N}}(M) + \\left| E \\right|$ is the total bit-length of the input.\n\\end{algorithm}\n\n\\begin{proof}[Proof of correctness]\\PRFR{Mar 7th}\n\tFirst, we show that $$\\mathrm{cl}(X) = X\\cup \\SET{e\\in E\\backslash X ~\\middle|~\\vphantom{A^A} \\mathrm{rk}(X\\cup\\SET{e}) = \\mathrm{rk}(X)}.$$\n\tThe closure of $X$ is the intersection\n\tof all flats $F\\in {\\mathcal{F}}(M)$ with $X\\subseteq F$ (Definition~\\ref{def:clM}),\n\tand flats are those sets $F\\subseteq E$, such that $\\mathrm{rk}(F\\cup\\SET{e}) > \\mathrm{rk}(F)$\n\tholds for all $e\\in E\\backslash F$ (Definition~\\ref{def:FcalM}).\n\tSince $\\mathrm{rk}(X) = \\mathrm{rk}(\\mathrm{cl}(X))$ (Lemma~\\ref{lem:clKeepsRank}), we obtain that for all $x\\in E\\backslash X$ with $\\mathrm{rk}(X\\cup\\SET{x})=\\mathrm{rk}(X)$,\n\twe have the implication $X \\subseteq F \\Rightarrow x\\in F$ for all flats $F\\in{\\mathcal{F}}(M)$. Consequently, $x\\in \\mathrm{cl}(X)$ whenever\n\t$\\mathrm{rk}(X\\cup\\SET{x})=\\mathrm{rk}(X)$. On the other hand, if $\\mathrm{rk}(X\\cup\\SET{x}) > \\mathrm{rk}(X)$, then $\\mathrm{cl}(X\\cup\\SET{x})\\in {\\mathcal{F}}(M)$ is the\n\tsmallest flat that contains $X\\cup\\SET{x}$, but $\\mathrm{rk}(\\mathrm{cl}(X\\cup\\SET{x})) > \\mathrm{rk}(\\mathrm{cl}(X))$, thus $x\\notin \\mathrm{cl}(X)$ whenever \n\t$\\mathrm{rk}(X\\cup\\SET{x})\\not= \\mathrm{rk}(X)$.\n\n\t\\noindent\n\tLet $E\\backslash X = \\dSET{e_1,e_2,\\ldots,e_k}$ with the implicit order of occurrence in the ``{\\ttfamily for $e$}''-loop,\n\tand let $i\\in \\SET{1,2,\\ldots,k}$ be the index of the iteration of the loop corresponding to $e$, i.e. $e_i = e$.\n\t The invariant at the \n\t``{\\ttfamily next $e$}''-instruction with regard to $C$ is $C= \\mathrm{cl}_M(X) \\cap \\left( X\\cup\\SET{e_1,e_2,\\ldots,e_i} \\right)$.\n\tThus the invariant at the ``{\\ttfamily end for $e$}''-instruction is\n\t $C = \\mathrm{cl}_M(X) \\cap \\left( X\\cup \\left( E\\backslash X \\right) \\right) = \\mathrm{cl}_M(X)$.\n\\end{proof}\n\n\\needspace{3\\baselineskip}\n\\begin{algorithm}\\index{algorithm!test for strict gammoid}\\label{alg:strictGammoidTest}\\PRFR{Mar 7th} \\textbf{Naive Test for Strict Gammoids}\\\\\n\n\\noindent\n\\begin{tabularx}{\\textwidth}{rl}\n\t\\textbf{Input}& {\\em(1)} A matroid $M=(E,{\\mathcal{I}})$ given by $\\mathrm{b}(M)$.\\\\\n\n\t\\textbf{Output}& $1$, if $\\alpha_M \\geq 0$, or $0$ otherwise.\\\\\n\t& \\\\\n\t& \\ttfamily initialize $\\alpha_M \\in \\mathbb{Z}^{2^E}$ with $\\alpha_M \\equiv 0$ \\\\\n\t& \\ttfamily initialize ${\\mathcal{F}} \\in \\SET{0,1}^{2^E}$ with ${\\mathcal{F}} \\equiv 0$ \\\\\n\t& \\ttfamily for $k = 0 \\ldots \\left| E \\right|$ do\\\\*\n\t& \\ttfamily $\\quad $ for $X \\in \\binom{E}{k}$ do\\\\*\n\t& \\ttfamily $\\quad $$\\quad $ $a := k - \\mathrm{rk}_M(X)$\\\\\n\t& \\ttfamily $\\quad $$\\quad $ for $Y \\subsetneq X$ do\\\\*\n\t& \\ttfamily $\\quad $$\\quad $ $\\quad $ if ${\\mathcal{F}}(Y) = 1$ then do\\\\*\n\t& \\ttfamily $\\quad $$\\quad $ $\\quad $ $\\quad $ $a := a - \\alpha_M(Y)$\\\\*\n\t& \\ttfamily $\\quad $$\\quad $ $\\quad $ $\\quad $ if $a < 0$ then do \\\\*\n\t& \\ttfamily $\\quad $$\\quad $ $\\quad $ $\\quad $ $\\quad $ return $0$ and stop\\\\*\n\t& \\ttfamily $\\quad $$\\quad $ $\\quad $ $\\quad $ end if $a < 0$\\\\*\n\t& \\ttfamily $\\quad $$\\quad $ $\\quad $ end if ${\\mathcal{F}}(Y) = 1$\\\\*\n\t& \\ttfamily $\\quad $$\\quad $ $\\quad $ next $Y$\\\\*\n\t& \\ttfamily $\\quad $$\\quad $ end for $Y$\\\\\n\t& \\ttfamily $\\quad $$\\quad $ $\\alpha_M(X) := a$\\\\*\n\t& \\ttfamily $\\quad $$\\quad $ if $X = \\mathrm{cl}_M(X)$ then ${\\mathcal{F}}(X) := 1$\\\\*\n\t& \\ttfamily $\\quad $$\\quad $ next $X$\\\\*\n\t& \\ttfamily $\\quad $ end for $X$\\\\*\n\t& \\ttfamily $\\quad $ next $k$\\\\*\n\t& \\ttfamily end for $k$\\\\\n\t& \\ttfamily return $1$.\n\\end{tabularx}\n\n\\noindent\nThe algorithm calculates $\\alpha_M$ bottom-up using the recurrence relation and simultaneously keeping track of the family of flats of $M$.\nIf $\\alpha_M < 0$ at some point, then the algorithm stops early with a negative answer. Otherwise we have to calculate all values of $\\alpha_M$ in order to be sure that $M$ is a strict gammoid, therefore iterating $2^{\\left| E \\right|}$ different values of $X$.\nAll values that are assigned to $\\alpha_M(X)$ are non-negative integers that are bounded by $\\left| X \\right| - \\mathrm{rk}_M(X)$, because\nthe algorithm stops if $a<0$ before assigning the negative value to $\\alpha_M(X)$. For the same reason we \nhave $\\left| a \\right| \\leq \\left| X \\right|$. Thus the calculation of the correct value of $\\alpha_M(X)$ \nneeds at most $2^{\\left| X \\right|}$ subtractions, each with a run-time in $O(\\log(\\left| X \\right|))$, \nand $2^{\\left| X \\right|}$ tests of the flat property in $O(1)$. In order to \ndetermine the value of $\\alpha_M(X)$ for a single instance $X$,\n we need $O\\left( \\left( \\log (\\left| X \\right|) + 1 \\right) \\cdot 2^{\\left| X \\right|} \\right) = O\\left(\\log(\\left| E \\right|)\\cdot 2^{\\left| E \\right|}\\right)$-time. \n The flat book-keeping needs one closure operation that takes\n$O(\\left| E \\right|^2\\cdot {\\mathbf{N}}(M))$, and $\\left| E \\right|$ bit-comparisons, thus the book-keeping is in $O\\left( \\left| E \\right|^2\\cdot {\\mathbf{N}}(M) \\right)$. If $M$ is a strict gammoid, \nthis has to be done for all $2^{\\left| E \\right|}$ subsets $X\\subseteq E$, \nthus the total run-time is \n\\[O\\left(\\log(\\left| E \\right|)\\cdot 2^{2 \\left| E \\right|} + \\left| E \\right|^2\\cdot {\\mathbf{N}}(M)\\cdot 2^{\\left| E \\right|}\\right) = O\\left(\\log(n)\\cdot 2^{2n}\\right)\\] where $n= {\\mathbf{N}}(M)$ is the bit-length of the input. \n\\end{algorithm}\n\n\\begin{proof}[Proof of correctness]\\PRFR{Mar 7th}\nIt suffices to show that the algorithm correctly computes the values $\\alpha_M(X)$, and that the algorithm returns $1$, if $\\alpha_M(X)\\geq 0$ holds for all $X\\subseteq E$, and $0$ otherwise (Corollary~\\ref{cor:MasonAlpha}).\nLet ${\\mathcal{X}} = \\dSET{X_1,X_2,\\ldots, X_{2^{\\left| E \\right|}}}$ be the family of all subsets of $E$ in the order of occurrence with respect to the\n``{\\ttfamily for $X$}''-instruction, and let $i\\in \\SET{1,2,\\ldots,2^{\\left| E \\right|}}$ such that $X = X_i$.\n The invariant at the ``$ a := k - \\mathrm{rk}_M(X)$''-instruction is, that for all $j\\in\\SET{1,2,\\ldots,i-1}$ the value of $\\alpha_M(X_j)$\n is correctly assigned and non-negative,\n and that we have that ${\\mathcal{F}}(X_j) = 1$ if and only if $X_j$ is a flat. Furthermore, $a = \\left| X_i \\right| - \\mathrm{rk}_M(X_i)$.\n Now let ${\\mathcal{Y}} = \\dSET{Y_1,Y_2,\\ldots,Y_K}$ be the proper subsets of $X$ in their order of occurrence with respect to the\n``{\\ttfamily for $Y$}''-instruction, and let $k\\in\\SET{1,2,\\ldots,K}$ be the index such that $Y = Y_k$.\nThe invariant at the ``{\\ttfamily next $Y$}''-instruction is\n $$a = \\left| X_i \\right| - \\mathrm{rk}_M(X_i) - \\sum_{F\\in{\\mathcal{F}}(M)\\cap \\SET{Y_1,Y_2,\\ldots,Y_k},\\, F\\subsetneq X} \\alpha_M(F).$$\nThus the invariant at the ``{\\ttfamily end for $Y$}''-instruction is $a = \\alpha_M(X_i) \\geq 0$,\nand the invariant at the ``{\\ttfamily return $1$}''-instruction is that $M$ is a strict gammoid.\nThe invariant at the ``{\\ttfamily return $0$ and stop}''-instruction is that\n$$ a = \\left| X_i \\right| - \\mathrm{rk}_M(X_i) - \\sum_{F\\in{\\mathcal{F}}(M)\\cap \\SET{Y_1,Y_2,\\ldots,Y_k},\\, F\\subsetneq X} \\alpha_M(F) < 0,$$\nand since $\\alpha_M(Y)\\geq 0$ for all $Y\\subsetneq X$, we may conclude that $\\alpha_M(X_i) \\leq a < 0$, which implies that\n$M$ is not a strict gammoid.\nTherefore the output of the algorithm is $1$ if $M$ is a strict gammoid, and $0$ if $M$ is not a strict gammoid.\n\\end{proof}\n\n\\begin{corollary}\\label{cor:decideTransversalOrStrictGammoid}\\PRFR{Mar 7th}\n\tGiven a matroid $M$ via its encoding $\\mathrm{b}(M)$, we can decide whether $M$ is a transversal matroid and whether $M$ is a strict gammoid in\n\t$O\\left(\\log({\\mathbf{N}}(M))2^{2{\\mathbf{N}}(M)}\\right)$ time.\n\\end{corollary}\n\\begin{proof}\\PRFR{Mar 7th}\n\tWe may use Algorithm~\\ref{alg:strictGammoidTest} on $M$ to test whether $M$ is a strict gammoid,\n\tand on $M^\\ast$ in order to test whether $M$ is a transversal matroid. The encoding $\\mathrm{b}(M^\\ast)$ can be obtained in $O({\\mathbf{N}}(M))$ time:\n\tthe location of the first zero has to be moved from $\\mathrm{rk}_M(E)+1$ to $\\left| E \\right| - \\mathrm{rk}_M(E) + 1$, and the encoding of the bases in $\\mathrm{b}(M)$ has to be brought in reverse order to obtain an encoding of $M^\\ast$. Then we can test whether $M^\\ast$ is a strict gammoid to obtain the result (Lemma~\\ref{lem:dualOfStrictGammoidIsTransversal}).\n\\end{proof}\n\n\n\\subsection{Special Cases}\n\n\\PRFR{Mar 7th}\nIn this section, we give a quick overview over some special classes of matroids\nand gammoids, where there is an easy way to answer the question\nwhether a matroid, that exhibits the additional properties,\nis a gammoid with other special properties --- or whether it is not.\n\n\\begin{proposition}[\\cite{IP73}, Proposition~4.8 and Corollary~4.9]\\label{prop:cornerCases}\\PRFR{Feb 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid. Then\n\t\\begin{enumerate}\\renewcommand{\\theenumi}{(\\roman{enumi})}\\renewcommand{\\labelenumi}{\\theenumi} \n\t\t\\item If $\\mathrm{rk}_M(E) \\leq 2$, then $M$ is a strict gammoid.\n\t\t\\item If $\\mathrm{rk}_M(E) = 3$, then $M$ is a gammoid if and only if $M$ is a strict gammoid.\n\t\t\\item If $\\mathrm{rk}_M(E) = \\left| E \\right| - 3$, then $M$ is a gammoid if and only if $M$ is a transversal matroid.\n\t\t\\item If $\\mathrm{rk}_M(E) \\geq \\left| E \\right| - 2$, then $M$ is a transversal matroid.\n\t\\end{enumerate}\n\\end{proposition}\n\n\\PRFR{Mar 7th}\n\\noindent We omit the proof here as it does not provide any further guidance for the unconstrained problem of deciding whether\na given matroid is a gammoid or not. The reader interested in a proof of this proposition should read A.W.~Ingleton and M.J.~Piff's paper \\cite{IP73}.\nCertainly, we should keep in mind the consequence of this proposition: We can expect the most general flavor of gammoids to unfold only\n with matroids $M=(E,{\\mathcal{I}})$ where $4 \\leq \\mathrm{rk}_M(E) \\leq \\left| E \\right| - 4$. For matroids with\n $\\mathrm{rk}_M(E) \\in \\SET{0,1,2,\\left| E \\right|-2,\\left| E \\right|-1,\\left| E \\right|}$, the answer is always that $M$ is a gammoid. For\n $\\mathrm{rk}_M(E)\\in\\SET{3,\\left| E \\right|-3}$, we may use Mason's $\\alpha$-criterion with respect to $M$, or $M^\\ast$, respectively,\n in order to decide whether $M$ is a gammoid (Corollary~\\ref{cor:MasonAlpha}) in $O\\left(\\log(n)2^{2n}\\right)$ time (Corollary~\\ref{cor:decideTransversalOrStrictGammoid}). \n\n\n\\bigskip \n\n\\PRFR{Mar 7th}\n\\noindent\n Instead of limiting the class of allowed input instances, we also might consider the related problem of determining\nwhether a given matroid $M$ belongs to some subclass ${\\mathcal{G}}_P$ of the class of gammoids, where ${\\mathcal{G}}_P$ consists of all gammoids\nthat have the additional property $P$. It is folklore, that the problem of deciding whether a given matroid $M$ belongs to \na minor-closed class of matroids ${\\mathcal{M}}_P$, which is characterized by finitely many excluded\nminors, has a solution algorithm that runs in polynomial time. First, we present the following theorem, that gives us a hint where\nsuch an excluded minor must appear in any $M\\notin{\\mathcal{M}}_P$.\n\n\\needspace{4\\baselineskip}\n\n\\begin{theorem}[Scum Theorem, \\cite{Ox11} p.113]\\label{thm:scum}\\PRFR{Mar 7th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid and let $N=(E',{\\mathcal{I}}')$ be a minor of $M$. There is a subset $Z\\subseteq E\\backslash E'$\n\tsuch that $M|' \\left( E\\backslash Z \\right)$ has the same rank as $N$, and such that \n\t\\[ N = \\left( M|' \\left( E\\backslash Z \\right) \\right)| E' .\\]\n\tIf $N$ has no loop, then we may choose $Z\\in{\\mathcal{F}}(M)$.\n\\end{theorem}\n\n\\noindent\nFor the proof, please refer to J.G.~Oxley's book \\cite{Ox11}. It is easy to see that given such a $Z\\subseteq E\\backslash E'$,\nevery base $Z' \\in {\\mathcal{B}}_M(Z)$ also has the property that $N = \\left( M|' \\left( E\\backslash Z' \\right) \\right)| E'$\n(Lemma~\\ref{lem:contractionBchoice}).\n\n\n\n\\needspace{3\\baselineskip}\n\\begin{algorithm}\\label{alg:isMinor}\\PRFR{Mar 7th}\\index{algorithm!minor} \\textbf{Test for Minor}\\\\\n\n\\noindent\n\\begin{tabularx}{\\textwidth}{rl}\n\t\\textbf{Input}& {\\em(1)} A matroid $M=(E,{\\mathcal{I}})$ given by $\\mathrm{b}(M)$.\\\\\n\t& {\\em(2)} A matroid $N=(E',{\\mathcal{I}}')$ given by $\\mathrm{b}(N)$.\\\\\n\t\\textbf{Output}& $1$, if $N$ is isomorphic to a minor of $M$,\\\\\n\t& $0$, otherwise.\\\\\n\t& \\\\\n\t& \\ttfamily if $\\mathrm{rk}_N(E',{\\mathcal{I}}') > \\mathrm{rk}_M(E,{\\mathcal{I}})$ then return $0$ and stop\\\\\n\t& \\ttfamily for $Z\\in \\binom{E}{\\mathrm{rk}_M(E)-\\mathrm{rk}_N(E')}$ do\\\\*\n\t& \\ttfamily $\\quad$ if $Z\\notin {\\mathcal{I}}$ then next $Z$\\\\\n\t& \\ttfamily $\\quad$ for every $\\phi\\colon E' \\longrightarrow E\\backslash Z \\text{\\rmfamily~injective map}$ do\\\\*\n\t& \\ttfamily $\\quad$ $\\quad$ for $X\\in \\binom{E'}{\\mathrm{rk}_N(E')}$ do\\\\*\n\t& \\ttfamily $\\quad$ $\\quad$ $\\quad$ if not $X\\in {\\mathcal{B}}(N) \\Leftrightarrow \\phi[X]\\cup Z \\in {\\mathcal{B}}(M)$ then next $\\phi$\\\\\n\t& \\ttfamily $\\quad$ $\\quad$ $\\quad$ next $X$\\\\*\n\t& \\ttfamily $\\quad$ $\\quad$ end for $X$ \\\\\n\t& \\ttfamily $\\quad$ $\\quad$ return $1$ and stop\\\\*\n\t& \\ttfamily $\\quad$ end for $\\phi$ \\\\\n\t& \\ttfamily $\\quad$ next $Z$ \\\\*\n\t& \\ttfamily end for $Z$\\\\\n\t& \\ttfamily return $0$.\n\\end{tabularx}\n\n\\PRFR{Mar 7th}\n\\noindent\nLet $n = {\\mathbf{N}}(M) + {\\mathbf{N}}(N)$, $n_M = {\\mathbf{N}}(M)$, and $n_N = {\\mathbf{N}}(N)$ be the respective encoding lengths.\nThe ``{\\ttfamily for $Z$}''-instruction loops through at most $\\binom{\\left| E \\right|}{\\mathrm{rk}_M(E) - \\mathrm{rk}_N(E')}$ iterations.\nFor $k,m\\in \\mathbb{N}$ with $k \\leq m$, we may estimate\n\\( \\binom{m}{k-1} \\leq k \\cdot \\binom{m}{k}\\), because we obtain a $(k-1)$-elementary subset of an $m$-elementary set $X$ by first\nchoosing an $k$-elementary subset of $X$ and then choosing one element to drop. This way we obtain all $(k-1)$-elementary subsets\nprovided that there is some $k$-elementary subset of $X$. Consequently, we may estimate the number of $Z$-iterations by\n\\begin{align*}\n\t\\binom{\\left| E \\right|}{\\mathrm{rk}_M(E) - \\mathrm{rk}_N(E')} & \\leq \\left| E \\right|^{\\mathrm{rk}_N(E')} \\cdot \\binom{\\left| E \\right|}{\\mathrm{rk}_M(E)}\n\t= O\\left( {\\left( n_M \\right)}^{n_N+1} \\right).\n\\end{align*}\nThe test whether $Z\\in{\\mathcal{I}}$ can be done in $O\\left( \\left( n_M \\right)^2 \\right)$ (Algorithm~\\ref{alg:indep}).\nThe generation of the injective maps $\\phi$ for a single instance of $Z$ as lookup \ntables has a combined run-time in $$O\\left(\\left| E \\right|^{\\left| E' \\right|}\\cdot \\log\\left( \\left| E \\right| \\right)\\right)\n=O\\left( \\left( n_M \\right)^{n_N}\\cdot \\log\\left( n_M \\right) \\right).$$\nThe \\mbox{``{\\ttfamily for $\\phi$}''}-instruction loops through at \nmost $\\left| E \\right|^{\\left| E' \\right|}= O\\left( \\left( n_M \\right)^{n_N} \\right)$ iterations,\nand the \\mbox{``{\\ttfamily for $X$}''}-instruction loops through at most $O(n_N)$ iterations.\nCalculating $\\phi[X]$ can be done with $\\left| X \\right| = \\mathrm{rk}_N(E') = O\\left( n_N \\right)$ table lookups and corresponding bit-set\noperations,\ncalculating $\\phi[X]\\cup Z$ can be achieved with an $\\left| E \\right|$-bit bitwise-or operation in $O\\left( n_M \\right)$-time.\nChecking whether $X\\in {\\mathcal{B}}(N) \\Leftrightarrow \\phi[X]\\cup Z\\in {\\mathcal{B}}(M)$ can be done in $O(1)$-time by bit comparison.\nThis yields a combined run-time of the algorithm in \n\\[ O\\left( \\left( n_M \\right)^{2n_N + 2}\\cdot \\left( n_N \\right)^2 \\right) = O\\left( n^{2n+4} \\right). \\qedhere \\]\n\\end{algorithm}\n\n\\noindent Thus deciding whether $M$ has a minor isomorphic to $N$ can be done in polynomial time with respect to ${\\mathbf{N}}(M)$ for a\nfixed matroid $N$.\n\n\\PRFR{Mar 7th}\n\\begin{proof}[Proof of correctness]\nAssume that $M=(E,{\\mathcal{I}})$ has a minor $L=(D,{\\mathcal{J}})$ that is isomorphic to $N=(E',{\\mathcal{I}}')$, \nthen there is a set $Z_L\\subseteq E\\backslash D$ such \nthat $L = \\left( M|' \\left(E\\backslash Z_L\\right) \\right)| D$ as guaranteed by the \nScum Theorem~\\ref{thm:scum}. Thus for every base $B_L$ of $Z_L$ in $M$,\nwe have the property, that a set $X\\subseteq D$ is a base of $L$ if and only if $B_L\\cup X$ \nis a base of $M$ (Lemma~\\ref{lem:contractionBchoice}). Furthermore, \nthe minor $L$ is isomorphic to $N$, if and only if there is a matroid isomorphism\n$\\phi'\\colon E' \\longrightarrow D$ between $L$ and $N$, i.e. a bijective map $\\phi'$ with the property that\n $\\phi'[X]\\in {\\mathcal{B}}(L) \\Leftrightarrow X\\in {\\mathcal{B}}(N)$ holds. Assume that $M$ has a minor $L$ isomorphic to $N$. Let further $\\phi'$\n be the corresponding matroid isomorphism, and let $Z_L\\subseteq E\\backslash D$ and $B_L\\subseteq Z_L$ \n be derived from the Scum Theorem~\\ref{thm:scum} \n as above. Since $B_L\\subseteq Z_L \\subseteq E\\backslash D$, we have $B_L\\cap D = \\emptyset$.\n By extension of the codomain of $\\phi'$ we obtain an injective map $\\hat\\phi\\colon E'\\longrightarrow E\\backslash B_L$,\n where $\\hat\\phi(e') = \\phi'(e')$ for all $e'\\in E'$. Either the algorithm returns $1$ early, \n or at some point, the ``{\\ttfamily for $Z$}''-instruction starts an iteration where\n $Z = B_L$ since $\\left| B_L \\right| = \\mathrm{rk}_M(Z_L) = \\mathrm{rk}_M(E) - \\mathrm{rk}_L(D) = \\mathrm{rk}_M(E) - \\mathrm{rk}_N(E')$. In this iteration, we have\n $Z=B_L\\in {\\mathcal{I}}$, therefore we enter the ``{\\ttfamily for $\\phi$}''-loop. Again, either the algorithm returns $1$ early, or\n we reach the iteration where $\\phi = \\hat\\phi$. In this iteration, we have the equivalence $X\\in {\\mathcal{B}}(N) \\Leftrightarrow \\phi[X]\\cup Z =\n \\phi'[X] \\cup B_L \\in {\\mathcal{B}}(M)$, therefore we reach the ``{\\ttfamily end for $X$}''-instruction. \n In the next instruction, we return the correct value $1$.\n\n \\noindent Now assume that $M$ has no minor isomorphic to $N$. If the algorithm reaches the ``{\\ttfamily return $0$}''-instruction\n the result is correct. We give an indirect argument for this to happen, and assume that the algorithm reaches the \n ``{\\ttfamily return $1$}''-instruction.\n But then the ``{\\ttfamily for $X$}''-loop must have finished without reaching the ``{\\ttfamily next $\\phi$}''-instruction.\n This, together with the property {\\em (B2)} that all bases of a matroid have the same cardinality, implies that\n $X\\in {\\mathcal{B}}(N) \\Leftrightarrow \\phi[X]\\cup Z\\in {\\mathcal{B}}(M)$ holds for all $X\\subseteq E'$.\n Thus $\\left( M|' \\left( E\\backslash Z \\right) \\right) | \\phi[E']$\n is a minor of $M$ isomorphic to $N$, contradicting our assumption that $M$ has no minor isomorphic to $N$.\n Therefore we may conclude that the algorithm returns $0$ if $M$ has no minor isomorphic to $N$. \n\\end{proof}\n\n\n\\begin{theorem}\\PRFR{Mar 7th}\n\tLet ${\\mathcal{G}}$ be a minor-closed class of matroids that is characterized by the excluded minors $N_1,N_2,\\ldots,N_k$,\n\tand let $K = \\max\\SET{{\\mathbf{N}}(N_1),{\\mathbf{N}}(N_2),\\ldots,{\\mathbf{N}}(N_k)}$ be the maximal encoding length of the excluded minors.\n\tIf $M=(E,{\\mathcal{I}})$ is an arbitrary matroid and $n = {\\mathbf{N}}(M)$ is its encoding length,\n\t then we may decide whether $M\\in {\\mathcal{G}}$ in $ O\\left( n^{K+2} \\right) $-time.\n\\end{theorem}\n\\begin{proof}\\PRFR{Mar 7th}\n\tFor each $i\\in\\SET{1,2,\\ldots,k}$ we may use Algorithm~\\ref{alg:isMinor} in order to test whether $M$ has a minor isomorphic to $N_i$\n\tin $O\\left( n^{2{\\mathbf{N}}(N_i)+2} \\right)$-time. On the other hand, $M\\in {\\mathcal{G}}$ if and only if $M$ has no minor isomorphic to one of\n\tthe matroids $N_1,N_2,\\ldots,N_k$. Thus if Algorithm~\\ref{alg:isMinor} returns $1$ for any $N_i$, then $M\\notin {\\mathcal{G}}$,\n\tand if Algorithm~\\ref{alg:isMinor} returns $0$ for all $N_i$, then $M\\in {\\mathcal{G}}$. Therefore we have to run at most $k$ tests \n\tin $O\\left( n^{2K+2} \\right)$-time in order to decide whether $M\\in{\\mathcal{G}}$.\n\\end{proof}\n\n\\noindent\nA consequence of this theorem is the following: Let $k\\in \\mathbb{N}$, then we may \ndecide in polynomial time whether a matroid $M$ \nis {\\em (a)} a gammoid with $\\mathrm{C}_V(M) \\leq k$ and {\\em (b)} a gammoid with $\\mathrm{C}_A(M) \\leq k$ (Remark~\\ref{rem:vKeasy} and Theorem~\\ref{thm:arcCquiteEasy}).\n\\subsection{The General Recognition Problem}\n\n\\PRFR{Mar 7th}\n\\noindent\nFor the rest of this chapter, \\label{sec:generalCase} we let ${\\mathcal{M}}$ be the class of all matroids. Now, let us investigate the problem $\\mathrm{Rec}\\Gamma_{\\mathcal{M}}$.\nIn order to present the most obvious algorithm that computes $\\Gamma_{\\mathcal{M}}(M)$,\n we need a way to verify whether a given vector $b\\in \\SET{0,1}^{\\binom{n}{r}}$\ncodes the bases of a rank-$r$ matroid on an $n$-elementary ground set.\n\n\\needspace{3\\baselineskip}\n\\begin{algorithm}\\PRFR{Mar 7th}\\label{alg:matroidTest}\\index{algorithm!test base axioms} \\textbf{Test Base Axioms}\\\\\n\n\\noindent\n\\begin{tabularx}{\\textwidth}{rl}\n\t\\textbf{Input}& {\\em(1)} $r\\in \\mathbb{N}$, given as unary encoded bit-stream.\\\\\n\t& {\\em(2)} $(e-r)\\in \\mathbb{N}$, given as unary encoded bit-stream.\\\\\n\t\t&{\\em(3)} $B\\in \\SET{0,1}^{\\binom{\\SET{1,2,\\ldots,e}}{r}}$ family of $r$-elementary sets, as a vector of $2^{\\binom{e}{r}}$.\\\\\n\t\\textbf{Output}& $1$, if $B$ is the characteristic vector of a family of bases of a matroid\\\\\n\t& ~~~~~ with rank $r$,\\\\\n\t& $0$, otherwise.\\\\\n\t& \\\\\n\t& \\ttfamily $g := 0$ \\\\\n\t& \\ttfamily for $X \\in \\binom{\\SET{1,2,\\ldots,e}}{r}$ do \\\\*\n\t& \\ttfamily $\\quad $ if $B(X) = 0$ then next $X$\\\\*\n\t& \\ttfamily $\\quad $ $g := 1$\\\\\n\t& \\ttfamily $\\quad $ for $Y \\in \\binom{\\SET{1,2,\\ldots,e}}{r}$ do \\\\*\n\t& \\ttfamily $\\quad $ $\\quad $ if $X=Y$ or $B(Y) = 0$ then next $Y$\\\\*\n\t& \\ttfamily $\\quad $ $\\quad $ for $x\\in X\\backslash Y$ do\\\\\n\t& \\ttfamily $\\quad $ $\\quad $ $\\quad $ for $y \\in Y\\backslash X$ do \\\\*\n\t& \\ttfamily $\\quad $ $\\quad $ $\\quad $ $\\quad $ if $B\\left( \\left(X\\BSET{x} \\right)\\cup\\SET{y} \\right) = 1$ then next $x$\\\\*\n\t& \\ttfamily $\\quad $ $\\quad $ $\\quad $ $\\quad $ next $y$\\\\*\n\t& \\ttfamily $\\quad $ $\\quad $ $\\quad $ end for $y$\\\\*\n\t& \\ttfamily $\\quad $ $\\quad $ $\\quad $ return $0$ and stop\\\\\n\t& \\ttfamily $\\quad $ $\\quad $ end for $x$\\\\*\n\t& \\ttfamily $\\quad $ $\\quad $ next $Y$\\\\\n\t& \\ttfamily $\\quad $ end for $Y$\\\\*\n\t& \\ttfamily $\\quad $ next $X$\\\\\n\t& \\ttfamily end for $X$\\\\*\n\t& \\ttfamily return $g$.\n\\end{tabularx}\n\n\\PRFR{Mar 7th}\n\\noindent\nObserve that the input resembles the format of an encoding of a rank-$r$ matroid defined on an $e$-elementary ground set,\nthe total bit-length of the input is $n = 2 + e + \\binom{e}{r}$.\nA rough estimate of the run-time is the following: the ``{\\ttfamily for $X$}''-instruction iterates over $\\binom{e}{r}$ sets,\nthe ``{\\ttfamily for $Y$}''-instruction iterates over $\\binom{e}{r}$ sets, too, the ``{\\ttfamily for $x$}''-instruction iterates over $\\leq r$ elements of $X$,\nthe ``{\\ttfamily for $y$}''-instruction iterates over $\\leq r$ elements of $Y$. In total, we have to do less than\n\\[ \\binom{e}{r}\\cdot \\binom{e}{r} \\cdot r^2 + \\binom{e}{r}\\cdot \\binom{e}{r} + \\binom{e}{r} \\]\nbit-comparisons involving the vector $B$. Thus the run-time is in \n $O\\left( r^2\\cdot \\binom{e}{r}^2 \\right) =\n O\\left( n^3 \\right)$, since $r^2 = O\\left( \\binom{e}{r} \\right) = O(n)$.\n\\end{algorithm}\n\\begin{proof}[Proof of correctness]\\PRFR{Mar 7th}\nClearly, the input format guarantees that {\\em (B2)} holds for all inputs.\nFor every matroid $M$ of rank $r$ on the ground set $\\SET{1,2,\\ldots,e}$,\nat least one set $X\\in \\binom{\\SET{1,2,\\ldots,e}}{r}$ must be a base of $M$, \nand the variable $g$ obviously keeps track of the existence of this set, and consequently, whether {\\em (B1)} holds.\nIn other words, upon reaching the ``{\\ttfamily return $g$}''-instruction, $g = 0$ if and only if $B \\equiv 0$,\ni.e. $B$ is the zero vector.\n Let $X,Y\\in\\binom{\\SET{1,2,\\ldots,e}}{r}$. The ``{\\ttfamily for $x$}''-instruction is reached for $X$ and $Y$\nif and only if $B(X)=B(Y)=1$, i.e. $X$ and $Y$ are supposed to be bases of the input matroid candidate.\nThe loops ``{\\ttfamily for $x$}'' and ``{\\ttfamily for $y$}'' test whether $\\left( X\\BSET{x}\\right)\\cup{y}$ is a base. If for a given $x\\in X\\backslash Y$\nan exchange partner $y\\in Y\\backslash X$ is found, the ``{\\ttfamily next $x$}''-instruction is reached. Otherwise, if no $y\\in Y\\backslash X$\nhas this property for a given $x\\in X\\backslash Y$, the ``{\\ttfamily end for $y$}''-instruction is reached. In this case, $B$ violates\nthe base exchange axiom {\\em (B3)} and therefore the input candidate does not correspond to a matroid. In this case,\nthe output of the algorithm is $0$. When the algorithm reaches the ``{\\ttfamily end for $X$}''-instruction,\nthen it is established that the axiom {\\em (B3)} holds for the input candidate. In this case, the input candidate is a matroid\nif and only if $B\\not\\equiv 0$, which is correctly reflected by the value of $g$. Thus the output of the algorithm is $1$ if the input\nbase vector candidate is a base vector of a matroid of rank $r$ with $e$ elements, and $0$ otherwise.\n\\end{proof}\n\n\\PRFR{Mar 7th}\n\\noindent Given any matroid $M=(E,{\\mathcal{I}})$, we can combine Remark~\\ref{rem:upperBoundForV} and the Algorithms~\\ref{alg:matroidTest} \nand \\ref{alg:strictGammoidTest}\nwith the brute-force exhaustive search algorithm\nin order to compute $\\Gamma_{\\mathcal{M}}(M)$: We generate all candidate families of subsets of $\\binom{E'}{\\mathrm{rk}_M(E)}$\nwith respect to a set $E'$ of\ncardinality $\\mathrm{rk}_M(E)^2\\cdot \\left| E \\right| + \\mathrm{rk}_M(E) + \\left| E \\right|$ with $E\\subseteq E'$, that\ncoincide with ${\\mathcal{B}}(M)$ when intersected with $2^E$. Then we use Algorithm~\\ref{alg:matroidTest} in order to determine\nwhether the generated family corresponds to an actual matroid $M'$ on $E'$. If this is the case, we test whether the\ngenerated matroid $M'$ is a strict gammoid. If so, then $M$ is a gammoid, and $M'$ certifies this.\nOtherwise we continue until we exhausted all possibilities to generate candidate families. If we have not found any strict gammoid among\nthe candidates, then $M$ is not a gammoid. \n\n\\needspace{3\\baselineskip}\n\\begin{algorithm}\\PRFR{Mar 7th}\\label{alg:gammoidTest}\\index{algorithm!brute-force $\\Gamma_{{\\mathcal{M}}}(M)$}\n\\textbf{Compute $\\Gamma_{{\\mathcal{M}}}(M)$ (Brute-Force Search)}\\\\\n\n\\noindent\n\\begin{tabularx}{\\textwidth}{rl}\n\t\\textbf{Input}& {\\em(1)} $M=(E,{\\mathcal{I}})$ matroid, given by its encoding $\\mathrm{b}(M)$.\\\\\n\t\\textbf{Output}& $1$ if $M$ is a gammoid,\\\\\n\t& $0$ otherwise.\\\\\n\t& \\\\\n\t& \\ttfamily let $E' := \\SET{1,2,\\ldots, \\mathrm{rk}_M(E)^2 \\cdot \\left| E \\right| + \\mathrm{rk}_M(E) + \\left| E \\right|}$\\\\\n\t& \\ttfamily let ${\\mathcal{E}} := \\binom{E'}{\\mathrm{rk}_M(E)}$\\\\\n\t& \\ttfamily let ${\\mathcal{Y}} := \\SET{Y\\in 2^{{\\mathcal{E}}} ~\\middle|~ Y\\cap 2^{\\SET{1,2,\\ldots,\\left| E \\right|}} = {\\mathcal{B}}(M)}$\\\\\n\t& \\ttfamily for $B \\in {\\mathcal{Y}}$ do \\\\\n\t& \\ttfamily \\quad if {\\rmfamily $B$ satisfies the base axioms} then do \\\\\n\t& \\ttfamily \\quad \\quad let \\rmfamily\n\t\t $N := (E',\\SET{I\\subseteq E'~\\middle|~ \\exists X\\in {\\mathcal{E}}\\colon\\,I\\subseteq X{\\mathop{\\text{~and~}}} B(X) = 1})$\\\\\n\t& \\ttfamily \\quad \\quad if {\\rmfamily $N$ is a strict gammoid} then return $1$ and stop\\\\\n\t& \\ttfamily \\quad end if {\\rmfamily $B$ satisfies the base axioms}\\\\\n\t& \\ttfamily end for $B$ \\\\\n\t& \\ttfamily return $0$. \n\\end{tabularx}\n\n\\PRFR{Mar 7th}\n\\noindent\nIn the worst case $M$ is not a gammoid and we have to iterate over\n $$\\displaystyle 2^{\\left| {\\mathcal{E}} \\right| - \\binom{\\left| E \\right|}{\\mathrm{rk}_M(E)}} = O\\left( 2^{\\left| E' \\right|^{\\mathrm{rk}_M(E)}} \\right)$$\npossible values for $B$ in the ``{\\ttfamily for $B$}''-loop.\nThe test whether $B$ corresponds to a matroid can be carried out by Algorithm~\\ref{alg:matroidTest} and is possible within\n $$O\\left( \\mathrm{rk}_M(E)^2\\cdot \\binom{|E'|}{\\mathrm{rk}_M(E)} \\right)\\text{-time}.$$\nThe test whether $N$ is a strict gammoid may be done with Algorithm~\\ref{alg:strictGammoidTest} and therefore can be done\nin $$O\\left(\\log\\left( \\left| E' \\right| \\right)\\cdot 2^{2 \\left| E' \\right|} + \n\\left| E' \\right|^2\\cdot {\\mathbf{N}}(N)\\cdot 2^{\\left| E' \\right|}\\right) \\text{-time}.$$\nLet $n={\\mathbf{N}}(M)$ be the bit-length of $\\mathrm{b}(M)$, clearly $\\left| E \\right| \\leq n$ and $\\mathrm{rk}_M(E) \\leq n$.\nWe have\n$\\left| E' \\right| \\leq n^3 + 2n$. Thus one test of the base axioms can be done in\n$$ O\\left( n^2\\cdot \\binom{n^3 + 2n}{n} \\right)=O\\left( n^{2.001n} \\right) \\text{-time.}$$\nSince the bit-length ${\\mathbf{N}}(N) \\leq 2^{\\left| E' \\right|} \\leq 2^{n^3 + 2n}$, we obtain that \neach strict gammoid test can be\ncarried out in \n$$ O\\left( n^6\\cdot 2^{2n^3 + 4n} \\right) \\text{-time}.$$\nR.~Pendavingh and J.~van~der~Pol give the following upper bound for the number of matroids $m(k,r)$ on $k$-elementary ground sets\nwith rank $r$ in \\cite{PP17}\n\\[ \\log(m(k,r)) \\leq \\frac{1}{k-r+1} \\binom{k}{r} \\cdot \\log(c\\cdot(k-r+1))\\]\nunder the mild condition that $r \\geq 3$ and $k \\geq r + 12$, and where $c$ denotes a constant factor that does not depend on $k$ or $r$.\nWe can use this bound in order to determine how often we have to decide whether $N$ is a strict gammoid or not,\nlet $i$ denote the number of strict gammoid tests, then \n\\begin{align*}\n \\log(i) & \\leq \\log(m(n^3 + 2n, \\mathrm{rk}_M(E)))\n \\\\i & = O\\left( 2^ {n^{\\left( 3\\mathrm{rk}_M(E)-3 \\right)}\\cdot\\log\\left( c\\cdot\\left( n^3+2n \\right) \\right)} \\right)\\\\\n\n & = O\\left( 2^{\\left( n^{3\\mathrm{rk}_M(E)} \\right)} \\right)\n = O\\left( 2^{n^{3n}} \\right)\n .\n \\end{align*}\n A naive upper bound for the strict gammoid tests can derived from the number of $B$-iterations, it is\n \\[ O\\left( 2^{{(n^3+ 2n)}^{3n}} \\right) = O\\left( \\prod_{i=0}^{3n} 2^{{\\binom{3n}{i}\\cdot(n^{2i + 3n})}} \\right), \\]\n and it obviously is a looser upper bound than the one derived from \\cite{PP17}, since it has a factor $2^{\\left( n^{9n} \\right)}$.\nThus we have to account\n\\[ O\\left( 2^{{(n^3+ 2n)}^{3n}}\\cdot n^{2.001n} \\right) \\]\nfor the base exchange axiom tests and\n\\[ O\\left( n^6\\cdot 2^{n^{3n}+2n^3 + 4n} \\right) \\]\nfor the strict gammoid tests. Clearly, \n$ O\\left( n^6\\cdot 2^{n^{3n}+2n^3 + 4n} \\right) = O\\left( 2^{{(n^3+ 2n)}^{3n}} \\right)$,\nso we may estimate the total-run time of the algorithm to be in\n\\[ O\\left( 2^{{(n^3+ 2n)}^{3n}}\\cdot n^{2.001n} \\right) = O\\left( 2^{\\left( n^{9n+1} \\right)} \\right) . \\qedhere \\]\n\\end{algorithm}\n\n\\begin{proof}[Proof of correctness]\\PRFR{Mar 7th}\nIf $M$ is a gammoid, then there is a representation $(D,T,E)$ with $D=(V,A)$ \nsuch that $\\left| V \\right| \\leq \\mathrm{rk}_M(E)^2 \\cdot \\left| E \\right| + \\mathrm{rk}_M(E) + \\left| E \\right| = \\left| E' \\right|$ \n(Remark~\\ref{rem:upperBoundForV}). But then $N = \\Gamma(D,T,V)$ is a strict gammoid with the property $N| E = \\Gamma(D,T,E) = M$,\nthus $\\SET{X\\subseteq E~\\middle|~X\\in {\\mathcal{B}}(N)} = {\\mathcal{B}}(M)$. The ``{\\ttfamily for $B$}''-instruction iterates through all \npossible and impossible $B = {\\mathcal{B}}(N)$ with that restriction-property, then tests whether $B$ is indeed a matroid base family, and then tests whether the\ncorresponding matroid is a strict gammoid. If so, the algorithm gives the truthful output $1$. If no such $B$ is found, the algorithm\nreturns $0$, and by the above consideration we may conclude that in this case $M$ is not a gammoid.\n\\end{proof}\n\n\n\\noindent\nNo one would expect that the brute-force method would be of any practical use for determining whether a given matroid is a gammoid,\nand it apparently is not.\nOne obvious problem with Algorithm~\\ref{alg:gammoidTest} is that it does not make use of any of the structural results for\nmatroid extensions, instead it guesses matroid extensions, and this takes so much time that the actual testing for the\nstrict gammoid property in $O\\left( 2^{n^{3n+1}} \\right)$ does not have a significant impact on the estimation. The other obvious\nproblem is that we are not using any information from the $\\alpha_M$-vector in order to guide our search for a strict gammoid\nextensions of $M$. Furthermore, it seems to be excessive to compute all $\\alpha$-vectors of strict gammoid extension candidates\nfrom scratch.\\footnote\n There is a second brute-force search method for finding the strict gammoid extension of a gammoid $M=(E,{\\mathcal{I}})$,\nwhich guesses the arcs of a digraph, then calculates the ranks for all $\\mathrm{rk}_M(E)$-elementary\nsubsets of $E$ of the gammoid represented by the candidate digraph:\n if all of these values are correct, then $M$ is a gammoid represented by the candidate digraph, if otherwise we run out of candidates,\n then $M$ is not a gammoid. This method is clearly better than Algorithm~\\ref{alg:gammoidTest} as it does eliminate the\n check whether a candidate is indeed a matroid. With the currently known bounds for arcs and vertices there are still\n $\\left(\\sum_{k=0}^{r}\\binom{n^3+2n}{k}\\right)^{n^3+n}$ candidate digraphs on $n^3+2n$ vertices (Remark~\\ref{rem:upperBoundForV}) with at most $r=O(n)$ leaving arcs per non-target vertex (Theorem~\\ref{thm:IPEssentialStars}),\n\n\n\n\n\n\n\n\n\n\n\n so this brute-force method is still not a practical solution --- and the computation of the bases of a given strict gammoid\n involves a more complicated algorithm, therefore we will not give more details for this method. Of course, \n the refined brute-search method still wastes a lot of\ntime because usually a considerable amount of digraphs represent the same gammoid.}\nWe give a straight-forward back-tracking algorithm that also computes $\\Gamma_{{\\mathcal{M}}}(M)$. \n\n\\needspace{3\\baselineskip}\n\\begin{algorithm}\\PRFR{Mar 7th}\\label{alg:gammoidBackTrack}\\index{algorithm!backtracking $\\Gamma_{{\\mathcal{M}}}(M)$}\\textbf{Compute $\\Gamma_{{\\mathcal{M}}}(M)$ (Digraph Backtracking)}\\\\\n\n\\noindent\n\\begin{tabularx}{\\textwidth}{rl}\n\t\\textbf{Input}& {\\em(1)} $M=(E,{\\mathcal{I}})$ matroid, given by its encoding $\\mathrm{b}(M)$.\\\\\n\t\\textbf{Output}& $1$ if $M$ is a gammoid,\\\\\n\t& $0$ otherwise.\\\\\n\t& \\\\\n\t& \\ttfamily let $V$ be \\rmfamily a set with $E\\subseteq V$ and $\\left| V \\right| = \\mathrm{rk}_M(E)^2\\cdot \\left| E \\right| + \\mathrm{rk}_M(E) + \\left| E \\right|$\\\\\n\t& \\ttfamily let $\\dSET{(u_1,v_1), (u_2,v_2),\\ldots, (u_{\\left| V \\right|^2-\\left| V \\right|},v_{\\left| V \\right|^2-\\left| V \\right|})} \n\t = V\\times V \\backslash \\SET{(v,v)~\\middle|~v\\in V}$\\\\\n\t& \\ttfamily let $K := (V, V\\times V)$\\\\\n\t& \\ttfamily let $B$ be \\rmfamily an arbitrary base of $M$\\\\\n\t& \\ttfamily declare state variable $A \\subseteq V\\times V$\\\\\n\t& \\ttfamily declare state variable $i\\in\\SET{1,2,\\ldots,\\left| V \\right|^2-\\left| V \\right|}$\\\\\n\t& \\ttfamily declare state variable ${\\mathcal{P}} \\subseteq {\\mathbf{P}}\\left( K \\right)$\\\\\n\t& \\ttfamily declare state variable ${\\mathcal{R}} \\subseteq 2^{{\\mathbf{P}}\\left( K \\right)}$\\\\\n\t& \\ttfamily declare state variable ${\\mathcal{B}} \\subseteq \\binom{V}{\\mathrm{rk}_M(E)}$\\\\\n\t& \\ttfamily $A := \\emptyset$\\\\\n\t& \\ttfamily $i := 1$\\\\\n\t& \\ttfamily ${\\mathcal{P}} := \\SET{v \\in {\\mathbf{P}}(K)~\\middle|~\\vphantom{A^A}v\\in V}$\\\\\n\t& \\ttfamily ${\\mathcal{R}} := \\SET{ \\SET{b\\in {\\mathbf{P}}(K)~\\middle|~\\vphantom{A^A} b\\in B} }$\\\\\n\t& \\ttfamily ${\\mathcal{B}} := \\SET{ B }$\\\\\n\t& \\ttfamily push state to stack\\\\\n\t& \\ttfamily $d := 1$\\\\\n\t& \\ttfamily while $d > 0$ do \\\\\n\t& \\ttfamily $\\quad$ if ${\\mathcal{B}} = {\\mathcal{B}}(M)$ return $1$ and stop\\\\\n\t& \\ttfamily $\\quad$ if $i > \\left| V \\right|^2 - \\left| V \\right|$ or ${\\mathcal{B}} \\not \\subseteq {\\mathcal{B}}(M)$ then do \\\\\n\t& \\ttfamily $\\quad$ $\\quad$ pop state from stack\\\\\n\t& \\ttfamily $\\quad$ $\\quad$ $d := d - 1$\\\\\n\t& \\ttfamily $\\quad$ $\\quad$ $i := i + 1$\\\\\n\t& \\ttfamily $\\quad$ else do\\\\\n\t& \\ttfamily $\\quad$ $\\quad$ push state to stack\\\\\n\t& \\ttfamily $\\quad$ $\\quad$ $d := d + 1$\\\\\n\t& \\ttfamily $\\quad$ $\\quad$ ${\\mathcal{P}}' := \\SET{lr ~\\middle|~\\vphantom{A^A} l,r\\in {\\mathcal{P}},\\,l_{-1}=u_i,\\,r_{1}=v_i,\\,\\left| l \\right|\\cap \\left| r \\right|=\\emptyset}$\\\\\n\t& \\ttfamily $\\quad$ $\\quad$ ${\\mathcal{R}}' := \\SET{\\left( R\\BSET{r} \\right)\\cup\\SET{l.r} ~\\middle|~ \n\t\t\t\t\t\t\t\t\t\\begin{array}{l} R\\in {\\mathcal{R}},\\, r\\in R,\\,l\\in {\\mathcal{P}}',\\,l_{-1} = r_1,\\\\\n\t\t\t\t\t\t\t\t\t\\left| l \\right|\\cap \\left( \\bigcup_{p\\in R} \\left| p \\right| \\right) = \\SET{r_1},\\, l_1\\in E \\end{array}}$\\\\\n\t& \\ttfamily $\\quad$ $\\quad$ ${\\mathcal{B}}' := \\SET{ \\SET{p_1 ~\\middle|~ p\\in R} ~\\middle|~ \\vphantom{A^A}R\\in {\\mathcal{R}}'}$\\\\\n\t& \\ttfamily $\\quad$ $\\quad$ $A := A\\cup\\SET{(u_i,v_i)}$\\\\\n\t& \\ttfamily $\\quad$ $\\quad$ ${\\mathcal{P}} := {\\mathcal{P}} \\cup {\\mathcal{P}}'$\\\\\n\t& \\ttfamily $\\quad$ $\\quad$ ${\\mathcal{R}} := {\\mathcal{R}} \\cup {\\mathcal{R}}'$\\\\\n\t& \\ttfamily $\\quad$ $\\quad$ ${\\mathcal{B}} := {\\mathcal{B}} \\cup {\\mathcal{B}}'$\\\\\n\t& \\ttfamily $\\quad$ end if\\\\\n\t& \\ttfamily end while $d > 0$\\\\\n\t& \\ttfamily return $0$.\n\\end{tabularx}\n\n\\PRFR{Mar 7th}\n\\noindent\nWe give a rough estimate of the worst-case run-time behavior of this algorithm relative to the run-time of two major blocks of instructions.\nFirst, let $\\phi(d)$ denote a worst-case run-time estimate for the instructions inclusively between the ``{\\ttfamily push state stack}''-instruction\nand the ``{\\ttfamily ${\\mathcal{B}} := {\\mathcal{B}}\\cup {\\mathcal{B}}'$}''-instruction. This is the time it takes to update the paths, maximal routings, and bases of the digraph\nwhen adding the arc $(u_i,v_i)$, and this operation clearly depends on the number of arcs in $A\\BSET{(u_i,v_i)}$, which is a function of the value of $d$ at\nthe start of the instruction block. We would expect $\\phi(d)$ to grow with $\\left| {\\mathcal{P}} \\right|$ and $\\left| {\\mathcal{R}} \\right|$. Clearly, we have the\nvery loose upper bound\n$\\left| {\\mathcal{P}} \\right| \\leq \\left| V \\right| + d!$, since every non-trivial path consists of a non-repeating sequence of arcs with further constraints.\\footnote{P.~Seymour and B.D.~Sullivan give an upper bound for the number of $4$-vertex paths in digraphs without a cycle walk of length $\\leq 4$, which is $\\frac{4}{75}\\left| V \\right|^4$ \\cite{SS10}.} We further have $\\left| {\\mathcal{R}} \\right| \\leq d!$ because we may associate a routing $R\\in{\\mathcal{R}}$ with\na non-repeating sequence of arcs obtained from its paths: if $R = \\dSET{p_1,\\ldots,p_r}$, \nwe first list all arcs of $p_1$ in the order of appearance, then the arcs of $p_2$, and so on until we reach $p_r$, \nand then we list all arcs from $A$ that are not traversed by any path $p\\in R$. Since all $R\\in {\\mathcal{R}}$ route onto $B$, we can reconstruct $R$ from the\narc sequence we just constructed. Again, this bound is very loose.\nFurthermore, let $\\psi(d)$ denote a worst-case run-time estimate for the instructions inclusively between the ``{\\ttfamily pop state stack}''-instruction\nand the ``{$i:=i+1$}''-instruction. For the worst-case analysis, we assume that the backtracking method does traverse \nevery digraph candidate $(V,A)$ with $A\\subseteq V\\times V \\BSET{(v,v)~\\middle|~ v\\in V}$ --- which clearly is impossible for any input matroid $M$.\nWith this assumption we obtain a run-time in\n\\[ O\\left( \\sum_{i=0}^{2n^3-1} \\binom{2n^3}{i} \\left( \\phi(i) + \\psi(i+1) \\right) \\right). \\]\nIt is clear that this estimation is overly pessimistic and does not convey a realistic picture of the run-time behavior of the digraph backtracking algorithm.\nTherefore we implemented a version of this algorithm in {\\ttfamily SageMath} \nand measured its performance on a few sample inputs (see Listing~\\ref{lst:isGammoidSage}).\nIt is an open research task to\nfind conditions and estimates for how often the above algorithm\ndoes prune a large chunk of candidate solutions, as well as good implementations of the update procedure,\nthat exceeds the scope of this work.\n\\end{algorithm}\n\n\n\n\\begin{proof}[Proof of correctness]\\PRFR{Mar 7th}\nWe have the following invariants at the ``{\\ttfamily while $d > 0$}''-instruction: $d = \\left| A \\right|$,\nlet $D=(V,A)$, then\n${\\mathcal{P}} = {\\mathbf{P}}(D)$, ${\\mathcal{R}}$ is the family of all linkings from a subset of $E$ onto $B$ in $D$, and ${\\mathcal{B}}$ is the family of subsets of $E$\nthat can be linked onto $B$ by a routing $R\\in{\\mathcal{R}}$. \nFurthermore, the stack consists of $d$ sets of previously pushed assignments of the\nvariables $A, i, {\\mathcal{P}}, {\\mathcal{R}}, {\\mathcal{B}}$.\nThe instructions in the ``{\\ttfamily while $d > 0$}''-loop recursively test or dismiss all digraphs $D'=(V,A')$\nfor the property $\\Gamma(D',B,E) = M$. First, the algorithm tests whether there is a digraph $D'$ representing $M$ with $(u_i,v_i)\\in A$;\nif it can be ruled out that there is such a digraph $D'$, the algorithm tests whether there is a digraph $D'$ representing $M$ with $(u_i,v_i)\\notin A$.\nOn the other hand, if the loop finds a representation, it returns $1$ and the algorithm ends.\nTherefore,\nif we reach the ``{\\ttfamily end while $d > 0$}''-instruction, we may conclude that there is no digraph on $V$ that represents $M$. \nWith Remark~\\ref{rem:upperBoundForV} and Theorem~\\ref{thm:gammoidRepresentationWithBaseTerminals} we then may conclude that $M$ is not a gammoid,\nand the next instruction correctly returns $0$.\n\nNow let us show in detail that the ``{\\ttfamily while $d > 0$}''-loop indeed has the property stated above.\nClearly, if ${\\mathcal{B}} = {\\mathcal{B}}(M)$, then $\\Gamma(D,B,E) = M$ and therefore we may safely return $1$.\nFirst, if there is some $X\\in{\\mathcal{B}}$\nwhich is not a base of $M$, then there is a routing $X\\double{\\rightarrow} B$ in every digraph $D'=(V,A')$ with $A\\subseteq A'$,\nconsequently $X$ is independent in $\\Gamma(D',B,E)$ for all such $D'$.\nThus we may dismiss all such candidate digraphs. The same holds if $i > \\left| V \\right|^2 - \\left| V \\right|$, in this case\nwe are out of arcs that we may add to $A$, but ${\\mathcal{B}} \\subsetneq {\\mathcal{B}}(M)$, i.e. there is still a base $Y$ of $M$ which is not a base of $\\Gamma(D,B,E)$.\nIn other words, if $M$ is a gammoid, then there is an arc $a\\in A$ that obstructs the addition of some other arcs, one of which is needed to represent $M$.\nIn both cases, we have to undo our last addition of an arc. We achieve this by popping the assignments of $A, i, {\\mathcal{P}}, {\\mathcal{R}}, {\\mathcal{B}}$ from the stack,\nwhich were pushed before the last arc had been added. Afterwards, we have to decrease $d$ in order to reflect the new stack size, and increase $i$ in order to\nprevent adding the same arc again.\nNow assume that neither $i>\\left| V \\right|^2 - \\left| V \\right|$ nor ${\\mathcal{B}} \\not\\subseteq {\\mathcal{B}}(M)$, thus we enter the ``{\\ttfamily else do}''-branch of\nthe second {\\ttfamily if}-instruction in \nthe ``{\\ttfamily while $d>0$}''-loop. In this case there is a base $Y$ of $M$ that is\nnot a base of $\\Gamma(D,B,E)$. \nWe try to fix this by adding the arc $(u_i,v_i)$ to $A$, after we pushed the current state to the stack and adjusted $d$ accordingly.\nLet $D=(V,A)$ denote the digraph before adding the arc, i.e. $(u_i,v_i)\\notin A$, and let $D'=(V,A\\cup\\SET{(u_i,v_i)})$.\nClearly ${\\mathbf{P}}(D') \\backslash {\\mathbf{P}}(D)$ consists of all paths $p\\in {\\mathbf{P}}(D')$ with $(u_i,v_i)\\in \\left| p \\right|_A$. But every such $p$ can be\nwritten as $lr$ where $l,r\\in {\\mathbf{P}}(D)$ with $\\left| l \\right|\\cap \\left| r \\right|=\\emptyset$ and such that $l$ ends in $u_i$ and $r$ starts in $v_i$.\nRemember that ${\\mathcal{P}} = {\\mathbf{P}}(D)$, thus ${\\mathcal{P}}' = {\\mathbf{P}}(D') \\backslash {\\mathbf{P}}(D)$. Now let $R\\colon X\\double{\\rightarrow} B$ with $X\\subseteq E$ be a routing in $D'$ \nwhich is not a routing in $D$,\nthen $R\\cap {\\mathcal{P}}' \\not= \\emptyset$. If we cut off the path of $R$ that uses the new arc $(u_i,v_i)$ at $u_i$, we obtain a routing that is also a routing in\n$D$. Therefore, all routings of $D'$ that start in a subset $X\\subseteq E$\nand that are not routings of $D$ are members of the family ${\\mathcal{R}}'$. Consequently, all bases of $\\Gamma(D',B,E)$\nthat are not bases of $\\Gamma(D,B,E)$ are members of ${\\mathcal{B}}'$. Thus the above invariants at the ``{\\ttfamily while $d > 0$}''-instruction\nhold after the update operations on $A,{\\mathcal{P}},{\\mathcal{R}},{\\mathcal{B}}$. The correctness of the algorithm is therefore established.\n\\end{proof}\n\n\\begin{remark}\\label{rem:backTrackSlow}\\PRFR{Mar 7th}\nAlgorithm~\\ref{alg:gammoidBackTrack} is obviously faster than the brute-force search method, \nas it does not generate non-matroid solution candidates.\n Yet, it still has to dismiss all\npossibilities of arranging arcs in a big digraph in order to determine that $M$ is not a gammoid. The dismissal of a chunk of solution candidates\nmay only take place as soon\nas it can be proven, that every gammoid corresponding to a digraph that contains a certain set of arcs\nhas some independent set $X\\subseteq E$ which is dependent in $M$.\\footnote{A tempting modification of Algorithm~\\ref{alg:gammoidBackTrack}\nwould be to check whether $\\SET{ \\SET{p_1 ~\\middle|~ p\\in R} \\cap E ~\\middle|~ R\\in {\\mathcal{R}}'} \\not\\subseteq {\\mathcal{I}}$\nholds instead of ${\\mathcal{B}} \\not\\subseteq {\\mathcal{B}}(M)$, but the Augmentation Lemma~\\ref{lem:augmentation} implies that this does not occur any earlier than\n ${\\mathcal{B}}\\not\\subseteq {\\mathcal{B}}(M)$.}\n Alas, this happens quite late in the process: At least as long as none of the auxiliary vertices\nin $V\\backslash E$ has a leaving arc that enters any $e\\in E$, there is no way to detect excess connectivity in partial solutions.\n Therefore Algorithm~\\ref{alg:gammoidBackTrack}\ntraverses all candidate arc sets $A$ that cover \nmore than $2^{\\left( \\left| V \\right| - 1 \\right)\\cdot \\left| V\\backslash E \\right|}$\ndigraphs without loop-arcs on $V$ with $A\\cap \\left( \t\\left( V\\backslash E \\right)\\times E \\right) = \\emptyset$.\nThis implies a lower bound of the run-time for all inputs $M=(E,{\\mathcal{I}})$ with $M$ not a gammoid\\footnote{\nFor instance, we may use the arbitrary large non-gammoids $M(K_4)\\oplus \\left( \\SET{X}, 2^X \\right)$\nfor growing finite sets $X$ to approach this run-time bound (see also Example~\\ref{ex:MK4}).} \nin\n\\( \\Omega \\left( 2^{\\left| E \\right|^{5.999}} \\right)\\).\n\nWe implemented a less naive version of Algorithm~\\ref{alg:gammoidBackTrack} (see Listing~\\ref{lst:isGammoidSage}),\nwhere we use an implicit linear order on $V=\\dSET{\\hat v_1,\\hat v_2,\\ldots,\\hat v_m}$\nsuch that $E = \\SET{\\hat v_1,\\hat v_2,\\ldots, \\hat v_{\\left| E \\right|}}$ holds. \nWe keep track of the smallest index $i_0$ that belongs to a vertex $\\hat v_{i_0}\\in V\\backslash E$\nthat is not entered by any arc $a\\in A$. Let $i,j\\in \\SET{1,2,\\ldots, \\left| V \\right|^2 - \\left| V \\right|}$\nand let $(u_i,v_i) = (\\hat v_{i_1}, \\hat v_{i_2})$ and $(u_j,v_j) = (\\hat v_{j_1}, \\hat v_{j_2})$.\nWe require that the implicit linear order on the arcs has the property\nthat we have $\\max\\SET{i_1,i_2} < \\max\\SET{j_1,j_2}$ or \n($\\max\\SET{i_1,i_2} = \\max\\SET{j_1,j_2}$ holds, and either $i_2 = j_1 = \\max\\SET{i_1,i_2}$ or $\\min\\SET{i_1,i_2} < \\min\\SET{j_1,j_2}$\nholds), if and only if $i < j$ holds.\n In other words, we enumerate $V\\times V\\BSET{(v,v)~\\middle|~v\\in V}$ in the following order:\n $(\\hat v_1, \\hat v_2),$ $ (\\hat v_2, \\hat v_1),$ $ (\\hat v_1, \\hat v_3),$\n $ (\\hat v_2, \\hat v_3),$ $ (\\hat v_3, \\hat v_1),$ $ (\\hat v_3, \\hat v_2),$ $ (\\hat v_1, \\hat v_4),\\ldots$\n Now we may implement a shortcut and backtrack as soon as $i_2 > i_0$ holds for $(u_i,v_i) = (\\hat v_{i_1}, \\hat v_{i_2})$.\n The rationale behind this is that if we have to add a new arc that enters a previously unentered vertex, we can always choose the\n vertex entered to be the one with the lowest index among all unentered vertices. Although the improvement corresponding to this adjustment is\n quite measurable in practice,\n the algorithm still has to try more than \n $2^{\\left| V\\backslash E \\right|^2 - \\left| V\\backslash E \\right|}$\n \n --- e.g. the number of digraphs $D'=(V',A')$ on $V'=V\\backslash E$ \n with $A'\\subseteq \\left( V'\\times V' \\right)\\BSET{(v,v) ~\\middle|~ v\\in V'}$ ---\n different candidate arc sets that have the property $A\\cap \\left( \\left(V\\backslash E\\right) \\times E \\right) = \\emptyset$ \n \n \n for every $Q = A\\cap \\left( E \\times \\left( V\\backslash E \\right) \\right)$ \n with $Q_A\\subseteq Q$ \n\n where we let $Q_A =\\SET{ (\\hat v_1, w) ~\\middle|~w\\in V\\backslash E}$. Since there \n are $2^{(\\left| E \\right| - 1)\\cdot \\left| V\\backslash E \\right|}$ different possibilities for \nsuch $Q$, \n\n\n\n the adjusted algorithm still exposes the \\( \\Omega \\left( 2^{\\left| E \\right|^{5.999}} \\right)\\) behavior.\n\n \\noindent\n In theory, there is a possibility to speed up the algorithm a little further, since Theorem~\\ref{thm:IPEssentialStars}\n guarantees that no vertex has more than $\\mathrm{rk}_M(E)$ leaving arcs. Clearly, the problem that the backtracking information\n is only available late in the process is not remedied by limiting the number of arcs leaving each vertex.\n The number of digraph candidates,\n that have to be processed before any target is connected, still is at least\n \\[ \\left( \\sum_{k=0}^{\\mathrm{rk}_M(E)} \\binom{\\mathrm{rk}_M(E)^2\\cdot \\left| E \\right| + \\left| E \\right|}{k} \\right)^\n {\\mathrm{rk}_M(E)^2\\cdot \\left| E \\right| } = \\Omega\\left( 2^{\\mathrm{rk}_M(E)^2\\cdot \\left| E \\right|} \\right) .\\]\n Thus such an adjusted algorithm still exposes \\( \\Omega \\left( 2^{\\left| E \\right|^{2.999}} \\right)\\) behavior.\n When we implemented forced bounds on the number of leaving arcs, the run-time actually increased.\n \n\\end{remark}\n\n\n\\PRFR{Mar 7th}\n\\noindent\nTherefore it is clearly indicated that we examine how Mason's criterion and matroid extensions play along with each other\nin order to gain better understanding of the problem. This understanding is an essential milestone for\nthe research in better algorithms for determining $\\Gamma_{{\\mathcal{M}}}(M)$.\nBefore we devote ourselves to that, we want to make a remark on potentially more easy subclasses of gammoids.\n\n\\begin{remark}\n\tLet $k\\in \\mathbb{N}$. For the subclasses ${\\mathcal{W}}^k$ that consists of all gammoids $G$ with $\\mathrm{W}^k(G) \\leq 1$,\n\tthe problem of deciding class membership\n\tappears to be dramatically more easy. Let $M=(E,{\\mathcal{I}})$ be a matroid. If $M\\in {\\mathcal{W}}^k$ there is a representation\n\tusing at most $k\\cdot \\left| E \\right|$ arcs. Therefore there is a representation with at most $2k\\cdot\\left| E \\right|$\n\tauxiliary vertices. So if $M\\in {\\mathcal{W}}^k$, then $M=\\Gamma(D,T,E)$ where $D=(V,A)$ with\n\t $\\left| V \\right| \\leq (2k+1)\\cdot\t\\left| E \\right|$\n\t and $\\left| A \\right| \\leq k\\cdot \\left| E \\right|$.\n\t Thus there are at most\n\t \\[ \\sum_{i=0}^{k\\cdot\\left| E \\right|} \\binom{9k^2\\cdot\\left| E \\right|^2}{i} \\]\n\t candidate digraphs for $M$ that we may have to regard.\n\t Furthermore, \n\t $M$ is not in ${\\mathcal{W}}^k$ as soon as $M| X$ cannot be represented with a digraph on $3k\\cdot \\left| X \\right|$\n\t vertices with at most $k\\cdot \\left| X \\right|$ arcs, this may open up possibilities for effective divide and conquer approaches.\t \n\\end{remark}\n\n\\subsection{Violations of Mason's $\\alpha$-Criterion}\n\nSome of the ideas and results presented in this section have been published in I.~Albrecht's \n{\\em On Finding New Excluded Minors for Gammoids} \\cite{Al17}, \nwhere an equivalent yet different version of Mason's $\\alpha$-criterion is used.\n\n\\begin{definition}\\label{def:aViolation}\\PRFR{Feb 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, $V\\subseteq E$. $V$ shall be an\n\t \\deftext[aM-violation@$\\alpha_M$-violation]{$\\bm \\alpha_{\\bm M}$-violation},\n\tif $\\alpha_M(V) < 0$ and for all $V'\\subsetneq V$, $\\alpha_M(V') \\geq 0$.\n\tThe \\deftextX{family of all $\\bm \\alpha_{\\bm M}$-violations} is denoted by \\label{n:VM}\n\t\\[ {\\mathcal{V}}(M) = \\SET{ V\\subseteq E ~\\middle|~ \\alpha_M(V) < 0 {\\mathop{\\text{~and~}}}\\, \\forall V'\\subsetneq V\\colon\\, \\alpha_M(V') \\geq 0}. \\qedhere\\]\n\\end{definition}\n\n\\noindent Clearly, an $\\alpha_M$ violation is an inclusion minimal set $X$, for which the inequality $\\alpha_M(X) \\geq 0$ \ndoes not hold. \n\n\n\\begin{corollary}\\PRFR{Feb 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid. Then $M$ is a strict gammoid, if and only if ${\\mathcal{V}}(M) = \\emptyset$.\n\\end{corollary}\n\\begin{proof}\\PRFR{Feb 15th}\nImmediate from Corollary~\\ref{cor:MasonAlpha} and Definition~\\ref{def:aViolation}.\n\\end{proof}\n\n\n\\needspace{4\\baselineskip}\n\n\\vspace*{-\\baselineskip}\n\\begin{wrapfigure}{r}{5cm}\n\\vspace{\\baselineskip}\n~~\\includegraphics{biApex}\n\\end{wrapfigure}\n~\n \n\n\\begin{example}\\label{ex:BiApexMatroid}\\PRFR{Apr 4th}\n\tLet $k\\in \\mathbb{N}\\BSET{0,1}$ be an arbitrary choice, let\n\t $$X= \\SET{\\vphantom{A^A}(i,j)\\in \\mathbb{N}\\times\\mathbb{N} ~\\middle|~ i,j < k,\\, j\\in \\SET{i,(i+1)\\,\\,\\mathrm{mod}\\, k}},$$\n\tand let $E = X\\cup\\dSET{a,b}$ where $a,b\\notin X$.\n\tFurthermore, let $${\\mathcal{H}}_a = \\SET{\\vphantom{A^A}\\SET{a,\\left( i,i \\right),\\left( i,(i+1)\\,\\,\\mathrm{mod}\\, k \\right)}~\\middle|~ i\\in \\mathbb{N},\\,i2$.\n\tLet us consider the case where $k>2$, and let $N = \\left( M^\\ast \\right)|\\left( E\\BSET{b} \\right)$.\n\tThen we also have $\\alpha_N(E\\BSET{b}) = -\\left( k - 2 \\right)$, ${\\mathcal{V}}(N) = \\SET{E\\BSET{b}}$, and $N$ is the dual of a strict gammoid \n\t--- so $N$ is a transversal matroid.\n\\end{example}\n\n\\begin{remark}\\PRFR{Apr 4th}\\label{rem:BiApex}\n\tLet ${\\mathcal{M}}$ be the class of matroids where $M\\in{\\mathcal{M}}$ if\n\tand only if $M$ is the matroid constructed in Example~\\ref{ex:BiApexMatroid} for some $k\\in\\mathbb{N}\\BSET{0,1}$. Then ${\\mathcal{M}}$ is an infinite family of\n\texcluded minors of rank $3$ for the family of gammoids.\n\tThe derived family ${\\mathcal{M}}^\\ast = \\SET{M^\\ast ~\\middle|~ M\\in {\\mathcal{M}}}$ is also an infinite family of excluded minors and yields excluded minors with\n\trank $2k - 1$. We see that excluded minors for the class of gammoids\n\tmay have multiple $\\alpha_M$-violations, and that the value of $\\alpha_M(E)$ for $E\\in{\\mathcal{V}}(M)$\n\tmay become arbitrarily low for excluded minors of the class of gammoids as well as for gammoids that are non-strict.\n\tThe matroids $N = \\left( M^\\ast \\right)|\\left( E\\BSET{b} \\right)$ and $M^\\ast$ have very similar $\\alpha$-violation structure:\n\t$M^\\ast$ contains two violations, and if we restrict $M^\\ast$ to any of these two violations, we obtain $N$ -- thus $M^\\ast$ essentially \n\thas two isomorphic copies of the unique $\\alpha_N$-violation. Consequently, we cannot decide whether a matroid is a gammoid or not by just considering\n\tone violation of $M$ at a time, instead we have to consider the interaction between violations in $M$ as well.\\footnote{We may consider the property that\n\ta matroid is a gammoid to be global with respect to the proper violation-restrictions $M| X$ for $X\\in{\\mathcal{V}}(M)$ in the same sense\n\tas the chromatic number of a graph is a global property with respect to proper induced sub-graphs.}\n\\end{remark}\n\n\n\\noindent Before we start developing the theory of $\\alpha_M$-violations, we should familiarize ourselves some more with\nthe two different kinds of violations that arise in matroids --- violations in non-gammoids and violations in gammoids that are not strict.\n\n\\needspace{7\\baselineskip}\n\n\\vspace*{-\\baselineskip}\n\\begin{wrapfigure}{r}{5cm}\n\\vspace{\\baselineskip}\n\\begin{centering}~\n\\includegraphics{P8ppDelOne}\n\\end{centering}%\n\\vspace*{-1\\baselineskip}\n\\end{wrapfigure}\n~\n \n\n\n\\begin{example}\\label{ex:violationGammoid}\\PRFR{Feb 15th}\n\tWe examine the situation with respect to the gammoid $M=\\Gamma(D,T,E)$ with the ground set $E=\\dSET{a,b,c,d,e,f,g}$ as presented in Example~\\ref{ex:nonStrictGammoid}.\n\tWe have ${\\mathcal{F}}(M)\\backslash {\\mathcal{I}} = \\{\\SET{a,b,c,e},$ $\\SET{a,b,d,f},$ $\\SET{b,c,d,g},$ $\\SET{d,e,f,g},$ $E\\}$ and, \n\tclearly, ${\\mathcal{V}}(M) = \\SET{E}$\n\tand we have $\\alpha_M(E) = -1$.\n\tBut since we know that $M=\\Gamma(D,T,V)| E$, we know that the violation $E$ can be resolved by adding the elements $x$ and $y$ to the matroid $M$. \n\tLet $M_x = \\Gamma(D,T,E\\cup\\SET{x}) = (E\\cup\\SET{x},{\\mathcal{I}}_x)$. Then the new dependent flats of $M_x$ with respect to $M$ are\n\t\\begin{align*} \\left( {\\mathcal{F}}(M_x)\\backslash {\\mathcal{I}}_x\\right) \\backslash \\left( {\\mathcal{F}}(M)\\backslash {\\mathcal{I}}\\right) = \\{ &\n\t\\SET{a,b,x},\\SET{d,f,x},\\SET{a,b,g,x},\\SET{c,d,f,x},\\\\\n\t&\\SET{a,b,c,e,x},\\SET{a,b,d,f,x},\\SET{d,e,f,g,x}, E\\cup\\SET{x}\\},\n\t\\end{align*}\n\tand the dependent flats of $M$ that vanish in $M_x$ are\n\t\\[ \\left( {\\mathcal{F}}(M)\\backslash {\\mathcal{I}}\\right) \\backslash \\left( {\\mathcal{F}}(M_x)\\backslash {\\mathcal{I}}_x\\right) = \\SET{\\SET{a,b,c,e},\\SET{a,b,d,f},\\SET{d,e,f,g},E}.\\]\n\tWe now have\n\t\\begin{align*}\\alpha_{M_x}\\left(\\SET{a,b,x}\\right) = \\alpha_{M_x}\\left(\\SET{d,f,x}\\right) = \\alpha_{M_x}\\left(\\SET{b,c,d,g}\\right)\n\t= \\alpha_{M_x}\\left( \\SET{a,b,g,x} \\right) & =\\\\\n\t \\alpha_{M_x}\\left(\\SET{a,b,c,e}\\right) = \\alpha_{M_x}\\left(\\SET{a,b,d,f}\\right) = \\alpha_{M_x}\\left(\\SET{d,e,f,g}\\right)& = \\hphantom{-}1,\\\\\n\t\\alpha_{M_x}\\left(\\SET{a,b,c,e,x}\\right) = 5 - 3 - \\alpha_{M_x}\\left( \\SET{a,b,x} \\right) & = \\hphantom{-}1,\\\\\n\t\\alpha_{M_x}\\left(\\SET{d,e,f,g,x}\\right) = 5 - 3 - \\alpha_{M_x}\\left( \\SET{d,f,x} \\right) & = \\hphantom{-}1,\\\\\n\t\\alpha_{M_x}\\left(\\SET{a,b,d,f,x}\\right) = 5 - 3 - \\alpha_{M_x}\\left( \\SET{a,b,x} \\right) - \\alpha_{M_x}\\left( \\SET{d,f,x} \\right) & = \\hphantom{-}0,\\\\\n\t\\alpha_{M_x}(E) = 7 - 4 - \\alpha_{M_x}\\left( \\SET{b,c,d,g} \\right) & = \\hphantom{-}2, \\text{~and}\\\\\n\t\\alpha_{M_x}\\left( E\\cup\\SET{x} \\right) = 8 - 4\n\t - \\alpha_{M_x}\\left( \\SET{a,b,x} \\right) \n\t - \\alpha_{M_x}\\left( \\SET{d,f,x} \\right)\n\t - \\alpha_{M_x}\\left( \\SET{d,e,f,g,x} \\right) &\\\\\n\t -\\, \\alpha_{M_x}\\left( \\SET{a,b,c,e,x} \\right)\n\t - \\alpha_{M_x}\\left( \\SET{b,d,f,g} \\right)\n\t &= -1.\n\t\\end{align*}\n\tThus ${\\mathcal{V}}(M_x) = \\SET{E\\cup\\SET{x}}$. So we still have a violation if we just add $x$ to the ground set of the gammoid, and it is easy to tell from the symmetric design of $D$, that the same holds when we would just add $y$. Although $E$ is no longer a violation,\n\twe seem to just have shifted the problem to $E\\cup\\SET{x} = \\mathrm{cl}_{M_x}(E)$. \n\tYet, we made some progress by adding $x$ to $M$: $M_x$ has a modular cut that is generated by three rank $2$ flats, namely ${\\mathcal{C}}_y = \\SET{F\\in {\\mathcal{F}}(M_x) ~\\middle|~ \\SET{b,c}\\subseteq F {\\mathop{\\text{~or~}}} \\SET{d,g}\\subseteq F {\\mathop{\\text{~or~}}} \\SET{e,x}\\subseteq F}$,\n\twhereas $M$ has no such modular cut. So the violation $E\\cup\\SET{x}$ of $M_x$ is less rigid than the violation $E$ of $M$.\n\tNow let $N= \\Gamma(D,T,V)$. Then ${\\mathcal{V}}(N) = \\emptyset$, and $\\alpha_N(X) = 1$, if $X\\in \\SET{\\SET{a,b,x},\\SET{b,c,y},\\SET{d,f,x},\\SET{d,g,y},\\SET{e,x,y}}$, otherwise $\\alpha_N(X) = 0$. So we see how the gammoid $M$ violates Mason's $\\alpha$-criterion: by deleting $x$ and $y$, the nullity of the rank $2$ flats disappears, and so a common reason for nullity in the hyperplanes goes below the radar, resulting in excess negative terms for $\\alpha_M(E)$, which then create an $\\alpha_M$-violation.\n\\end{example}\n\n\\begin{example}\\label{ex:violationNonGammoid}\\label{ex:MK4}\\PRFR{Feb 15th}\n\tLet us now consider the matroid $M(K_4) = (E,{\\mathcal{I}})$ which shall be defined on the ground set $E = \\dSET{a,b,c,d,e,f}$ and which has\n\tthe following circuits\n\t$${\\mathcal{C}}(M(K_4)) = \\SET{\\vphantom{A^A}\\SET{a,b,d}, \\SET{a,c,e}, \\SET{b,c,f}, \\SET{d,e,f}}.$$\n\tEvery circuit of $M(K_4)$ is also a hyperplane of $M(K_4)$, and therefore a flat.\n\tWe calculate\n\t\\begin{align*}\n\t\t\\alpha_{M(K_4)}\\left( \\SET{a,b,d} \\right) = \\alpha_{M(K_4)}\\left( \\SET{a,c,e} \\right) &=\\\\ \\alpha_{M(K_4)}\\left( \\SET{b,c,f} \\right) =\n\t\t\\alpha_{M(K_4)}\\left( \\SET{d,e,f} \\right) &= 1\\\\\n\t\t\\alpha_{M(K_4)}\\left( E \\right) = 6 - 3 - 4\\cdot 1 &= -1.\n\t\\end{align*}\n\tThus ${\\mathcal{V}}(M(K_4)) = \\SET{E}$ and by Proposition~\\ref{prop:cornerCases} and Corollary~\\ref{cor:MasonAlpha}, we obtain that $M(K_4)$ is\n\tnot a gammoid. Unlike Example~\\ref{ex:violationGammoid}, the $\\alpha_{M(K_4)}$-violation $E$ does not allow any particular progress\n\tby adding elements to $M(K_4)$. First, consider an extension $N\\in {\\mathcal{X}}(M(K_4),g)$, that corresponds to a modular cut\n\t $\\SET{F\\in{\\mathcal{F}}(M)\\mid g\\in \\mathrm{cl}_N(F)}$ which is the principal filter of a flat $F_g \\in {\\mathcal{F}}(M(K_4))$ in ${\\mathcal{F}}(M(K_4))$.\n\t Such an extension always has the violation\n\t $E\\cup\\SET{g}\\in {\\mathcal{V}}(N)$. Furthermore, we do not gain any headroom in the sense of allowing new qualities of modular cuts\n\t that are not available with respect to $M(K_4)$:\n\t If $A,B\\in{\\mathcal{F}}(N)$\n\t is not a modular pair in $N$, then $A\\BSET{g}, B\\BSET{g}$ is not a modular pair in $M(K_4)$.\n\t The modular cuts of $M(K_4)$ that are not principal filters in ${\\mathcal{F}}(M(K_4))$ are the cuts of ${\\mathcal{F}}(M(K_4))$ generated by\n\t the two- and three-elementary subsets of $Q=\\SET{\\SET{a,f},\\SET{b,e},\\SET{c,d}}$.\n\t\n\t\n\t\n\t Now none of the hyperplanes of $M(K_4)$ belong to such cuts, because no hyperplane is a subset of any element of $Q$.\n\t Therefore, the hyperplanes of $M(K_4)$ are still flats in the corresponding extension, and so $E$ is still a violation.\n\\end{example}\n\n\n\n\\subsection{The $\\alpha$-Invariant and Single Element Extensions}\\label{sec:alphaNExt}\n\\noindent In this section we take a look at how single element extensions of a matroid interact with the $\\alpha$-invariant.\nWe start with an easy observation.\n\n\\begin{lemma}\\label{lem:necPropForVanishingExt}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, $e\\notin E$, $N\\in{\\mathcal{X}}(M,e)$ be a single-element extension of $M$ such that\n\t$ C = \\SET{F\\in{\\mathcal{F}}(M)~\\middle|~ e\\in \\mathrm{cl}_N(F)}$. Let further $X\\subseteq E$ such that\n\t$\\alpha_N(X) \\geq 0$.\n\t Then there is a set $C_0 \\subseteq \\SET{F\\in C~\\middle|~ F\\subsetneq X}$ such that\n\t\\[ \\forall F\\in C_0\\colon\\, \\alpha_M(F) > 0 \\quad{\\mathop{\\text{~and~}}}\\quad \\sum_{F\\in C_0} \\alpha_M(F) \\geq -\\alpha_M(X).\\]\n\\end{lemma}\n\\begin{proof}\n\tClearly, $\\alpha_N(X) - \\alpha_M(X) \\geq - \\alpha_M(X)$. It follows from Lemma~\\ref{lem:flatsOfExtension} \n\tand Definition~\\ref{def:alphaM} that\n\t $\\alpha_N(Y) = \\alpha_M(Y)$ for all $Y\\subseteq E$\n\twith $\\mathrm{cl}_M(Y)\\notin C$. Furthermore, $\\SET{F\\in{\\mathcal{F}}(N)~\\middle|~F\\subsetneq X,\\,F\\in C} = \\emptyset$. \n\t\\begin{align*}\n\t\t\\alpha_N(X) -\\alpha_M(X) & = \\hphantom{-} \\left| X \\right| - \\mathrm{rk}_N(X) -\n\t\t\t \\sum_{F\\in{\\mathcal{F}}(N),\\,F\\subsetneq X} \\alpha_N(F) \\\\ & \\hphantom{ = }\\,\\, - \\left| X \\right| + \\mathrm{rk}_M(X) + \n\t\t\t \\sum_{F\\in{\\mathcal{F}}(M),\\,F\\subsetneq X} \\alpha_M(F) \\\\\n\t\t\t\t\t& = - \\left( \\sum_{F\\in{\\mathcal{F}}(N),\\,F\\subsetneq X} \\alpha_N(F) \\right)\n\t\t\t\t\t + \\left( \\sum_{F\\in{\\mathcal{F}}(M),\\,F\\subsetneq X} \\alpha_M(F) \\right) \\\\\n\t\t\t\t& = \\sum_{F\\in C,\\,F\\subsetneq X} \\alpha_M(F) \\leq \\sum_{F \\in C_0} \\alpha_M(F) \\\\\n\t\\end{align*}\n\twhere $C_0 = \\SET{F\\in C ~\\middle|~,\\,F\\subsetneq X,\\,\\alpha_M(F) > 0}$ is a subset of $C$ with the desired property.\n\\end{proof}\n\n\\noindent Unfortunately, if $M$ is a matroid and $C$ is a modular cut that satisfies the consequent of\n Lemma~\\ref{lem:necPropForVanishingExt}\nwith respect to every $X\\subseteq E$, and if $N\\in {\\mathcal{X}}(M,e)$ is the extension corresponding to $C$,\nthen $N$ may still not be a strict gammoid.\nOn the other hand, if there is a subset $X\\subseteq E$ which violates the consequent of Lemma~\\ref{lem:necPropForVanishingExt}, \nwe know that $N$ is definitely not a strict gammoid. If we tried to extend a given matroid in order to obtain a strict gammoid, then\nit would be quite natural to first try modular cuts which satisfy the consequent of Lemma~\\ref{lem:necPropForVanishingExt}\nfor as many $X\\subseteq E$ with $\\alpha_M(X) < 0$ as possible.\n\n\n\\begin{definition}\\label{def:AlphaPoset}\\PRFR{Feb 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid. We define the \\deftext[aM-poset@$\\alpha_M$-poset]{$\\bm\\alpha_{\\bm M}$-poset}\n\tas the pair $(\\mathsf{A}_M, \\sqsubseteq_M)$ \\label{n:alphaposet} where $\\mathsf{A}_M = 2^E$ and where for all $X,Y\\in \\mathsf{A}_M$\n\t\\[ X \\sqsubseteq_M Y \\quad\\Longleftrightarrow\\quad X = Y {\\mathop{\\text{~or~}}} X\\in {\\mathcal{F}}(M,Y) \\]\n\tholds. If $M$ is clear from the context, we also write $\\mathsf{A}$ for $\\mathsf{A}_M$ and $\\sqsubseteq$ for $\\sqsubseteq_M$.\n\\end{definition}\n\\begin{remark}\\PRFR{Feb 15th}\n\t$(\\mathsf{A}_M,\\sqsubseteq)$ is obviously a poset: for all $X\\in \\mathsf{A}_M$ we have $X\\sqsubseteq X$. Furthermore, if \n\t$X \\sqsubseteq Y$ and $Y\\sqsubseteq X$ holds\n\tfor $X,Y\\in\\mathsf{A}_M$, then $X = Y$ must hold because all elements of ${\\mathcal{F}}(M,Y)$ are proper subsets of $Y$\n\tand therefore $X\\in{\\mathcal{F}}(M,Y)$ and $Y\\in{\\mathcal{F}}(M,X)$ contradict each other. Now let $X,Y,Z\\in \\mathsf{A}_M$\n\tsuch that $X \\sqsubseteq Y$ and $Y\\sqsubseteq Z$. If $X=Y$ or $Y=Z$, there is nothing to show. Otherwise,\n\t$X \\sqsubseteq Y \\sqsubseteq Z$ implies $X,Y\\in{\\mathcal{F}}(M)$. Since $X\\subsetneq Y \\subsetneq Z$ we obtain $X\\in {\\mathcal{F}}(M,Z)$,\n\tthus $X\\sqsubseteq Z$.\n\\end{remark}\n\n\\needspace{4\\baselineskip}\n\\begin{lemma}\\label{lem:alphaMoebius}\\PRFR{Feb 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, and let\n\t\\[ \\nu\\colon 2^E \\longrightarrow \\mathbb{Z},\\,X\\mapsto \\left| X \\right| - \\mathrm{rk}(X) .\\]\n\tThen \\[\\alpha_M = \\nu \\ast \\mu_{\\mathsf{A}} \\]\n\twhere $\\mu_{\\mathsf{A}}$ is the M\u00f6bius-function of the $\\alpha_M$-poset $(\\mathsf{A}, \\sqsubseteq)$.\n\\end{lemma}\n\\begin{proof}\\PRFR{Feb 15th}\\PRFR{ ~~ + sage}\n\tFrom the recurrence relation of the $\\alpha$-invariant (Definition~\\ref{def:alphaM}) \n\tand the definition of the $\\alpha$-poset (Definition~\\ref{def:AlphaPoset}) we obtain\n\t\\[ \\nu(X) = \\left| X \\right| - \\mathrm{rk}(X) = \\alpha(X) + \\sum_{F\\in{\\mathcal{F}}(M,X)}\\alpha(F) = \\sum_{Y\\sqsubseteq X} \\alpha(Y) \\]\n\tfor all $X\\subseteq E$. The zeta-matrix of $(\\mathsf{A},\\sqsubseteq)$ (Definition~\\ref{def:zetaMatrix}) \n\tallows us to write\n\t\\[ \\nu = \\alpha \\ast \\zeta_\\mathsf{A}.\\]\n\tWe multiply with the M\u00f6bius-function of $(\\mathsf{A},\\sqsubseteq)$ and use Lemma~\\ref{lem:moebiusInversion} in order to obtain\n\t\\[\n\t\t\\nu \\ast \\mu_\\mathsf{A} = \\alpha \\ast \\zeta_\\mathsf{A} \\ast \\mu_\\mathsf{A} = \\alpha \\ast \\mathrm{id}_\\mathbb{Z}\\left( 2^E \\right) = \\alpha.\n\t\t\\qedhere\n\t\\] \n\\end{proof}\n\n\n\n\\begin{corollary}\\label{cor:muAlphaNFromMuAlphaM}\\PRFR{Feb 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, $e\\notin E$, and $N\\in{\\mathcal{X}}(M,e)$ a single element extension of $M$. Then\n\t\\[\\alpha_N|_{2^E} \\,\\,\\,= \\,\\,\\, \\alpha_M \\ast \\zeta_{\\mathsf{A}_M}\\ast \\left( \\mu_{\\mathsf{A}_N} | 2^E\\times 2^E \\right) .\\]\n\\end{corollary}\n\\begin{proof}\\PRFR{Feb 15th}\\PRFR{ ~~ + sage}\n\tLet $\\nu_N\\in\\mathbb{Z}^{2^E}$ and $\\nu_M\\in \\mathbb{Z}^{2^{E\\cup\\SET{e}}}$ be the maps where \n\t$\\nu_M(X) = \\left| X \\right| - \\mathrm{rk}_M(X)$ and $\\nu_N(X) = \\left| X \\right| - \\mathrm{rk}_N(X)$ \n\tholds for all $X\\subseteq E$, or $X\\subseteq E\\cup\\SET{e}$, respectively.\n\t Then $\\nu_N|_{2^E} = \\nu_M = \\alpha_M\\ast \\zeta_{\\mathsf{A}_M}$ because $N$ is an extension of $M$. Furthermore, for $X\\subseteq E$ and $Y\\subseteq E\\cup\\SET{e}$\n\t with $e\\in Y$, we have $Y\\not\\sqsubseteq_N X$, and therefore $\\mu_{\\mathsf{A}_N}(Y,X) = 0$ $(\\ast)$,\n\t thus we may restrict the equation from Lemma~\\ref{lem:alphaMoebius}\n\t in the following way:\n\t \\begin{align*}\n\t \\alpha_N|_{2^E} \\,\\,\\, &= \\,\\,\\, (\\nu_N \\ast \\mu_{\\mathsf{A}_N} )|_{2^E}\n\t \\quad\\quad\\quad\\quad\\quad \\\n\t \\,\\,\\, = \\,\\,\\, \\nu_N \\ast \\left( \\mu_{\\mathsf{A}_N} | 2^{E\\cup\\SET{e}} \\times 2^E \\right) \\\\\n\t & \\stackrel{(\\ast)}{=} \\,\\,\\ \\left(\\nu_N|_{2^E}\\right) \\ast \\left( \\mu_{\\mathsf{A}_N} | 2^E\\times 2^E \\right)\n\t \\,\\,\\, = \\,\\,\\ \\alpha_M \\ast \\zeta_{\\mathsf{A}_M}\\ast \\left( \\mu_{\\mathsf{A}_N} | 2^E\\times 2^E \\right). \n\t \\qedhere\n\t \\end{align*} \n\\end{proof}\n\n\\noindent\nLet us explain the above equations a little further.\n\t Here, we interpret $\\alpha_N|_{2^E}$ as a vector in the $2^{\\left| E \\right|}$-dimensional $\\mathbb{Z}$-module $\\mathbb{Z}^{2^E}$.\n\t The term $\\nu_N \\ast \\mu_{\\mathsf{A}_N}$ denotes\n\t a vector of the $\\mathbb{Z}$-module $\\mathbb{Z}^{2^{E\\cup\\SET{e}}}$, and for all $X\\subseteq E\\cup\\SET{e}$,\n\t $$\\left(\\nu_N \\ast \\mu_{\\mathsf{A}_N}\\right)(X) = \\sum_{W\\subseteq E\\cup\\SET{e}} \\nu_N(W)\\cdot \\mu_{\\mathsf{A}_N}(W,X) = \\alpha_N(X)$$ \n\t by Lemma~\\ref{lem:alphaMoebius}, therefore the equation also holds for the vector restricted to $\\mathbb{Z}^{2^E}$.\n\t The vector $\\nu_N \\ast \\left( \\mu_{\\mathsf{A}_N} | 2^{E\\cup\\SET{e}} \\times 2^E \\right)$ on the right arises by first restricting\n\t $\\mu_{\\mathsf{A}_N}$ to $2^{E\\cup\\SET{e}}\\times 2^E$, effectively dropping all rows ${\\left( \\mu_{\\mathsf{A}_N} \\right)}_R$ from $\\mu_{\\mathsf{A}_N}$ where\n\t $e\\in R\\subseteq E\\cup\\SET{e}$, and only afterwards calculating the product. For all $X\\subseteq E$, we still have to compute\n\t \\[ \\left( \\nu_N \\ast \\left( \\mu_{\\mathsf{A}_N} | 2^{E\\cup\\SET{e}} \\times 2^E \\right) \\right)(X) = \\sum_{W\\subseteq E\\cup\\SET{e}} \\nu_N(W)\\cdot \\mu_{\\mathsf{A}_N}(W,X).\\]\n\t For the next equation, we need the property $(\\ast)$ that allows us to drop all the summands that belong to $W\\subseteq E\\cup\\SET{e}$\n\t with $e\\in W$ on the left-hand side:\n\t \\begin{align*} \\sum_{W\\subseteq E\\cup\\SET{e}} \\nu_N(W)\\cdot \\mu_{\\mathsf{A}_N}(W,X) \n\t & \\stackrel{(\\ast)}{=} \\sum_{W\\subseteq E} \\nu_N(W)\\cdot \\mu_{\\mathsf{A}_N}(W,X)\\\\ & =\n\t \\left( \\left(\\nu_N|_{2^E}\\right) \\ast \\left( \\mu_{\\mathsf{A}_N} | 2^E\\times 2^E \\right) \\right)(X) .\n\t \\end{align*}\n\t \n\n\\begin{lemma}\\label{lem:AlphaPosetDownsetsExtension}\\PRFR{Feb 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, $e\\notin E$, $N\\in{\\mathcal{X}}(M,e)$ a single element extension of $M$, and \n\t$C = \\SET{F\\in{\\mathcal{F}}(M)~\\middle|~e\\in\\mathrm{cl}_N(F)}$ the corresponding modular cut.\n\tFurther, let $(\\mathsf{A}_M,\\sqsubseteq_M)$ be the $\\alpha_M$-poset, and $(\\mathsf{A}_N,\\sqsubseteq_N)$ \n\tbe the $\\alpha_N$-poset.\n\tThen for all $X\\subseteq E$ and all $Y\\subseteq E\\cup\\SET{e}$ with $X\\not= Y$\n\t\\[ X\\sqsubseteq_N Y \\quad\\Longleftrightarrow\\quad X \\sqsubseteq_M Y {\\mathop{\\text{~and~}}} X\\notin C. \\]\n\\end{lemma}\n\n\\begin{proof}\\PRFR{Feb 15th}\\PRFR{ ~~ + sage}\n\tLemma~\\ref{lem:flatsOfExtension} yields ${\\mathcal{F}}(N)\\cap 2^E = {\\mathcal{F}}(M)\\backslash C$ and the statement of this lemma follows from Definition~\\ref{def:AlphaPoset}.\n\\end{proof}\n\n\n\\begin{lemma}\\label{lem:AlphaPosetDownsetsExtensionWithE}\\PRFR{Feb 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, $e\\notin E$, $N\\in{\\mathcal{X}}(M,e)$ be an extension of $M$,\n\t$C = \\SET{F\\in{\\mathcal{F}}(M)~\\middle|~ e\\in \\mathrm{cl}_N(F)}$ the corresponding modular cut, and\n\tlet $\\mathsf{A}_M$ and $\\mathsf{A}_N$ denote the $\\alpha_M$- and $\\alpha_N$-posets,\n\trespectively.\n\tThen for all $X,Y\\subseteq E$, we have\n\t\\begin{enumerate}\\renewcommand{\\theenumi}{(\\roman{enumi})}\\renewcommand{\\labelenumi}{\\theenumi}\n\t\\item \\( X\\sqsubseteq_{\\mathsf{A}_N} Y\\cup\\SET{e} \\) holds if and only if $X\\sqsubseteq_{\\mathsf{A}_M} Y$ and $X\\notin C$.\\\\\n\tFurthermore, if $Y\\sqsubseteq_{\\mathsf{A}_N} Y\\cup\\SET{e}$ then $Y\\cup\\SET{e}\\in {\\mathcal{F}}(N)$.\n\n\t\\item \\( X\\cup\\SET{e} \\sqsubseteq_{\\mathsf{A}_N} Y\\cup\\SET{e}\\) holds if and only if $X\\sqsubseteq_{\\mathsf{A}_M} Y$ and $X\\notin \\partial C$\n\twhere \\[ \\partial C = \\SET{F\\in{\\mathcal{F}}(M)\\backslash C ~\\middle|~ \\exists x\\in E\\backslash F\\colon\\,\\mathrm{cl}_M(F\\cup\\SET{x}) \\in C}. \\]\n\t\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\\PRFR{Feb 15th}\nThis is clear from Lemma~\\ref{lem:flatsOfExtension} and Definition~\\ref{def:AlphaPoset}, too.\n\\end{proof}\n\n\n\\begin{remark}\\label{rem:moebiusAMstableBelowC}\\PRFR{Feb 15th}\nAs we have just seen, the $\\mathsf{A}_N$-down-sets of subsets of $E$ are the corresponding down-sets \nof the $\\alpha_M$-poset $\\mathsf{A}_M$ where the upper part, that corresponds to the modular cut $C$ of the \nsingle element extension $N$ of $M$, has been cut off. \nSince the values of the M\u00f6bius-function $\\mu_{P}(X,Y)$ for an arbitrary poset $P$\nonly depend on the $P$-down-sets of elements of the $P$-down-set of $Y$ (Definition~\\ref{def:moebiusFunction}), \nwe see that for $X\\subseteq E$ and $Y\\subseteq E$ with $C\\cap 2^Y \\subseteq \\SET{Y}$ \nwe have $\\mu_{\\mathsf{A}_M}(X,Y) = \\mu_{\\mathsf{A}_N}(X,Y)$ and consequently\n$\\alpha_M(Y) = \\alpha_N(Y)$.\n\\end{remark}\n\n\\begin{corollary}\\label{cor:muAlphaAsSumWithDelta}\\PRFR{Feb 15th}\nLet $M=(E,{\\mathcal{I}})$ be a matroid, $e\\notin E$, $N\\in{\\mathcal{X}}(M,e)$ a single element extension of $M$, and \n\t$C = \\SET{F\\in{\\mathcal{F}}(M)~\\middle|~e\\in\\mathrm{cl}_N(F)}$ the corresponding modular cut.\n\tThen for all $X,Y\\subseteq E$\n\t\\[ \\mu_{\\mathsf{A}_N}(X,Y) = \\mu_{\\mathsf{A}_M}(X,Y) + \\sum_{Z\\in C,\\,X\\subseteq Z\\subsetneq Y} \\mu_{\\mathsf{A}_M}(X,Z).\\]\n\\end{corollary}\n\\begin{proof}\\PRFR{Feb 15th}\\PRFR{~~ + sage}\n\tThe first equation is a direct consequence of Lemma~\\ref{lem:AlphaPosetDownsetsExtension} and Remark~\\ref{rem:moebiusAMstableBelowC}:\n\t\\begin{align*} - \\sum_{X\\sqsubseteq_{M} Z \\sqsubset_{M} Y} \\mu_{\\mathsf{A}_M}(X,Z) \\,\\,\\,=\\,\\,\\, &\n\t- \\left( \\sum_{X\\sqsubseteq_{N} Z \\sqsubset_{N} Y} \\mu_{\\mathsf{A}_N}(X,Z) \\right)\n\t\\\\ & - \n\t\\left( \\sum_{X\\sqsubseteq_{M} Z \\sqsubset_{M} Y, Z\\in C} \\mu_{\\mathsf{A}_M}(X,Z) \\right)\n\t\\end{align*}\n\tholds for all $X,Y\\subseteq E$. Thus we may expand the terms $\\mu_{\\mathsf{A}_N}(X,Y)$ and $\\mu_{\\mathsf{A}_M}(X,Y)$ with Definition~\\ref{def:alphaM}, and then cancel in the above equation.\n\\end{proof}\n\n\n\\begin{definition}\\label{def:DeltaAlphaInvariant}\\PRFR{Feb 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid and let ${\\mathcal{M}}(M)$ be the class of all modular cuts of $M$.\\label{n:Deltaalphainvariant}\n\tThe \\deftext[da-invariant of M@$\\Delta\\alpha$-invariant of $M$]{$\\bm \\Delta \\bm \\alpha$-invariant of $\\bm M$}\n\tshall be defined as\n\t\\[\n\t\t\\Delta \\alpha_M \\colon {\\mathcal{M}}(M)\\times 2^E \\longrightarrow \\mathbb{Z},\\]\\[\n\t\t(C,X) \\mapsto \\sum_{Y\\subsetneq X} \\left( \\left( \\left| Y \\right| - \\mathrm{rk}(Y) \\right) \\cdot \\sum_{Z\\in C,\\,Y\\subseteq Z \\subsetneq X} \\mu_{\\mathsf{A}}(Y,Z)\\right),\n\t\\]\n\twhere $\\mu_\\mathsf{A}$ denotes the M\u00f6bius-function of the $\\alpha_M$-poset.\n\tIf the matroid $M$ is clear from the context, we will denote $\\Delta\\alpha_M$ simply by $\\Delta\\alpha$.\n\\end{definition}\n\n\n\n\\begin{lemma}\\label{lem:alphaNXThroughDeltaalphaM}\\PRFR{Feb 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, $e\\notin E$, $N\\in{\\mathcal{X}}(M,e)$ be an extension of $M$,\n\t$C = \\SET{F\\in{\\mathcal{F}}(M)~\\middle|~ e\\in \\mathrm{cl}_N(F)}$ the corresponding modular cut, and\n\t$X\\subseteq E$.\n\tThen \\[ \\alpha_N(X) = \\alpha_M(X) + \\Delta\\alpha_M(C,X).\\]\n\\end{lemma}\n\\begin{proof}\\PRFR{Feb 15th}\\PRFR{~~ + sage}\n\tLet $X\\subseteq E$, and let $\\mathsf{A}_M$ and $\\mathsf{A}_N$ denote the $\\alpha_M$- and $\\alpha_N$-posets, respectively.\n\tFrom Corollary~\\ref{cor:muAlphaNFromMuAlphaM} and Lemmas~\\ref{lem:alphaMoebius} and \\ref{lem:moebiusInversion} we obtain the equation\n\t\\[ \\alpha_N(X) = \\sum_{Y\\subseteq X} \\left( \\left( \\left| Y \\right| - \\mathrm{rk}_M(Y) \\right)\\cdot \\mu_{\\mathsf{A}_N}(Y,X)\\right) .\\]\n\tCorollary~\\ref{cor:muAlphaAsSumWithDelta} yields\n\t\\[ \\mu_{\\mathsf{A}_N}(Y,X) = \\mu_{\\mathsf{A}_M}(Y,X) + \\sum_{Z\\in C,\\,Y\\subseteq Z \\subsetneq X} \\mu_{\\mathsf{A}_M}(Y,Z) \\]\n\tand therefore applying the distributive law of $\\mathbb{Z}$ together with Definition~\\ref{def:DeltaAlphaInvariant} yields \n\tthe desired equation\n\t\\begin{align*}\n\t\t \\alpha_N(X) \\,\\,\\,=\\,\\,\\, & \\sum_{Y\\subseteq X} \n\t\t \\left( \\left( \\left| Y \\right| - \\mathrm{rk}_M(Y) \\right)\\cdot\\left(\\mu_{\\mathsf{A}_M}(Y,X)\n\t\t + \\sum_{Z\\in C,\\,Y\\subseteq Z \\subsetneq X} \\mu_{\\mathsf{A}_M}(Y,Z)\\right) \\right) \n\t\t \\\\\n\t\t = \\,\\,\\,& \\alpha_M(X) + \\Delta\\alpha_M(C,X). \\qedhere\n\t\\end{align*}\n\\end{proof}\n\n\\needspace{6\\baselineskip}\n\n\\begin{lemma}\\label{lem:alphaNXbelowC}\\PRFR{Feb 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, $e\\notin E$, $N\\in{\\mathcal{X}}(M,e)$ be an extension of $M$,\n\t$C = \\SET{F\\in{\\mathcal{F}}(M)~\\middle|~ e\\in \\mathrm{cl}_N(F)}$ the corresponding modular cut, and\n\t$X\\subseteq E$ such that\n\t $\\mathrm{rk}_M \\left( F\\cap X' \\right) < \\mathrm{rk}_M(F)$ for all $F\\in C$ and all proper subsets $X'\\subsetneq X$.\n\tThen $$ \\alpha_N(X\\cup\\SET{e}) = \\begin{cases}[r] 0 & \\quad\\text{if~} X\\in {\\mathcal{F}}(M) {\\mathop{\\text{~and~}}} \\mathrm{cl}_M(X)\\notin C, \\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t \\alpha_M(X) & \\quad\\text{if~} X\\notin {\\mathcal{F}}(M) {\\mathop{\\text{~and~}}} \\mathrm{cl}_M(X)\\notin C,\\\\\n\t\t\t\t\t\t\t\t\t\t\t\t 1 + \\alpha_M(X) & \\quad\\text{if~} \\mathrm{cl}_M(X)\\in C.\\\\\n\t\t\t\t\t\t\t\t\t\t\t\t \\end{cases} $$\n\\end{lemma}\n\\begin{proof}\\PRFR{Feb 15th}\\PRFR{~~ + sage}\n\tLet $(\\mathsf{A}_M,\\sqsubseteq_M)$ and $(\\mathsf{A}_N,\\sqsubseteq_N)$ denote the $\\alpha_M$- and $\\alpha_N$-poset, respectively.\n\tLet $W\\subseteq X$, then $W$ satisfies the premises of this lemma whenever $X$ satisfies the premises.\n\tFurthermore, if for some $F\\in C$ the equality $\\mathrm{rk}_M(F\\cap X) = \\mathrm{rk}_M(F)$ holds,\n\tthen\n\t\\linebreak\n\t $F = \\mathrm{cl}_M(X)$ and conversely, if $\\mathrm{cl}_M(X)\\notin C$, then $\\mathrm{rk}_M(F\\cap X) < \\mathrm{rk}_M(F)$ for all $F\\in C$.\n\n\n\tNow, we prove the statement for all $X\\in{\\mathcal{F}}(M)$ with $\\mathrm{cl}_M(X)\\notin C$ by induction on $\\mathrm{rk}_M(X)$.\n\tLet $O=\\mathrm{cl}_M(\\emptyset)$ be the unique rank-$0$ flat of $M$.\n\t Then\n\t the down-set $\\downarrow_{\\mathsf{A}_N}\\left( O\\cup\\SET{e} \\right) = \\SET{O,O\\cup\\SET{e}}$.\n\tThus, by Definitions~\\ref{def:alphaM} and \\ref{def:AlphaPoset},\n\twe have \\begin{align*}\n\t\t\\alpha_N(O\\cup \\SET{e}) = & \\left| O\\cup\\SET{e} \\right| - \\mathrm{rk}_N(O\\cup \\SET{e}) - \\alpha_N(O)\\\\\n\t=& \\left| O \\right| + 1 - 1 - \\left( \\left| O \\right| - \\mathrm{rk}_N(O) \\right) = 0.\n\t\\end{align*}\n\tNow let $X\\in{\\mathcal{F}}(M)$ be a flat with $\\mathrm{rk}_M(X) > 0$. Lemma~\\ref{lem:AlphaPosetDownsetsExtensionWithE}\n\tyields that\n\t$$\\downarrow_{\\mathsf{A}_N} \\left( X\\cup\\SET{e} \\right) = \\SET{F,F\\cup\\SET{e}~\\middle|~ F\\in\\,\\, \\downarrow_{\\mathsf{A}_M} X}.$$\n\tNote that $X\\cup\\SET{e}$ may or may not be a flat in $N$, as we have $X\\cup\\SET{e}\\notin{\\mathcal{F}}(N)$ if $X\\in{\\mathcal{F}}(M)$ \n\tand $X$ is covered by a flat from $C$ --- but $X\\cup\\SET{e}$ is still an element of the above down-set.\n\tThe assumption, that $\\mathrm{rk}_M \\left( F\\cap X \\right) < \\mathrm{rk}_M(F)$ for all $F\\in C$, guarantees that all $F\\in{\\mathcal{F}}(M,X)$\n\tare flats of $N$, too.\n\tFurthermore, we have\n\t\\[ \\alpha_N(X\\cup\\SET{e}) = \\left| X\\cup\\SET{e} \\right| - \\mathrm{rk}_N(X\\cup\\SET{e}) - \\sum_{F\\sqsubset_N X} \\alpha_N(F).\\]\n\tUsing the induction hypothesis, we obtain\n\t\\begin{align*}\n\t \\alpha_N(X\\cup\\SET{e}) = &\n\t \\left| X\\cup\\SET{e} \\right| - \\mathrm{rk}_N(X\\cup\\SET{e}) - \\left( \\sum_{F\\sqsubset_N X} \\alpha_N(F) \\right) - \\alpha_N(X) \n\t \\\\\n\t = & \\left| X \\right| - \\mathrm{rk}_N(X) -\\left( \\sum_{F\\sqsubset_N X} \\alpha_N(F) \\right)\n\t - \\left( \\left| X \\right| - \\mathrm{rk}_N(X) - \\sum_{F\\sqsubset_N X} \\alpha_N(F) \\right) \\\\ = & \\,\\,0. \\\\\n\t\\end{align*}\n\tNow let $X\\subseteq E$ with $\\mathrm{cl}_M(X) \\notin C$ and $X\\notin {\\mathcal{F}}(M)$.\n\tThen\n\t$$\\downarrow_{\\mathsf{A}_N} \\left( X\\cup\\SET{e} \\right) = \\SET{F,F\\cup\\SET{e}~\\middle|~ F\\in\\,\\, \\downarrow_{\\mathsf{A}_M} X}\\BSET{X}.$$\n\tAnalogously to the above calculation we obtain\n\t\\begin{align*}\n\t \\alpha_N(X\\cup\\SET{e}) = & \\left| X\\cup\\SET{e} \\right| - \\mathrm{rk}_N(X\\cup\\SET{e}) - \\sum_{F\\sqsubset_N X} \\alpha_N(F)\n\t \\\\ = &\\,\\, \\alpha_N(X) = \\alpha_M(X), \\end{align*}\n\t where the last equation is due to the fact that $F\\not\\subseteq X$ holds for all $F\\in C$,\n\t which implies that $N| X = M| X$ and therefore\n\t $\\alpha_N(X) = \\alpha_{N| X}(X) = \\alpha_{M| X}(X) = \\alpha_M(X)$\n\t (Definition~\\ref{def:alphaM}).\t\n\n\t \\noindent\n\t Now assume that $\\mathrm{cl}_M(X)\\in C$. \n\t If $X\\in{\\mathcal{F}}(M)$, then $e\\in\\mathrm{cl}_N(X)$, thus $X\\notin {\\mathcal{F}}(N)$. Otherwise $X\\notin{\\mathcal{F}}(M)$ and therefore $X\\notin{\\mathcal{F}}(N)$, too. \n\t In both cases we obtain that\n\t $$\\SET{F\\subseteq E\\cup\\SET{e}~\\middle|~\\vphantom{A^A} F \\sqsubset_N X\\cup\\SET{e}}\n\t = \\SET{F,F\\cup\\SET{e}~\\middle|~ F\\in\\,\\, \\downarrow_{\\mathsf{A}_M} X} \\BSET{\\vphantom{A^A}X, X\\cup\\SET{e}}.$$\n\t Furthermore, for all $X'\\subsetneq X$ we have $\\mathrm{cl}_M(X')\\notin C$, \n\t because $\\mathrm{rk}_M \\left( F\\cap X' \\right) < \\mathrm{rk}_M(F)$ for all $F\\in C$.\n\t This implies that if $F\\cup\\SET{e} \\sqsubset_{\\mathsf{A}_N} X$ for some $F\\in{\\mathcal{F}}(M)$, then $\\alpha_N(F\\cup\\SET{e}) = 0$.\n\t Consequently, with Lemma~\\ref{lem:flatsOfExtension}, we obtain\n\t \\[ \\sum_{F\\sqsubset_N X\\cup\\SET{e}} \\alpha_N(F) = \\sum_{F\\sqsubset_N X\\cup\\SET{e},\\,e\\notin F} \\alpha_N(F)\n\t = \\sum_{F\\sqsubset_N X} \\alpha_N(F) = \\sum_{F\\sqsubset_M X} \\alpha_M(F).\\]\n\t Since $e\\in\\mathrm{cl}_N(X)$, we have $\\mathrm{rk}_N(X\\cup\\SET{e}) = \\mathrm{rk}_N(X)$. This yields the desired equation\n\t \\begin{align*}\n\t \t \\alpha_N(X\\cup\\SET{e}) & = \\left| X\\cup\\SET{e} \\right| - \\mathrm{rk}_N(X\\cup\\SET{e}) - \\sum_{F\\sqsubset_N X\\cup\\SET{e}} \\alpha_N(F) \\\\\n\t \t & = 1 + \\left| X \\right| - \\mathrm{rk}_M(X) - \\sum_{F\\sqsubset_M X} \\alpha_M(F) = 1 + \\alpha_M(X). \\qedhere\n\t \t\\end{align*}\n\\end{proof}\n\n\\noindent In order to determine the values of $\\alpha_N(X)$ of the extension $N$ of $M$ by $e$ when $e\\in X$ and $e\\in\\mathrm{cl}_N(X\\BSET{e})$,\n\t\twe have to keep track of the flats $F$ of $M$ that are proper subsets $X$ with the additional property that $e\\in\\mathrm{cl}_N(F)$.\n\\begin{definition}\\PRFR{Feb 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid and let $C\\in{\\mathcal{M}}(M)$ be a modular cut of $M$.\n\tWe define the \\deftext[extension poset of $C$ with respect to $M$]{extension poset of $\\bm C$ with respect to $\\bm M$}\n\tas the pair \\label{n:BetaMC} $\\left(\\mathsf{B}_M^C, \\sqsubseteq_M^C \\right)$ where\n\t\\( \\mathsf{B}_M^C = 2^E \\) and where\n\t\\[ X \\sqsubseteq_M^C Y \\quad \\Longleftrightarrow\\quad X = Y {\\mathop{\\text{~or~}}} \\left( X\\in C {\\mathop{\\text{~and~}}} X\\subseteq Y\\right)\\]\n\tholds for all $X,Y\\subseteq E$. If $M$ is clear from the context, we will denote $\\mathsf{B}_M^C$ by $\\mathsf{B}^C$ and\n\t$\\sqsubseteq_M^C$ by $\\sqsubseteq^C$, too.\n\\end{definition}\n\\begin{remark}\\PRFR{Feb 15th}\n\tClearly, $\\sqsubseteq_M^C$ is reflexive, the anti-symmetry of $\\mathsf{B}_M^C$ follows from the anti-symmetry of $\\subseteq$.\n\tLet $X \\sqsubset_M^C Y \\sqsubset_M^C Z$. Then $X,Y\\in C$ and $X\\subsetneq Y\\subsetneq Z$. Therefore $X\\sqsubset_M^C Z$ holds,\n\tand $\\mathsf{B}_M^C$ is indeed a poset.\n\\end{remark}\n\n \\needspace{8\\baselineskip}\n\\begin{definition}\\PRFR{Feb 15th}\n\t\\label{def:DeltaPAlphaInvariant}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid and let ${\\mathcal{M}}(M)$ be the class of all modular cuts of $M$.\\label{n:DeltaPalphainvariant}\n\tThe \\deftext[da-invariantX of M@$\\tilde{\\Delta}\\alpha$-invariant of $M$]{$\\bm \\tilde{\\Delta} \\bm \\alpha$-invariant of $\\bm M$}\n\tshall be defined as\n\t\\[\n\t\t\\tilde{\\Delta} \\alpha_M \\colon {\\mathcal{M}}(M)\\times 2^E \\longrightarrow \\mathbb{Z},\\]\\[\n\t\t(C,X) \\mapsto \\begin{cases}[r]\n\t\t\t\t\t\t\t- \\alpha_M(X) &\\quad\\text{if~} X\\in {\\mathcal{F}}(M) {\\mathop{\\text{~and~}}} \\mathrm{cl}_M(X)\\notin C, \\\\\n\t\t\t\t\t\t\t0 & \\quad\\text{if~} X\\notin {\\mathcal{F}}(M) {\\mathop{\\text{~and~}}} \\mathrm{cl}_M(X)\\notin C,\\\\\n\t\t\t\t\t\t\t1 - \\displaystyle \\sum_{F \\sqsubset^C X} \\tilde{\\Delta} \\alpha_M(C,F) & \\quad\\text{otherwise,}\n\t\t\t\t\t\\end{cases}\n\t\\]\n\twhere $\\left( \\mathsf{B}^C, \\sqsubseteq^C \\right)$ denotes the extension poset of $C$ with respect to $M$.\n\tIf the matroid $M$ is clear from the context, we will denote $\\tilde{\\Delta}\\alpha_M$ simply by $\\tilde{\\Delta}\\alpha$.\n\\end{definition}\n\n\\begin{lemma}\\PRFR{Feb 15th}\\label{lem:DPalpha}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, $e\\notin E$, $N\\in{\\mathcal{X}}(M,e)$ be an extension of $M$,\n\t$C = \\SET{F\\in{\\mathcal{F}}(M)~\\middle|~ e\\in \\mathrm{cl}_N(F)}$ the corresponding modular cut.\n\tThen $$ \n\t\t\\alpha_N(X\\cup\\SET{e}) = \\alpha_M(X) + \\tilde{\\Delta}\\alpha_M(C,X). $$\n\\end{lemma}\n\\begin{proof}\\PRFR{Feb 15th}\\PRFR{~~ + sage}\n\tLet $X\\subseteq E$. The cases where $\\mathrm{cl}_M(X)\\notin C$ are covered by Lemma~\\ref{lem:alphaNXbelowC}.\n\tFurthermore, if $X$ is $\\subseteq$-minimal with the property that $\\mathrm{cl}_M(X)\\in C$, then $\\downarrow_{\\mathsf{B}_M^C} X = \\SET{X}$\n\tand therefore $\\tilde{\\Delta}\\alpha(C,X) = 1 = \\alpha_N(X\\cup\\SET{e}) - \\alpha_M(X)$ holds, too, by Lemma~\\ref{lem:alphaNXbelowC}.\n\tFor the general case where $\\mathrm{cl}_M(X)\\in C$, remember that we saw in the proof of Lemma~\\ref{lem:alphaMoebius} that the\n\tequations\n\t\\[ \\left| X \\right| - \\mathrm{rk}_M(X) = \\sum_{F\\sqsubseteq_M X} \\alpha_M(F) \\]\n\tand\n\t\\[ \\left| X\\cup\\SET{e} \\right| - \\mathrm{rk}_N(X\\cup\\SET{e}) = \\sum_{F\\sqsubseteq_{N} X\\cup\\SET{e}} \\alpha_N(F) \\]\n\thold. Thus we obtain\n\t\\[ (\\ast)\\quad \\left( \\sum_{F\\sqsubseteq_{M} X} \\alpha_M(F) \\right) + 1 = \\sum_{F\\sqsubseteq_{N} X\\cup\\SET{e}} \\alpha_N(F).\\]\n\tWe prove the missing part of the statement by induction on the length $k$ \n\tof a maximal chain $C_1 \\subsetneq C_2 \\subsetneq \\ldots \\subsetneq C_k \\subsetneq X$ with $C_1,\\ldots,C_k\\in C$.\n\tThe base case with $k=0$ has been established above. \n\tUsing Lemma~\\ref{lem:flatsOfExtension} we obtain that\n\t\\( \\downarrow_{\\mathsf{A}_N} \\left( X\\cup\\SET{e} \\right) = Q \\mathbin{\\dot{\\cup}} R \\mathbin{\\dot{\\cup}} S \\mathbin{\\dot{\\cup}} T \\)\n\twhere \\allowdisplaybreaks\n\t\\begin{align*}\n\t\tQ & = \\SET{Y ~\\middle|~\\vphantom{A^A} Y\\in {\\mathcal{F}}(M)\\backslash C,\\,Y\\subseteq X}, \\\\\n\t\tR & = \\SET{Y\\cup\\SET{e} ~\\middle|~\\vphantom{A^A} Y\\in {\\mathcal{F}}(M)\\backslash C,\\,Y\\subsetneq X,\\,\\forall f\\in E\\backslash Y\\colon\\,\\mathrm{cl}_M(Y\\cup\\SET{f})\\notin C}, \\\\\n\t\tS & = \\SET{Y\\cup\\SET{e} ~\\middle|~\\vphantom{A^A} Y\\in C,\\,Y\\subsetneq X}, \\text{~and}\\\\\n\t\tT & = \\SET{X\\cup\\SET{e}\\vphantom{A^A}}.\n\t\\end{align*}\n\tClearly, $Q\\subseteq\\,\\,\\downarrow_{\\mathsf{A}_M} X$, and\n\tLemma~\\ref{lem:alphaNXThroughDeltaalphaM} and Definition~\\ref{def:DeltaAlphaInvariant} yield that \\[\n\t\t\\sum_{F\\in Q} \\alpha_N(F) = \\sum_{F\\in Q} \\alpha_M(F).\n\t\\]\n\tLemma~\\ref{lem:alphaNXbelowC} yields that $ \\sum_{F\\in R} \\alpha_N(F) = 0$. All $F\\in S$ \n\thave $F\\BSET{e}\\in C$ with $F\\BSET{e}\\subsetneq X$\n\tand therefore those sets $F\\BSET{e}$ have shorter maximal descending\n\tchains in $C$ than $X$. \n\tThe induction hypothesis applied to each summand yields that\n\t\\[ \\sum_{F\\in S} \\alpha_N(F) = \\sum_{F\\in S} \\left( \\alpha_M(F\\BSET{e}) + \\tilde{\\Delta}\\alpha_M(C,F\\BSET{e}) \\right).\\]\n\tFurthermore, observe that $X\\notin Q$ because $\\mathrm{cl}_M(X) \\in C$ holds, and so we have the equivalence\n\t\\[ F \\sqsubset_M X \\quad\\Longleftrightarrow \\quad F\\in Q {\\mathop{\\text{~or~}}} F\\cup\\SET{e} \\in S \\]\n\tfor all $F\\subseteq E$: Elements $F$ of the $\\mathsf{A}_M$-down-set of $X$ have either\n\t $F\\in {\\mathcal{F}}(M)\\backslash C$ or $F\\in C$, thus either $F\\in Q$ or $F\\cup\\SET{e} \\in S$.\n\t Therefore we may cancel the corresponding summands of $\\downarrow_{\\mathsf{A}_M} X$ and drop the zero summands from $R$\n\t in the equation $(\\ast)$ and obtain\n\t \\begin{align*}\n\t \\alpha_M(X) + 1& = \\alpha_N(X) + \\sum_{F\\in S} \\tilde{\\Delta}\\alpha_M(C,F\\BSET{e}) .\\\\\n\t \\end{align*}\n\t Since all $F\\in S$ have $e\\in F$, and since \n\t \\[ \\SET{F\\BSET{e} \\vphantom{A^A}~\\middle|~ F\\in S} = \\SET{F\\in C \\vphantom{A^A}~\\middle|~ F\\subsetneq X} = \n\t \\SET{F\\subseteq E~\\middle|~ \\vphantom{A^A}F \\sqsubset^C X} \\]\n\t we obtain the desired equation\n\t \\[ \\alpha_N(X\\cup\\SET{e}) \\,\\,\\,=\\,\\,\\, \\alpha_M(X) + 1 - \\sum_{F\\sqsubset^C X} \\tilde{\\Delta}\\alpha_M(C,F) \\,\\,\\,=\\,\\,\\, \\alpha_M(X) + \\tilde{\\Delta}\\alpha_M(C,X). \\qedhere \\]\n\\end{proof}\n\n\\noindent\nWe implemented and tested the performance of determining the $\\alpha_N$-invariant for single element extensions $N\\in {\\mathcal{X}}(M,e)$ by means of the formulas\ngiven in Lemmas~\\ref{lem:alphaNXbelowC} and \\ref{lem:DPalpha}.\nFor details, please refer to Listing~\\ref{lst:measureDeltaAlpha}.\n\n\n\\section{Matroid Tableaux}\n\n\\PRFR{Mar 29th}\n\\noindent In this section, we present a general framework for the decision of $\\mathrm{Rec}\\Gamma_{{\\mathcal{M}}}$ instances\\footnote{Remember that in this chapter starting from Section~\\ref{sec:generalCase}, ${\\mathcal{M}}$ denotes the class of all matroids.}\nby searching the domain of matroids defined on ground sets with bounded cardinality by the means of tableaux and derivations. \n\n\n\\needspace{5\\baselineskip}\n\\begin{definition}\\PRFR{Mar 29th}\n\tA \\deftext{matroid tableau} is a tuple \\label{n:mattab} ${\\mathbf{T}} = (G,{\\mathcal{G}},{\\mathcal{M}},{\\mathcal{X}},\\simeq)$ where\n\t\\begin{enumerate}\\renewcommand{\\theenumi}{(\\roman{enumi})}\\renewcommand{\\labelenumi}{\\theenumi}\n\t\t\\item $G$ is a matroid, called the \\deftextX{goal of $\\bm {\\mathbf{T}}$},\n\t\t\\item ${\\mathcal{G}}$ is a family of matroids, called the \\deftextX{gammoids of $\\bm {\\mathbf{T}}$},\n\t\t\\item ${\\mathcal{M}}$ is a family of matroids, called the \\deftextX{intermediates of $\\bm {\\mathbf{T}}$},\n\t\t\\item ${\\mathcal{X}}$ is a family of matroids, called the \\deftextX{excluded matroids of $\\bm {\\mathbf{T}}$}, and where\n\t\t\\item $\\simeq$ is an equivalence relation on $\\SET{G'~\\middle|~ G'\\text{~is a minor of~}G}\\cup{\\mathcal{G}} \\cup {\\mathcal{M}} \\cup {\\mathcal{X}}$, called the \\deftextX{equivalence of $\\bm {\\mathbf{T}}$}. \\qedhere\n\t\\end{enumerate}\n\\end{definition}\n\n\\needspace{5\\baselineskip}\n\\begin{definition}\\label{def:validTableau}\\PRFR{Mar 29th}\n\tLet ${\\mathbf{T}} = (G,{\\mathcal{G}},{\\mathcal{M}},{\\mathcal{X}},\\simeq)$ be a matroid tableau.\n\t${\\mathbf{T}}$ shall be \\deftext[valid matroid tableau]{valid},\n\t\\begin{enumerate}\\renewcommand{\\theenumi}{(\\roman{enumi})}\\renewcommand{\\labelenumi}{\\theenumi}\n\t\\item if\n\tall matroids in ${\\mathcal{G}}$ are indeed gammoids,\n\t\\item if no matroid in ${\\mathcal{M}}$ is a strict gammoid,\n\t\\item if all matroids in ${\\mathcal{X}}$ are indeed matroids which are not gammoids, and\n\t\\item if for every equivalency classes $[M]_\\simeq$ of $\\simeq$ we have that either $[M]_\\simeq$ is fully contained in the class of gammoids\n\tor $[M]_\\simeq$ does not contain a gammoid. \\qedhere\n\\end{enumerate}\n\\end{definition}\n\n\\begin{definition}\\label{def:decisiveTableau}\\PRFR{Mar 29th}\n\tLet ${\\mathbf{T}} = (G,{\\mathcal{G}},{\\mathcal{M}},{\\mathcal{X}},\\simeq)$ be a matroid tableau. ${\\mathbf{T}}$ shall be \\deftext[decisive matroid tableau]{decisive},\n\tif ${\\mathbf{T}}$ is valid and \n\tif either of the following holds:\n\t\\begin{enumerate}\\renewcommand{\\theenumi}{(\\roman{enumi})}\\renewcommand{\\labelenumi}{\\theenumi} \n\t\\item There is a matroid $M\\in {\\mathcal{G}}$ such that $G \\simeq M$.\n\t\\item There is \n\ta matroid $X\\in {\\mathcal{X}}$ that is isomorphic to a minor of $G$.\n\t\\item For every extension $N=(E',{\\mathcal{I}}')$ of $G=(E,{\\mathcal{I}})$ \n\twith $$\\left| E' \\right| = \\mathrm{rk}_G(E)^2\\cdot \\left| E \\right| + \\mathrm{rk}_G(E) + \\left| E \\right|$$ there is a matroid $M\\in{\\mathcal{M}}$ that is\n\tisomorphic to $N$. \\qedhere\n\\end{enumerate}\n\\end{definition}\n\n\\needspace{3\\baselineskip}\n\\begin{lemma}\\label{lem:decisiveTableau}\\PRFR{Mar 29th}\n\tLet ${\\mathbf{T}} = (G,{\\mathcal{G}},{\\mathcal{M}},{\\mathcal{X}},\\simeq)$ be a decisive matroid tableau. Then $G$ is a gammoid if and only if there is a matroid $M\\in {\\mathcal{G}}$\n\tsuch that $G\\simeq M$.\n\\end{lemma}\n\\begin{proof}\\PRFR{Mar 29th}\n\tAssume that such an $M\\in {\\mathcal{G}}$ exists. From Definition~\\ref{def:validTableau}\n\twe obtain that $M$ is a gammoid, and that in this case $G\\simeq M$ implies that $G$ is a gammoid, too.\n\tNow assume that no $M\\in {\\mathcal{G}}$ has the property $G\\simeq M$. Since ${\\mathbf{T}}$ is decisive, either case {\\em (ii)} or {\\em (iii)} of\n\tDefinition~\\ref{def:decisiveTableau} holds. If case {\\em (ii)} holds, then $G$ cannot be a gammoid since it has a non-gammoid minor,\n\tbut the class of gammoids is closed under minors (Theorem~\\ref{thm:GammoidsClosedMinorsDuality}).\n\tIf case {\\em (iii)} holds but not case {\\em (ii)}, then no extension of $G=(E,{\\mathcal{I}})$ with $k=\\mathrm{rk}_G(E)^2\\cdot \\left| E \\right| $ $+\\, \\mathrm{rk}_G(E) + \\left| E \\right|$\n\telements is a strict gammoid. Now assume that $G$ is a gammoid, then there is a digraph $D=(V,A)$ with $\\left| V \\right| \\leq k$ vertices, such that\n\t$G = \\Gamma(D,T,E)$\n\tfor some $T\\subseteq V$ (Remark~\\ref{rem:upperBoundForV}). \n\tLet $N' = \\Gamma(D,T,V)\\oplus (\\SET{\\left| V \\right|,\\left| V \\right|+1,\\ldots,k},\\SET{\\emptyset})$. \n\tClearly, $N'$ is an extension of $G$ on a ground set with $k$ elements, which is also a strict gammoid, a contradiction to the assumption that $N'$ is \n\tisomorphic to some $N\\in {\\mathcal{M}}$, since ${\\mathcal{M}}$ is a family which consists of matroids that are not strict gammoids. Therefore we may conclude that in case {\\em (iii)} the matroid $G$ is not a gammoid.\n\\end{proof}\n\n\\subsection{Valid Derivations}\n\n\\PRFR{Mar 29th}\n\\noindent A \\deftext{derivation} is an operation on a finite number of input tableaux and possible additional parameters with constraints\nthat produces an output tableau. Furthermore,\na derivation is \\deftext[valid derivation]{valid}, if the output tableau is valid for all sets of valid input tableaux and possible additional parameters that\nsatisfy the constraints. The valid derivations presented here are fairly straight-forward consequences of the concepts presented earlier in this work.\n\n\\begin{definition}\\PRFR{Mar 29th}\n\tLet ${\\mathbf{T}}_i = (G_i,{\\mathcal{G}}_i,{\\mathcal{M}}_i,{\\mathcal{X}}_i,\\simeq^{(i)})$ be matroid tableaux for $i\\in \\SET{1,2,\\ldots,n}$.\n\tThe \\deftext{joint tableau} shall be the matroid tableaux \\label{n:jointTableau}\n\t\\[\\bigcup_{i=1}^{n} {\\mathbf{T}}_i = (G_1,{\\mathcal{G}},{\\mathcal{M}},{\\mathcal{X}},\\simeq)\\]\n\twhere \\[ {\\mathcal{G}} = \\bigcup_{i=1}^n {\\mathcal{G}}_i,\\,\\,\\, {\\mathcal{M}} = \\bigcup_{i=1}^n {\\mathcal{M}}_i,\\,\\,\\, {\\mathcal{X}} = \\bigcup_{i=1}^n {\\mathcal{X}}_i, \\]\n\tand where $\\simeq$ is the smallest equivalence relation such that $M \\simeq^{(i)} N$ implies $M \\simeq N$ for all $i\\in \\SET{1,2,\\ldots,n}$.\n\tIn other words, $\\simeq$ is the equivalence relation on the family of matroids\n\t $\\SET{G'~\\middle|~ G'\\text{~is a minor of~}G}\\cup{\\mathcal{G}} \\cup {\\mathcal{M}} \\cup {\\mathcal{X}}$ which is\n\tgenerated by the relations $ \\simeq^{(1)}, \\simeq^{(2)}, \\ldots, \\simeq^{(n)}$.\n\\end{definition}\n\n\\needspace{2\\baselineskip}\n\\begin{lemma}\\PRFR{Mar 29th}\n\tThe derivation of the joint tableau is valid.\n\\end{lemma}\n\\begin{proof}\\PRFR{Mar 29th}\n\tClearly, ${\\mathcal{G}}$, ${\\mathcal{M}}$, and ${\\mathcal{X}}$ inherit their desired properties of Definition~\\ref{def:validTableau} from\n\tthe valid input tableaux ${\\mathbf{T}}_i$ where $i\\in\\SET{1,2,\\ldots, n}$. Now let $M \\simeq N$ with $M\\not= N$.\n\tThen there are matroids $M_1,M_2,\\ldots,M_{k}$ and indexes $i_0,i_1,\\ldots,i_k\\in \\SET{1,2,\\ldots,n}$ such that\n\tthere is a chain of $\\simeq^{(i)}$-relations\n\t\\[ M \\simeq^{(i_0)} M_1 \\simeq^{(i_1)} M_2 \\simeq^{(i_2)} \\cdots \\simeq^{(i_{k-1})} M_k \\simeq^{(i_k)} N. \\]\n\tThe assumption that the input tableaux are valid yields that $M$ is a gammoid if and only if $M_1$ is a gammoid,\n\tif and only if $M_2$ is a gammoid, and so on. Therefore it follows that $M$ is a gammoid if and only if $N$ is a gammoid,\n\tthus $\\simeq$ has the desired property of Definition~\\ref{def:validTableau}. Consequently, $\\bigcup_{i=1}^n {\\mathbf{T}}_i$ is a valid tableau.\n\\end{proof}\n\n\\begin{definition}\\PRFR{Mar 29th}\n\tLet ${\\mathbf{T}} = (G,{\\mathcal{G}},{\\mathcal{M}},{\\mathcal{X}},\\simeq)$ and ${\\mathbf{T}}' = (G,{\\mathcal{G}}',{\\mathcal{M}}',{\\mathcal{X}}',\\simeq')$ be matroid tableaux.\n\tWe say that ${\\mathbf{T}}$ is a \\deftext[sub-tableau]{sub-tableau of ${\\mathbf{T}}\\bm'$} if ${\\mathcal{G}} \\subseteq {\\mathcal{G}}'$, ${\\mathcal{M}} \\subseteq {\\mathcal{M}}'$, and\n\t${\\mathcal{X}} \\subseteq {\\mathcal{X}}'$ holds, and if $M \\simeq N$ implies $M \\simeq' N$.\n\\end{definition}\n\n\\needspace{2\\baselineskip}\n\\begin{lemma}\\PRFR{Mar 29th}\n\tThe derivation of a sub-tableau is valid.\n\\end{lemma}\n\\begin{proof}\\PRFR{Mar 29th}\n\tClearly ${\\mathbf{T}}$ inherits the properties of Definition~\\ref{def:validTableau} from the validity of ${\\mathbf{T}}'$.\n\\end{proof}\n\n\\begin{definition}\\PRFR{Mar 29th}\n\tLet ${\\mathbf{T}} = (G,{\\mathcal{G}},{\\mathcal{M}},{\\mathcal{X}},\\simeq)$ be a matroid tableau. We shall call the\n\t\\label{n:expTab}\n\t tableau $[{\\mathbf{T}}]_\\simeq= (G,{\\mathcal{G}}',{\\mathcal{M}},{\\mathcal{X}}',\\simeq)$ \\deftext[expansion tableau]{expansion tableau of ${\\mathbf{T}}$} \n\twhenever \\[ {\\mathcal{G}}' = \\bigcup_{M\\in{\\mathcal{G}}} [M]_\\simeq \\quad{\\mathop{\\text{~and~}}}\\quad {\\mathcal{X}}' = \\bigcup_{M\\in{\\mathcal{X}}} [M]_\\simeq. \\qedhere\\]\n\\end{definition}\n\n\\needspace{2\\baselineskip}\n\\begin{lemma}\\PRFR{Mar 29th}\n\tThe derivation of the expansion tableau is valid.\n\\end{lemma}\n\\begin{proof}\\PRFR{Mar 29th}\n\tIf $M'\\in {\\mathcal{G}}'$, then there is some $M\\in {\\mathcal{G}}$ such that $M\\simeq M'$. Since we assume ${\\mathbf{T}}$ to be valid, we may infer that\n\t$M'$ is a gammoid if and only if $M$ is a gammoid, and the latter is the case since $M\\in{\\mathcal{G}}$. Therefore $M'$ is a gammoid.\n\tAn analogous argument yields that if $M'\\in {\\mathcal{X}}'$, then $M'$ is not a gammoid.\n\\end{proof}\n\n\\needspace{5\\baselineskip}\n\\begin{definition}\\PRFR{Mar 29th}\n Let ${\\mathbf{T}} = (G,{\\mathcal{G}},{\\mathcal{M}},{\\mathcal{X}},\\simeq)$ be a matroid tableau. We shall call the\n \\label{n:extTab}\n\t tableau $[{\\mathbf{T}}]_{\\equiv} = (G,{\\mathcal{G}}',{\\mathcal{M}}',{\\mathcal{X}}',\\simeq')$ \\deftext[extended tableau]{extended tableau of $\\bm {\\mathbf{T}}$}\n\t whenever $${\\mathcal{G}}' = {\\mathcal{G}} \\cup \\SET{M^\\ast~\\middle|~M\\in {\\mathcal{G}}},\\,\\,\\,\n\t \t\t {\\mathcal{X}}' = {\\mathcal{X}} \\cup \\SET{M^\\ast~\\middle|~M\\in {\\mathcal{X}}},\\,\\,\\, \n\t {\\mathcal{M}}' = {\\mathcal{M}} \\cup {\\mathcal{X}}',$$ and when \n\t $\\simeq'$ is the smallest equivalence relation that contains the \n\t relations $\\simeq$ and $\\sim$; where $M\\sim N$ if and only if $N$ is\n\t isomorphic to $M$ or $M^\\ast$.\n\\end{definition}\n\n\\needspace{2\\baselineskip}\n\\begin{lemma}\\PRFR{Mar 29th}\n\tThe derivation of the extended tableau is valid.\n\\end{lemma}\n\\begin{proof}\\PRFR{Mar 29th}\n\tBy Theorem~\\ref{thm:GammoidsClosedMinorsDuality} the class of gammoids is closed under duality, therefore a matroid $M$ is a gammoid if and only if $M^\\ast$ is a gammoid. So ${\\mathcal{G}}'$ and ${\\mathcal{X}}'$ inherit their desired properties of Definition~\\ref{def:validTableau} from the validity of ${\\mathbf{T}}$.\n\tIf $M\\in {\\mathcal{M}}'\\backslash {\\mathcal{M}}$, then $M\\in {\\mathcal{X}}'$, therefore $M$ cannot be a strict gammoid.\n\\end{proof}\n\n\n\n\n\n\\begin{definition}\\label{def:decisionTableau}\\PRFR{Mar 29th}\n\tLet ${\\mathbf{T}} = (G,{\\mathcal{G}},{\\mathcal{M}},{\\mathcal{X}},\\simeq)$ be a decisive matroid tableau. The tableau\n\t\\label{n:concTab}\n\t ${\\mathbf{T}}! = (G,{\\mathcal{G}}',{\\mathcal{M}},{\\mathcal{X}}',\\simeq)$ shall be the \\deftext[conclusion tableau]{conclusion tableau for ${\\mathbf{T}}$} if either\n\t\\begin{enumerate}\\renewcommand{\\theenumi}{(\\roman{enumi})}\\renewcommand{\\labelenumi}{\\theenumi}\n\t\\item ${\\mathcal{G}}' = {\\mathcal{G}} \\cup \\SET{G'~\\middle|~ G'\\text{~is a minor of~} G}$, ${\\mathcal{X}}' = {\\mathcal{X}}$, and the tableau ${\\mathbf{T}}$ \n\tsatisfies case {(i)} of Definition~\\ref{def:decisiveTableau}; or\n\t\\item ${\\mathcal{G}}' = {\\mathcal{G}}$, ${\\mathcal{X}}' = {\\mathcal{X}}\\cup\\SET{G}$, and ${\\mathbf{T}}$ satisfies case {(ii)} or {(iii)} of Definition~\\ref{def:decisiveTableau}. \\qedhere\n\t\\end{enumerate}\n\\end{definition}\n\n\\needspace{2\\baselineskip}\n\\begin{corollary}\\PRFR{Mar 29th}\n\tThe derivation of the conclusion tableau is valid.\n\\end{corollary}\n\\begin{proof}\\PRFR{Mar 29th}\n\tEasy consequence of Lemma~\\ref{lem:decisiveTableau}.\n\\end{proof}\n\n\\begin{definition}\\PRFR{Mar 29th}\n\tLet ${\\mathbf{T}} = (G,{\\mathcal{G}},{\\mathcal{M}},{\\mathcal{X}},\\simeq)$ be a matroid tableau, let $M_1=(E_1,{\\mathcal{I}}_1)$ and $M_2=(E_2,{\\mathcal{I}}_2)$ be matroids of the\n\ttableau, i.e.\n\t\\[ \\SET{M_1,M_2}\\subseteq \\SET{G'~\\middle|~ G'\\text{~is a minor of~}G}\\cup{\\mathcal{G}} \\cup {\\mathcal{M}} \\cup {\\mathcal{X}} .\\]\n\tFurthermore, let $E_1'$ and $E_2'$ be finite sets,\n\t$D_1=(V_1,A_1)$ and $D_2=(V_2,A_2)$ be digraphs such that $E_1 \\cup E_2' \\subseteq V_1$ \n\tand $E_1'\\cup E_2\\subseteq V_2$, and such that\n\tthe induced matroid \\( I(D_1,M_1,E_2')\\) is isomorphic to \\( M_2 \\) \n\tand the the induced matroid \\( I(D_2,M_2,E_1') \\) is isomorphic to \\(M_1\\).\n\tThe tableau\n\t\\label{n:idTab}\n\t $${\\mathbf{T}}(M_1\\simeq M_2) = (G,{\\mathcal{G}},{\\mathcal{M}},{\\mathcal{X}},\\simeq')$$ is called \n\t\\deftext[identified tableau]{identified tableau for ${\\mathbf{T}}$ with respect to $\\bm M_1$ and $\\bm M_2$} if\n\tthe relation\n\t$\\simeq'$ is the smallest equivalence relation, such that $M_1\\simeq' M_2$ holds, and such that $M'\\simeq N'$ implies $M'\\simeq' N'$.\n\\end{definition}\n\n\\needspace{2\\baselineskip}\n\\begin{lemma}\\PRFR{Mar 29th}\n\tThe derivation of an identified tableau is valid.\n\\end{lemma}\n\\begin{proof}\\PRFR{Mar 29th}\n\tFrom Lemma~\\ref{lem:digraphInducedGammoidIfTisGammoid} we obtain that $M_2' = I(D_1,M_1,E_2')$ is a gammoid if $M_1$ is a gammoid,\n\tand that $M_1' = I(D_2,M_2,E_1')$ is a gammoid if $M_2$ is a gammoid. Therefore $M_1$ is a gammoid if and only if $M_2$ is a gammoid.\n\tConsequently, $\\simeq'$ satisfies the properties of Definition~\\ref{def:validTableau}, and thus the identified tableau ${\\mathbf{T}}(M_1 \\simeq M_2)$ is valid\n\tfor every valid input tableau ${\\mathbf{T}}$.\n\\end{proof}\n\n\\subsection{Valid Tableaux}\n\n\\PRFR{Mar 29th}\n\\noindent In this section we present a variety of valid tableaux which may be used as inputs for valid derivation operations.\nTrivially, if $M$ is a gammoid, then $(M,\\SET{M},\\emptyset,\\emptyset,\\langle\\,\\rangle)$ is a valid tableau,\nand if $M$ is not a gammoid, then $(M,\\emptyset,\\emptyset,\\SET{M},\\langle\\,\\rangle)$ is a valid tableau;\n where $\\langle .\\rangle$ denotes the generated equivalence relation defined on the set of matroids occurring in the respective tableau. Thus exactly one of these two tableaux\nis valid. Unfortunately, in order to know which one is valid, we have to decide whether $M$ is a gammoid first ---\nin general this is not easier than determining $\\Gamma_{\\mathcal{M}}(M)$, but there are special cases which we should not ignore.\n\n\\begin{corollary}\\label{cor:strictGammoidTableau}\\PRFR{Mar 29th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid with $\\alpha_M \\geq 0$. Then the matroid tableau\n\t${\\mathbf{T}}$ is valid, where\n\t${\\mathbf{T}} = (M,{\\mathcal{G}},{\\mathcal{M}},{\\mathcal{X}},\\simeq)$ with\n\t\\( {\\mathcal{G}} = \\SET{M,M^\\ast}\\), ${\\mathcal{M}}= \\emptyset$, ${\\mathcal{X}}=\\emptyset$, and $M\\simeq N \\Leftrightarrow M=N$.\n\\end{corollary}\n\\begin{proof}\\PRFR{Mar 29th}\nSee Corollary~\\ref{cor:MasonAlpha}.\n\\end{proof}\n\n\n\\begin{corollary}\\PRFR{Mar 29th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid with $\\mathrm{rk}_M(X) = 3$, $X\\subseteq E$ with $\\alpha_M(X) < 0$. Then the matroid tableau\n\t${\\mathbf{T}}$ is valid, where\n\t${\\mathbf{T}} = (M,{\\mathcal{G}},{\\mathcal{M}},{\\mathcal{X}},\\simeq)$ with\n\t\\( {\\mathcal{G}} = \\emptyset\\), ${\\mathcal{M}}= \\emptyset$, ${\\mathcal{X}}=\\SET{M,M^\\ast}$, and $M\\simeq N \\Leftrightarrow M=N$.\n\\end{corollary}\n\\begin{proof}\\PRFR{Mar 29th}\nSee Corollary~\\ref{cor:MasonAlpha} and Proposition~\\ref{prop:cornerCases}.\n\\end{proof}\n\n\\begin{remark}\\label{rem:nonStrictGammoidTableau}\\PRFR{Mar 29th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, $X\\subseteq E$ with $\\alpha_M(X) < 0$. Then the matroid tableau\n\t${\\mathbf{T}}$ is valid, where\n\t${\\mathbf{T}} = (M,{\\mathcal{G}},{\\mathcal{M}},{\\mathcal{X}},\\simeq)$ with\n\t\\( {\\mathcal{G}} = \\emptyset\\), ${\\mathcal{M}}= \\SET{M}$, ${\\mathcal{X}}=\\emptyset$, and $M\\simeq N \\Leftrightarrow M=N$.\n\\end{remark}\n\n\\needspace{4\\baselineskip}\n\\begin{theorem}[\\cite{In77}, Theorem~13; \\cite{Brylawski1971}, \\cite{Brylawski1975}, \\cite{Ingleton1971}]\\label{thm:graphicGammoidsAreSeriesParallel}\\PRFR{Mar 29th}\n\tLet ${\\mathbb{F}}_2$ be the two-elementary field, $E,C$ finite sets, and let $\\mu \\in {\\mathbb{F}}_2^{E\\times C}$ be a matrix.\n\tThen $M(\\mu)$ is a gammoid if and only if there is no minor $N$ of $M(\\mu)$ which is isomorphic to $M(K_4)$.\n\tThe latter is the case if and only if $M(\\mu)$ is isomorphic to the polygon matroid of a series-parallel network.\n\\end{theorem}\n\n\\noindent For proofs of a sufficient set of implications which establish the equivalency stated, refer to\n \\cite{Brylawski1971}, \\cite{Brylawski1975}, and \\cite{Ingleton1971}.\n\n \\begin{theorem}[\\cite{Ox11}, Theorem~6.5.4]\\label{thm:binaryMatroids}\\PRFR{Mar 29th}\n \tLet $M=(E,{\\mathcal{I}})$ be a matroid. Then $M$ is isomorphic to $M(\\mu)$ for some matrix $\\mu \\in {\\mathbb{F}}_2^{E\\times C}$\n \tif and only if $M$ has no minor isomorphic to the uniform matroid $U_{2,4} = \\left( E',{\\mathcal{I}}' \\right)$,\n \twhere $E'= \\dSET{a,b,c,d}$ and \\linebreak ${\\mathcal{I}}' = \\SET{X\\subseteq E'~\\middle|~ \\left| X \\right| \\leq 2}$.\n \\end{theorem}\n \\noindent See \\cite{Ox11}, pp.193f, for a proof.\n\n\n\\begin{corollary}\\PRFR{Mar 29th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid.\n\tIf $M$ has no minor isomorphic to $M(K_4)$ and no minor isomorphic to $U_{2,4}$, then the matroid tableau\n\t${\\mathbf{T}}$ is valid, where\n\t${\\mathbf{T}} = (M,{\\mathcal{G}},{\\mathcal{M}},{\\mathcal{X}},\\simeq)$ with\n\t\\( {\\mathcal{G}} = \\SET{M,M^\\ast}\\), ${\\mathcal{M}}= \\emptyset$, ${\\mathcal{X}}=\\emptyset$, and $M\\simeq N \\Leftrightarrow M=N$.\n\\end{corollary}\n\\begin{proof}\\PRFR{Mar 29th}\nDirect consequence of Theorems~\\ref{thm:graphicGammoidsAreSeriesParallel} and \\ref{thm:binaryMatroids}.\n\\end{proof}\n\n\\begin{definition}\\PRFR{Mar 29th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid. Then $M$ shall be \\deftext{strongly base-orderable}, if for every pair of bases\n\t$B_1,B_2\\in{\\mathcal{B}}(M)$ there is a bijection $\\phi\\colon B_1\\longrightarrow B_2$ such that\n\t\\[ \\left( B_1\\backslash X \\right) \\cup \\phi[X] \\in {\\mathcal{B}}(M) \\]\n\tholds\n\tfor all $X\\subseteq B_1$. This property is also referred to as \\deftext{full exchange property}.\n\\end{definition}\n\n\\begin{lemma}[\\cite{M72}, Corollary~4.1.4]\\label{lem:GammoidsAreStronglyBaseOrderable}\\PRFR{Mar 29th}\n\tLet $M=(E,{\\mathcal{I}})$ be a gammoid. Then $M$ is strongly base-orderable.\n\\end{lemma}\n\\begin{proof}\\PRFR{Mar 29th}\n\tLet $B_1,B_2\\in {\\mathcal{B}}(M)$ be any two bases of $M$,\n\tand let $(D,B_1,E)$ be a representation of $M$ (Theorem~\\ref{thm:gammoidRepresentationWithBaseTerminals}).\n\tSince $B_2\\in {\\mathcal{I}}$ and $\\left| B_1 \\right| = \\left| B_2 \\right|$, there is a linking $R\\colon B_2\\double{\\rightarrow} B_1$ in $D$.\n\tLet $\\phi\\colon B_1\\longrightarrow B_2$ be the unique bijection with the property that $p_{1} = \\phi(p_{-1})$ for all $p\\in R$.\n\tLet $X\\subseteq B_1$, then the derived linking $$R_X = \\SET{p\\in R ~\\middle|~ p_1 \\in \\phi[X]} \\cup \\SET{b\\in B_1~\\middle|~b\\notin X}$$\n\tproves that\n\t$\\left( B_1\\backslash X \\right)\\cup \\phi[X] \\in {\\mathcal{B}}(M)$. Thus $M$ is strongly base-orderable.\n\\end{proof}\n\n\\begin{corollary}\\PRFR{Mar 29th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, $B_1,B_2\\in {\\mathcal{B}}(M)$ be bases of $M$ such that for every bijection\n\t$\\phi\\colon B_1\\backslash B_2 \\longrightarrow B_2\\backslash B_1$ there is a set $X\\subseteq B_1\\backslash B_2$\n\twith the property $ \\left( B_1\\backslash X \\right)\\cup \\phi[X] \\notin {\\mathcal{B}}(M)$. Then the matroid tableau\n\t${\\mathbf{T}}$ is valid, where\n\t${\\mathbf{T}} = (M,{\\mathcal{G}},{\\mathcal{M}},{\\mathcal{X}},\\simeq)$ with\n\t\\( {\\mathcal{G}} = \\emptyset\\), ${\\mathcal{M}}= \\emptyset$, ${\\mathcal{X}}=\\SET{M,M^\\ast}$, and $M\\simeq N \\Leftrightarrow M=N$.\n\\end{corollary}\n\\begin{proof}\\PRFR{Mar 29th}\n\tDirect consequence of the proof of Lemma~\\ref{lem:GammoidsAreStronglyBaseOrderable}.\n\\end{proof}\n\n\\needspace{4\\baselineskip}\n\\vspace*{-\\baselineskip}\n\\begin{wrapfigure}{r}{4.5cm}\n\\vspace{\\baselineskip}\n\\begin{centering}~~\n\\includegraphics{P7}\n\\end{centering}%\n\\vspace*{-\\baselineskip}\n\\end{wrapfigure}\n~\n \n\n\\begin{example}\\PRFR{Mar 29th}\n\tConsider the matroid $P_7$ (\\cite{Ox11}, p.644), its affine configuration is depicted on the right.\n\tIt is strongly base-orderable\n\tbut it is not a strict gammoid. \n\tSince $P_7$ has rank $3$, it follows that $P_7$ is a strongly base-orderable non-gammoid (Proposition~\\ref{prop:cornerCases}).\n\\end{example}\n\n\\begin{example}\\PRFR{Mar 29th}\n\tThe V\u00e1mos matroid (\\cite{Ox11}, p.649) is strongly base-orderable, but not representable over the reals $\\mathbb{R}$.\n\\end{example}\n\n\n\\begin{theorem}[\\cite{Ingleton69}, \\cite{MNW09}]\\label{thm:IngeltonsCondition}\\PRFR{Mar 29th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid such that there is a field ${\\mathbb{F}}$ and a matrix $\\mu\\in{\\mathbb{F}}^{E\\times C}$\n\twith $M= M(\\mu)$. Let further $W,X,Y,Z\\subseteq E$. Then\n\t\\begin{align*}\n\t\t\\mathrm{rk}(W) \\,+\\, &\\mathrm{rk}(X) + \\mathrm{rk}(W\\cup X\\cup Y) + \\mathrm{rk}(W\\cup X\\cup Z) + \\mathrm{rk}(Y\\cup Z) \\\\\n\t\t& \\leq \\mathrm{rk}(W\\cup X) + \\mathrm{rk}(W\\cup Y) + \\mathrm{rk}(W\\cup Z) + \\mathrm{rk}(X\\cup Y) + \\mathrm{rk}(X\\cup Z).\n\t\\end{align*}\n\\end{theorem}\n\n\\PRFR{Mar 31st}\n\\noindent\nFor a proof, see \\cite{Ingleton69}. We mention A.W.~Ingleton's theorem here\nbecause it has been used by D.~Mayhew in \\cite{Ma16} in order to prove\n that certain matroids are excluded minors of the class of gammoids.\nP.~Nelson and J.~van~der~Pol \\cite{NvdP17} showed that A.W.~Ingleton's necessary condition for representability over any field\nis rather weak: It is quite improbable for matroids on large ground sets that\na matroid which satisfies this condition is indeed representable over any field, because there are double-exponentially many\nmatroids satisfying the condition with respect to the cardinality of the ground set, \nyet there are only exponentially many representable\nmatroids with respect to the cardinality of the ground set. Furthermore, L.~Guill\u00e9, T.~Chan, and A.~Grant found a unique\nminimal subset of $\\frac{6^n}{4} - O(5^n)$ inequalities that imply A.W.~Ingleton's condition if satisfied \\cite{GCG11}.\n\n\\needspace{5\\baselineskip}\n\\begin{corollary}\\PRFR{Mar 29th}\\label{cor:ingletonTab}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, and let $W,X,Y,Z\\subseteq E$\n\tsuch that\n\t\\begin{align*}\n\t\t\\mathrm{rk}(W) \\,+\\, &\\mathrm{rk}(X) + \\mathrm{rk}(W\\cup X\\cup Y) + \\mathrm{rk}(W\\cup X\\cup Z) + \\mathrm{rk}(Y\\cup Z) \\\\\n\t\t& > \\mathrm{rk}(W\\cup X) + \\mathrm{rk}(W\\cup Y) + \\mathrm{rk}(W\\cup Z) + \\mathrm{rk}(X\\cup Y) + \\mathrm{rk}(X\\cup Z).\n\t\\end{align*}\n\tThen the matroid tableau\n\t${\\mathbf{T}}$ is valid, where\n\t${\\mathbf{T}} = (M,{\\mathcal{G}},{\\mathcal{M}},{\\mathcal{X}},\\simeq)$ with\n\t\\( {\\mathcal{G}} = \\emptyset\\), ${\\mathcal{M}}= \\emptyset$, ${\\mathcal{X}}=\\SET{M,M^\\ast}$, and $M\\simeq N \\Leftrightarrow M=N$.\n\\end{corollary}\n\\begin{proof}\\PRFR{Mar 29th}\n\tConsequence of Theorems~\\ref{thm:IngeltonsCondition} and \\ref{thm:gammoidOverR}.\n\\end{proof}\n\n\\PRFR{Mar 31st}\n\\noindent This strict inequality dualizes to\n\\begin{align*}\n\t\\nu(W') \\,+\\, & \\nu(X') + \\nu\\left( W'\\cap X'\\cap Y' \\right) + \n\t\\nu\\left( W'\\cap Y'\\cap Z' \\right) + \\nu\\left( Y'\\cap Z' \\right)\n\t\\\\\n\t < \\,& \\hphantom{+\\,} \\nu\\left( W'\\cap X' \\right)+ \\nu\\left( W'\\cap Y' \\right)+ \n\t\\nu\\left( W'\\cap Z' \\right) + \\nu\\left( X'\\cap Y' \\right)+ \\nu\\left( X'\\cap Z'\\right)\n\\end{align*}\nwhere $\\nu(X) = \\left| X \\right| - \\mathrm{rk}(X)$. A.~Cameron showed that $M$ satisfies A.W.~Ingleton's condition\nif and only if $M^\\ast$ satisfies it (\\cite{Ca14Msc}, Lemma~4.5, p.26), therefore the dualized inequality\ndoes not provide\nany valid matroid tableaux that cannot be derived using Corollary~\\ref{cor:ingletonTab}.\n\n\\subsection{Derivation of a Decisive Tableau}\n\n\\PRFR{Mar 29th}\n\\noindent First of all, it is clear that we may derive a decisive matroid tableau for any given matroid $G=(E,{\\mathcal{I}})$\nby simply determining all extensions of $G$ with $\\mathrm{rk}_G(E)^2\\cdot \\left| E \\right| + \\mathrm{rk}_G(E) + \\left| E \\right|$ \nelements (Remark~\\ref{rem:upperBoundForV}). For each such extension $N$, there is a valid tableau, which depends on whether $N$\nis a strict gammoid (Corollary~\\ref{cor:strictGammoidTableau}) or not (Remark~\\ref{rem:nonStrictGammoidTableau}).\nThus we may derive the joint tableau of all valid tableaux of the extensions of $G$. It is clear from Definition~\\ref{def:decisiveTableau} \nthat this joint tableau is decisive: either case {\\em (i)} or case {\\em (iii)} holds. Thus we may always decide $\\Gamma_{\\mathcal{M}}(G)$ using the\nmatroid tableau method. Unfortunately, we cannot guarantee that there is no excluded minor $X$ for the class of gammoids, where the only\nfeasible way to refute, that $X$ is a gammoid, requires to employ the tiresome case {\\em (iii)}. Now, let us provide a glimpse of the art\nof employing matroid tableaux.\n\n\\begin{example}\\label{ex:tab}\\PRFR{Mar 29th}\n\tConsider the matroid $G = G_{8,4,1} = (E,{\\mathcal{I}})$ where $E=\\SET{1,2,\\ldots,8}$ and where ${\\mathcal{I}} = \\SET{\\vphantom{A^A}X\\subseteq E ~\\middle|~ \\left| X \\right|\\leq 4,\\,X\\notin {\\mathcal{H}}}$ with\n\t\\[{\\mathcal{H}} = \\SET{\\vphantom{A^A} \\SET{1, 3, 7, 8},\n \\SET{1, 5, 6, 8},\n \\SET{2, 3, 6, 8},\n \\SET{4, 5, 6, 7},\n \\SET{2, 4, 7, 8}}.\\]\n Clearly, $\\alpha_G(H) = 1$ for all $H\\in {\\mathcal{H}}$, and consequently $\\alpha_G(E) = 4-5 = -1$.\n The dual matroid $G^\\ast = (E,{\\mathcal{I}}^\\ast)$ has a similar structure:\n ${\\mathcal{I}}^\\ast = \\SET{\\vphantom{A^A}X\\subseteq E ~\\middle|~ \\left| X \\right|\\leq 4,\\,X\\notin {\\mathcal{H}}^\\ast}$ with\n\t\\[{\\mathcal{H}}^\\ast = \\SET{\\vphantom{A^A} \\SET{1, 2, 3, 8},\n \\SET{1, 3, 5, 6},\n \\SET{1, 4, 5, 7},\n \\SET{2, 3, 4, 7},\n \\SET{2, 4, 5, 6}}.\\]\n Thus $\\alpha_{G^\\ast}(H') = 1$ for all $H'\\in{\\mathcal{H}}^\\ast$, and so $\\alpha_{G^\\ast}(E) = 4-5 = -1$, too.\n It turns out that neither $G$ nor $G^\\ast$ have any minors of rank $3$ which are not strict gammoids.\n Furthermore, both $G$ and $G^\\ast$ are strongly base-orderable, and both $G$ and $G^\\ast$ have a $U_{2,4}$ minor.\n For the rest of this example, we will refer to `single-element extensions of the same rank' simply by the word `extension'.\n There are\n $11962$ different isomorphism classes of extensions of $G$, and $11495$ different isomorphism classes of\n extensions of $G^\\ast$. No extension of $G$ or $G^\\ast$ is a strict gammoid or a transversal matroid.\n $8643$ isomorphism classes of $G$-extensions either have non-gammoid rank-$3$ minors, or they are not strongly base-orderable, the same holds for\n $7892$ isomorphism classes of $G^\\ast$-extensions. This leaves $3319$ classes of $G$-extensions and $3603$ classes of $G^\\ast$-extensions\n which may or may not be classes of gammoids --- so extending and backtracking may not be our best approach here.\n\n \\noindent We have seen before that there is no easy way to decide whether $G$ or $G^\\ast$ is a gammoid, therefore we start with the\n valid tableaux $${\\mathbf{T}}_G = \\left( G,\\emptyset,\\SET{G},\\emptyset,\\langle \\, \\rangle \\right)\n {\\mathop{\\text{~and~}}} {\\mathbf{T}}_{G^\\ast} = \\left( G^\\ast,\\emptyset,\\SET{G^\\ast},\\emptyset,\\langle \\, \\rangle \\right),$$\n where $\\langle .\\rangle$ denotes the generated equivalence relation defined on the set of matroids occurring in the respective tableau.\n We may derive the extended joint tableau \n $$ {\\mathbf{T}}_1 = [{\\mathbf{T}}_G \\cup {\\mathbf{T}}_{G^\\ast}]_\\equiv = \\left( G, \\emptyset,\\SET{G,G^\\ast}, \\langle G\\simeq G^\\ast\\rangle \\right).$$\n Now observe that although $G$ is deflated, $G^\\ast$ is not deflated. We have\n \t \\begin{align*}C^\\ast_8 & = \\SET{\\vphantom{A^A}F\\in {\\mathcal{F}}\\left( G^\\ast| \\SET{1,2,\\ldots,7} \\right)~\\middle|~ 8 \\in \\mathrm{cl}_{G^\\ast}(F)}\n \t \\\\ & = \\SET{\\vphantom{A^A}F\\in {\\mathcal{F}}\\left( G^\\ast| \\SET{1,2,\\ldots,7} \\right)~\\middle|~\\SET{1,2,3}\\subseteq F}.\n \\end{align*}\n Let $G^\\ast_7 = G^\\ast| \\SET{1,2,\\ldots,7}$. We have $\\alpha_{G^\\ast_7}(\\SET{1,2,\\ldots,7}) = -1$, thus $G^\\ast_7$ is not a \n strict gammoid, and thus\n \\[ {\\mathbf{T}}_{G^\\ast_7} = \\left(G^\\ast_7, \\emptyset, \\SET{G^\\ast_7}, \\emptyset, \\langle\\,\\rangle\\right) \\]\n is a valid tableau. Since $G^\\ast_7$ is a deflate of $G^\\ast$, each of them is an induced matroid with respect to the other. Therefore\n we may identify $G^\\ast$ and $G^\\ast_7$ in the joint tableau\n \\[ {\\mathbf{T}}_2 = \\left( {\\mathbf{T}}_1\\cup {\\mathbf{T}}_{G^\\ast_7} \\right)(G^\\ast\\simeq G^\\ast_7) = \\left( G,\\emptyset,\\SET{G,G^\\ast,G^\\ast_7},\\emptyset,\\langle G \\simeq G^\\ast \\simeq G^\\ast_7\\rangle \\right).\\]\n Now let $G_7 = \\left( G^\\ast_7 \\right)^\\ast$, and we have $\\alpha_{G_7} \\geq 0$. Thus\n \\[ {\\mathbf{T}}_{G_7} = \\left( G_7, \\SET{G_7},\\emptyset, \\emptyset, \\langle\\,\\rangle \\right) \\]\n is a valid tableau. We now may derive the decisive tableau\n \\[ {\\mathbf{T}}_3 = [{\\mathbf{T}}_2 \\cup {\\mathbf{T}}_{G_7}]_\\equiv = \\left( G, \\SET{G_7}, \\SET{G,G^\\ast,G^\\ast_7,G_7},\\emptyset, \\langle G\\simeq G^\\ast \\simeq G^\\ast_7 \\simeq G_7\\rangle \\right) \\]\n where case {\\em (i)} of Definition~\\ref{def:decisiveTableau} holds. Consequently, $G$ is a gammoid.\n\\end{example}\n\n\\begin{figure}\n\\includegraphics[width=\\textwidth]{tableauEx}\n\\caption{\\label{fig:tab}Reconstruction of a representation of $G_{8,4,1}$ from the matroid tableaux in Example~\\ref{ex:tab}.}\n\\end{figure}\n\n\\PRFR{Mar 31st}\n\\noindent The representation of $G_{8,4,1}$ given in Figure~\\ref{fig:tab} can obviously be reduced by two vertices \nif we move both the vertices $6$ and $7$ one step along their only incident arcs and delete the now superfluous sources. \nSo $G_{8,4,1}$ may be represented with $11$ vertices,\nand it is still possible that there is a representation of $G_{8,4,1}$ with $10$ vertices. Clearly, $9$ vertices do not suffice since no\nsingle-element extension of $G_{8,4,1}$ is a strict gammoid.\n\n\\bigskip \n\\PRFR{Mar 31st}\n\\noindent\nBased on our experience, let us provide our best procedure for determining whether a given matroid $G=(E,{\\mathcal{I}})$ is a gammoid.\nWe start the procedure with the valid initial tableau ${\\mathbf{T}} := (G,\\emptyset,\\emptyset,\\emptyset,\\langle\\,\\rangle)$.\n\n\n\\begin{step}\\label{step:start}\\PRFR{Mar 31st}\n\tIf ${\\mathbf{T}}$ is decisive, stop.\n\\end{step}\n\n\\begin{step}\\PRFR{Mar 31st}\n\tChoose an intermediate goal\n\t$M\\in \\left( \\SET{G' ~\\middle|~ G'\\text{~is a minor of~}G} \\cup {\\mathcal{M}} \\right)\\backslash \\left( {\\mathcal{G}} \\cup {\\mathcal{X}} \\right)$,\n\tpreferably one with $M\\simeq G$ which is small both in rank and cardinality.\n\\end{step}\n\n\\begin{step}\\PRFR{Mar 31st}\n\tIf ${\\mathbf{T}}_M = (M,\\emptyset,\\emptyset,\\emptyset,\\langle\\,\\rangle)\\cup {\\mathbf{T}}$ is decisive,\n\tthen set ${\\mathbf{T}} := \\left[\\vphantom{A^A}[{\\mathbf{T}} \\cup \\left( {\\mathbf{T}}_M! \\right)]_\\equiv\\right]_\\simeq$\n\t and continue with Step~\\ref{step:start}.\n\\end{step}\n\n\\begin{step}\\label{step:choice}\n\tDetermine whether $M$ has a minor that is isomorphic to $M(K_4)$. If this is the case, then \n\t${\\mathbf{T}}_M = \\left( M, \\emptyset, \\emptyset, \\SET{M,M^\\ast}, \\langle\\,\\rangle \\right)$ is valid, we set\n\t ${\\mathbf{T}} := \\left[\\vphantom{A^A}[{\\mathbf{T}}\\cup {\\mathbf{T}}_M]_\\equiv \\right]_\\simeq$ and then\n\tcontinue with Step~\\ref{step:start}.\n\\end{step}\n\n\\noindent Since $M(K_4) = \\left( M(K_4) \\right)^\\ast$, we have that $M(K_4)$ is neither a minor of $M$ nor of $M^\\ast$\nwhen reaching the next step.\n\n\\begin{step}\\label{step:chosen}\\PRFR{Mar 31st}\n\tDetermine whether $M$ has a minor that is isomorphic to $U_{2,4}$. If this is not the case, then \n\t${\\mathbf{T}}_M = \\left( M, \\SET{M,M^\\ast}, \\emptyset, \\emptyset, \\langle\\,\\rangle \\right)$ is valid, we set\n\t ${\\mathbf{T}} := \\left[\\vphantom{A^A}[{\\mathbf{T}}\\cup {\\mathbf{T}}_M]_\\equiv \\right]_\\simeq$ and then\n\tcontinue with Step~\\ref{step:start}.\n\\end{step}\n\n\\noindent Since $U_{2,4} = \\left( U_{2,4} \\right)^\\ast$, we have that $U_{2,4}$ is neither a minor of $M$ nor of $M^\\ast$\nwhen reaching the next step.\n\n\\begin{step}\\PRFR{Mar 31st}\n\tIf $M\\in {\\mathcal{M}}$, continue immediately with Step~\\ref{step:nonStrict}.\n\tDetermine whether $\\alpha_M \\geq 0$. If this is the case, then \n${\\mathbf{T}}_M = \\left( M, \\SET{M,M^\\ast}, \\emptyset, \\emptyset, \\langle\\,\\rangle \\right)$ is valid, we set\n\t ${\\mathbf{T}} := \\left[\\vphantom{A^A}[{\\mathbf{T}}\\cup {\\mathbf{T}}_M]_\\equiv \\right]_\\simeq$ and\n\tcontinue with Step~\\ref{step:start}.\n\\end{step}\n\n\\begin{step}\\label{step:nonStrict}\\PRFR{Mar 31st}\n\tIf $M^\\ast\\in {\\mathcal{M}}$, continue immediately with Step~\\ref{step:baseorderable}.\n\tDetermine whether $\\alpha_{M^\\ast} \\geq 0$. If this is the case, then\n${\\mathbf{T}}_{M^\\ast} = \\left( M^\\ast, \\SET{M,M^\\ast}, \\emptyset, \\emptyset, \\langle\\,\\rangle \\right)$ is valid, we set\n\t ${\\mathbf{T}} := \\left[\\vphantom{A^A}[{\\mathbf{T}}\\cup {\\mathbf{T}}_{M^\\ast}]_\\equiv \\right]_\\simeq$ and\n\tcontinue with Step~\\ref{step:start}.\n\\end{step}\n\n\\begin{step}\\label{step:baseorderable}\\PRFR{Mar 31st}\n\tDetermine whether $M$ is strongly base-orderable. If this is not the case, then \n\t${\\mathbf{T}}_M = \\left( M, \\emptyset, \\emptyset, \\SET{M,M^\\ast}, \\langle\\,\\rangle \\right)$ is valid, we set\n\t ${\\mathbf{T}} := \\left[\\vphantom{A^A}[{\\mathbf{T}}\\cup {\\mathbf{T}}_M]_\\equiv \\right]_\\simeq$ and then\n\tcontinue with Step~\\ref{step:start}.\n\\end{step}\n\n\\noindent The class of strong base-orderable matroids is closed under duality and minors \\cite{Ingleton1971}, therefore $M^\\ast$ and\nall minors of $M$ and $M^\\ast$ are strongly base-orderable upon reaching the next step.\n\n\\begin{step}\\PRFR{Mar 31st}\n\tLet $M=(E,{\\mathcal{I}})$. Determine whether there is some $X\\in {\\mathcal{I}}$ with $\\left| X \\right| = \\mathrm{rk}_M(E) - 3$\n\tand some $Y\\subseteq E\\backslash X$ such that $\\alpha_{M|'\\left( E\\backslash X \\right)}(Y) < 0$. If this is the\n\tcase, then the tableau \n\t${\\mathbf{T}}_M = \\left( M, \\emptyset, \\emptyset, \\SET{M,M^\\ast}, \\langle\\,\\rangle \\right)$ is valid, we set\n\t ${\\mathbf{T}} := \\left[\\vphantom{A^A}[{\\mathbf{T}}\\cup {\\mathbf{T}}_M]_\\equiv \\right]_\\simeq$ and then\n\tcontinue with Step~\\ref{step:start}.\n\\end{step}\n\n\\begin{step}\\PRFR{Mar 31st}\n\tLet $M^\\ast=(E,{\\mathcal{I}}^\\ast)$. Determine whether there is some $X\\in {\\mathcal{I}}^\\ast$ with \\linebreak \n\t $\\left| X \\right| = \\mathrm{rk}_{M^\\ast}(E) - 3$\n\tand some $Y\\subseteq E\\backslash X$ such that $\\alpha_{M^\\ast|'\\left( E\\backslash X \\right)}(Y) < 0$. If this is the\n\tcase, then the tableau \n\t${\\mathbf{T}}_{M^\\ast} = \\left( M^\\ast, \\emptyset, \\emptyset, \\SET{M,M^\\ast}, \\langle\\,\\rangle \\right)$ is valid, we set\n\t ${\\mathbf{T}} := \\left[\\vphantom{A^A}[{\\mathbf{T}}\\cup {\\mathbf{T}}_{M^\\ast}]_\\equiv \\right]_\\simeq$ and then\n\tcontinue with Step~\\ref{step:start}.\n\\end{step}\n\n\\PRFR{Mar 31st}\n\\noindent The next step may be omitted or carried out sloppily\\footnote{It clearly would be sloppy to just consider $W$, $X$, $Y$, and $Z$\nwith $\\max\\SET{\\left| W \\right|,\\left| X \\right|,\\left| Y \\right|,\\left| Z \\right|} \\leq k$ for some $k\\in \\mathbb{Z}\\BSET{0,1}$, or even\nto just check whether $M$ has a V\u00e1mos-matroid as a minor.},\n because it may take a considerable amount of time for larger matroids and it does\nnot seem to be worth the computational effort in practice. \n\n\\begin{step} \\PRFR{Mar 31st}\n\tLet $M=(E,{\\mathcal{I}})$. Determine whether there are $W,X,Y,Z\\in {\\mathcal{I}}$ such that\n\t\\begin{align*}\n\t\t\\mathrm{rk}(W) + &\\mathrm{rk}(X) + \\mathrm{rk}(W\\cup X\\cup Y) + \\mathrm{rk}(W\\cup X\\cup Z) + \\mathrm{rk}(Y\\cup Z) \\\\\n\t\t& > \\mathrm{rk}(W\\cup X) + \\mathrm{rk}(W\\cup Y) + \\mathrm{rk}(W\\cup Z) + \\mathrm{rk}(X\\cup Y) + \\mathrm{rk}(X\\cup Z).\n\t\\end{align*}\n\tIf this is the\n\tcase, then the tableau \n\t${\\mathbf{T}}_{M^\\ast} = \\left( M^\\ast, \\emptyset, \\emptyset, \\SET{M,M^\\ast}, \\langle\\,\\rangle \\right)$ is valid, we set\n\t ${\\mathbf{T}} := \\left[\\vphantom{A^A}[{\\mathbf{T}}\\cup {\\mathbf{T}}_{M^\\ast}]_\\equiv \\right]_\\simeq$ and then\n\tcontinue with Step~\\ref{step:start}.\n\\end{step}\n\n\\begin{step}\\PRFR{Mar 31st}\n\tDetermine whether $M$ is deflated. If not, then find a deflate $N$ of $M$ with a ground set of minimal cardinality,\n\tset ${\\mathbf{T}} := \\left[\\vphantom{A^A}\\left[\\left( {\\mathbf{T}}\\cup{\\mathbf{T}}_N \\right)(M \\simeq N)\\right]_\\equiv \\right]_\\simeq$\n\twhere $${\\mathbf{T}}_N = \\begin{cases}[r]\n\t\t \\left( N,\\SET{N,N^\\ast},\\emptyset,\\emptyset,\\langle\\,\\rangle \\right)& \\quad \\text{if~}\\alpha_N \\geq 0,\\\\\n\t\t \\left( N,\\emptyset,\\SET{N},\\emptyset,\\langle\\,\\rangle \\right) & \\quad \\text{otherwise,}\n\t\t \\end{cases}$$ \n\t\tand continue with Step~\\ref{step:start}.\n\\end{step}\n\n\\begin{step}\\PRFR{Mar 31st}\\label{lastStep}\n\tDetermine whether $M^\\ast$ is deflated. If not, then find a deflate $N$ of $M^\\ast$ with a ground set of minimal cardinality,\n\tset ${\\mathbf{T}} := \\left[\\vphantom{A^A}\\left[\\left( {\\mathbf{T}}\\cup{\\mathbf{T}}_N \\right)(M^\\ast \\simeq N)\\right]_\\equiv \\right]_\\simeq$\n\twhere $${\\mathbf{T}}_N = \\begin{cases}[r]\n\t\t \\left( N,\\SET{N,N^\\ast},\\emptyset,\\emptyset,\\langle\\,\\rangle \\right)& \\quad \\text{if~}\\alpha_N \\geq 0,\\\\\n\t\t \\left( N,\\emptyset,\\SET{N},\\emptyset,\\langle\\,\\rangle \\right) & \\quad \\text{otherwise,}\n\t\t \\end{cases}$$ \n\t\tand continue with Step~\\ref{step:start}.\n\\end{step}\n\n\\PRFR{Mar 31st}\n\\noindent This is the point where we may try creative ways of determining whether $M$ is a gammoid or not.\nIf we are successful, then we augment the tableau ${\\mathbf{T}}$ accordingly, and continue with Step~\\ref{step:start}.\nIn the best case, we might guess a representation of $M$, or find a considerably smaller matroid $M'$ such that $M$ is induced from $M'$\nby some digraph $D$. In theory, it is also possible to find a known non-gammoid $X'$ that is induced from $M$ by some digraph $D$, which\nthen implies that $M$ must be a non-gammoid. Since the class of strongly base-orderable matroids is closed under matroid \ninduction by digraphs,\n$X'\\notin \\SET{M(K_4), P_7}$, because we know since Step~\\ref{step:baseorderable} that $M$ is strongly base-orderable.\nIn practice, we never managed to successfully show that some known excluded minor may be induced from a candidate matroid $M$ \nunder examination.\nCurrently, $P_8^=$ (\\cite{Ox11}, p.651) is the only \nexcluded minor for the class of gammoids (J.~Bonin, \\cite{joeP8}) that we know of, which is strongly base-orderable \nyet neither has rank or co-rank $3$. Furthermore,\n$P_8^=$ is isomorphic to its dual $\\left( P_8^=\\right)^\\ast$ which makes it a rather special matroid. \nTherefore we think it is reasonable to assume that the odds are clearly in favor of $M$ being a gammoid upon reaching the \nnext step.\\footnote{\nOr, more pessimistically, we might not know sufficiently general excluded minors for\nthe class of gammoids to assess the situation here more realistically.}\n\n\\needspace{4\\baselineskip}\n\\begin{step}\\label{step:exhaustion}\\PRFR{Mar 31st}\n\tTry to find an extension $N$ of $M$ with at most $\\mathrm{rk}_G(E)^2\\cdot \\left| E \\right| + \\mathrm{rk}_G(E) + \\left| E \\right|$ elements\n\tsuch that $N$ is not isomorphic to any $M'\\in {\\mathcal{G}} \\cup {\\mathcal{M}} \\cup {\\mathcal{X}}$.\n\tSet ${\\mathbf{T}} := \\left[\\vphantom{A^A}\\left[{\\mathbf{T}}\\cup{\\mathbf{T}}_N\\right]_\\equiv \\right]_\\simeq$\n\twhere $${\\mathbf{T}}_N = \\begin{cases}[r]\n\t\t \\left( N,\\SET{N,N^\\ast},\\emptyset,\\emptyset,\\langle\\,\\rangle \\right)& \\quad \\text{if~}\\alpha_N \\geq 0,\\\\\n\t\t \\left( N,\\emptyset,\\SET{N},\\emptyset,\\langle\\,\\rangle \\right) & \\quad \\text{otherwise,}\n\t\t \\end{cases}$$ \n\t\tand continue with Step~\\ref{step:start}.\n\tIf no such extension of $M$ exists, then set $M:= G$ and continue with Step~\\ref{step:chosen}.\n\\end{step}\n\n\\PRFR{Mar 31st}\n\\noindent Clearly, if we continue this process long enough, then Step~\\ref{step:exhaustion} ensures \nthat the tableau ${\\mathbf{T}}$ will eventually\nbecome decisive for $G$ by exhausting all isomorphism classes of extensions of $G$\nwith at most $\\mathrm{rk}_G(E)^2\\cdot \\left| E \\right| + \\mathrm{rk}_G(E) + \\left| E \\right|$ elements.\n\n\n\n\\subsection{Reconstructing $(D,T,E)$ if $\\Gamma_{\\mathcal{M}}(M) = 1$}\n\n\\subsection{Sufficient Conditions for $\\Gamma_{\\mathcal{M}}(M) = 0$}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{$M$-Stuck Families and $M$-Persistent Violations}\n\n\\section{Representation over $\\mathbb{R}$}\n\n\\noindent \n\\PRFR{Feb 15th}\nThere are many ways to arrive at the fact that every gammoid can be represented by a matrix\nover a field ${\\mathbb{K}}$ whenever ${\\mathbb{K}}$ has enough elements. Or, to be more precise, for every field ${\\mathbb{F}}$ and \nevery gammoid $M$ there\nis an extension field ${\\mathbb{K}}$ of ${\\mathbb{F}}$, such that $M$ can be represented by a matrix over ${\\mathbb{K}}$.\nFor the sake of simplicity, we only consider representations of gammoids over the field of the reals $\\mathbb{R}$.\nIn \\cite{Ar06}, F.~Ardila points out that the Lindstr\u00f6m Lemma yields an easy method to construct a matrix $\\mu\\in \\mathbb{R} ^{E\\times B}$ from the digraph $D=(V,A)$ such that $\\Gamma(D,T,E) = M(\\mu)$;\nthe construction is universal in the sense that it works with indeterminates and thus yields a representation over ${\\mathbb{F}}$ whenever these indeterminates can be replaced with elements from ${\\mathbb{F}}$\nwithout zeroing out any nonzero subdeterminants of $\\mu$.\n\n\\begin{definition}\\PRFR{Feb 15th}\n\tLet $D=(V,A)$ be a digraph and $w\\colon A\\longrightarrow \\mathbb{R}$.\n\tThen $w$ shall be called\n\t \\deftext[indeterminate weighting of D@indeterminate weighting of $D$]{indeterminate weighting of $\\bm D$},\n\twhenever the set $\\SET{w(a)\\mid a\\in A}$ is $\\mathbb{Z}$-independent.\n\\end{definition}\n\n\\begin{example}\\PRFR{Feb 15th}\n\tLet $D=(V,A)$ be any digraph, then $\\left| A \\right| < \\infty$. Thus there is a set $X\\subseteq \\mathbb{R}$\n\tthat is $\\mathbb{Z}$-independent with $\\left| X \\right| = \\left| A \\right|$ (Lemma~\\ref{lem:enoughZindependents}). Then\n\tany bijection $\\sigma\\colon A\\longrightarrow X$ induces an indeterminate weighting $w\\colon X\\longrightarrow \\mathbb{R}$\n\twith $w(x) = \\sigma(x)$, thus indeterminate weightings exist for all digraphs.\n\\end{example}\n\n\\begin{notation}\\label{n:prodp}\\PRFR{Feb 15th}\n\tLet $D=(V,A)$ be a digraph and $w\\colon A\\longrightarrow \\mathbb{R}$ be an indeterminate weighting of $D$. Let $q=(q_i)_{i=1}^n\\in {\\mathbf{W}}(D)$,\n\twe shall write \\[ \\prod q = \\prod_{i=1}^{n-1} w\\left( \\vphantom{A^A}(q_i,q_{i+1}) \\right). \\qedhere \\]\n\\end{notation}\n\n \\begin{lemma}[Lindstr\u00f6m \\cite{Li73}]\\label{lem:lindstrom}\\PRFR{Feb 15th}\n \tLet $D=(V,A)$ be an acyclic digraph, $n\\in \\mathbb{N}$ a natural number, $S=\\dSET{s_1,s_2,\\ldots,s_n}\\subseteq V$ and $T=\\dSET{t_1,t_2,\\ldots,t_n}\\subseteq V$ be equicardinal subsets of $V$, and let $w\\colon A\\longrightarrow \\mathbb{R}$ be an indeterminate weighting of $D$. Furthermore,\n \t$\\mu\\in \\mathbb{R}^{V\\times V}$ shall be the matrix with\n \t\\[ \\mu(u,v) = \\sum_{p\\in {\\mathbf{P}}(D;u,v)} \\prod p .\\]\n \n \tThen\n \t\\[ \\det \\left( \\mu| S\\times T \\right) = \\sum_{L\\colon S\\double{\\rightarrow} T} \\left( \\mathrm{sgn}(L) \n \t\\prod_{p\\in L} \\left( \\prod p \\right) \\right) \\]\n \twhere $\\mathrm{sgn}(L) = \\mathrm{sgn}(\\sigma)$ for the unique permutation $\\sigma\\in \\mathfrak{S}_n$ with\n \tthe property that for every $i\\in\\SET{1,2,\\ldots,n}$ there is a path $p\\in L$ with $p_1 = s_i$ and \n \t$p_{-1} = t_{\\sigma(i)}$.\n \tFurthermore, \\[ \\det \\left( \\mu| S\\times T \\right) = 0 \\] if and only if\n \tthere is no linking from $S$ to $T$ in $D$.\n \\end{lemma}\n\\noindent As suggested by F.~Ardila, we present the following bijective proof given by I.M.~Gessel and X.G.~Viennot \\cite{GV89p}. \n\\begin{proof}\\PRFR{Feb 15th} The Leibniz formula (Definition~\\ref{def:det}) yields\n\\begin{align*}\n\\det \\left( \\mu| S\\times T \\right) \n& = \\sum_{\\sigma\\in \\mathfrak{S}_n} \\mathrm{sgn}(\\sigma) \\prod_{i=1}^{n} \\mu(s_i,t_{\\sigma(i)}) \\\\\n& = \\sum_{\\sigma\\in \\mathfrak{S}_n} \\mathrm{sgn}(\\sigma) \\prod_{i=1}^{n} \\left( \n\t\t\t\t\\sum_{p\\in {\\mathbf{P}}{\\left(D;s_i,t_{\\sigma(i)}\\right)}} \\prod p \\right) \\\\\n& = \\sum_{\\sigma\\in \\mathfrak{S}_n} \\mathrm{sgn}(\\sigma)\\left( \\sum_{K\\in Q_\\sigma} \\,\\,\n\t\t\t\\prod_{p\\in K} \\left( \\prod p\\right) \\right),\n\\end{align*}\nwhere $$Q_\\sigma = \\SET{\\left. K \\in \\binom{{\\mathbf{P}}(D)}{n} \\,\\,\\right|\\,\\, \\forall i\\in \\SET{1,2,\\ldots,n}\\colon\\,\\exists p\\in K\\colon\\,p_1=s_i {\\mathop{\\text{~and~}}} p_{-1}=t_{\\sigma(i)} }\\,$$ consists of all families of paths connecting $s_i$ with $t_{\\sigma(i)}$ for all $i\\in \\SET{1,2,\\ldots,n}$.\n\n\n Clearly, for $\\sigma,\\tau\\in \\mathfrak{S}_n$ with $\\sigma\\not= \\tau$, the sets $Q_\\sigma\\cap Q_\\tau = \\emptyset$ are disjoint, therefore the following map with the domain $Q=\\bigcup_{\\sigma\\in \\mathfrak{S}_n} Q_\\sigma$ is well defined:\n \\[ \\mathrm{sgn}\\colon Q \\longrightarrow \\SET{-1,1},\\quad K\\mapsto \\mathrm{sgn}(\\sigma) \n \\quad\\text{~where~}\\sigma\\in\\mathfrak{S}_n\\text{~such that~}K\\in Q_\\sigma .\\]\n\n Thus we may write\n \\begin{align*}\n\\det \\left( \\mu| S\\times T \\right) & = \\sum_{K\\in Q} \\mathrm{sgn}(K) \\prod_{p\\in K} \\left( \\prod p \\right).\n\\end{align*}\nFurthermore, if $L\\colon S\\double{\\rightarrow} T$ is a linking from $S$ to $T$ in $D$, then $L\\in Q_\\sigma$ \nwhere $\\sigma\\in\\mathfrak{S}_n$ is the unique permutation mapping the indexes of the initial vertices of the paths in $L$ \nto the indexes of the terminal vertices of the paths in $L$. Let us denote the routings in $Q$ by\n\\[ R = \\SET{L\\in Q\\mid L\\text{~is a routing}}.\\]\nWe prove the first statement of the lemma by showing that there is a bijection\n$\\phi\\colon Q\\backslash R \\longrightarrow Q\\backslash R$, such that for all $K\\in Q\\backslash R$,\n$$\\prod_{p\\in K}\\left( \\prod p \\right) = \\prod_{p\\in \\phi(K)} \\left( \\prod p \\right)$$\nand $\\mathrm{sgn}(K) = -\\mathrm{sgn}(\\phi(K))$. We construct such a map $\\phi$ now.\nLet $$K' = \\SET{p\\in K~\\middle|~\\vphantom{A^A} \\exists q\\in K\\BSET{p}\\colon\\,\\left| p \\right|\\cap \\left| q \\right|\\not=\\emptyset}$$\nbe the set of paths in $K$ that meet a vertex of another path, clearly $\\left| K' \\right| \\geq 2$\nsince $K$ is not a routing. There is a total order on $K'$:\n let $p,q\\in K'$, then $p\\leq q$ if and only if $i\\leq j$ where $p_1 = s_i$ and $q_1 = s_j$.\nNow let $p=(p_i)_{i=1}^{n(p)}\\in K'$ be chosen \nsuch that $p$ is the minimal element with respect to the above order.\nLet $j(p)\\in \\SET{1,2,\\ldots,n(p)}$ be the smallest index, such that there is some $q\\in K'\\BSET{p}$\nwith $p_{j(p)}\\in \\left| q \\right|$. Now let\n $q=(q_i)_{i=1}^{n(q)}\\in\\SET{k\\in K'\\BSET{p}~\\middle|~\\vphantom{A^A} p_{j(p)}\\in \\left| q \\right|}$\n be the minimal choice with respect to the above order on $K'$, and let $j(q)\\in \\SET{1,2,\\ldots,n(q)}$\n such that $q_{j(q)} = p_{j(p)}$. Now let\n $p' = p_1 p_2\\ldots p_{j(p)} q_{j(q) + 1} q_{j(q) + 2} \\ldots q_{n(q)}$\n and $q' = q_1 q_2 \\ldots q_{j(q)} p_{j(p)+1} p_{j(p)+2}\\ldots p_{n(p)}$. Since $D$ is acyclic, all walks are paths in $D$, so ${\\mathbf{W}}(D) = {\\mathbf{P}}(D)$.\n Therefore we may set\n $\\phi(K) = \\left( K\\BSET{p,q}\\right)\\cup\\SET{p',q'} \\in Q\\backslash R$.\n Clearly, $\\phi(\\phi(K)) = K$, therefore $\\phi$ is bijective and self-inverse. Furthermore,\n if $K\\in Q_\\sigma$, then $\\phi(K) \\in Q_{\\sigma \\cdot (x y)}$ for a suitable cycle\n $(x y)\\in \\mathfrak{S}_n$. Thus $\\mathrm{sgn}(\\phi(K)) = \\mathrm{sgn}(\\sigma)\\mathrm{sgn}\\left( (x y) \\right) = - \\mathrm{sgn}(\\sigma) = -\\mathrm{sgn}(K)$. Clearly, $K$ and $\\phi(K)$ traverse the same arcs, therefore $\\prod_{p\\in K} (\\prod p) = \\prod_{p\\in \\phi(K)} (\\prod p)$.\nThe bijection $\\phi$ implies that the summands $K\\in Q\\backslash R$ add up to zero, thus we have $$\\det \\left( \\mu| S\\times T \\right) = \\sum_{L\\in R} \\mathrm{sgn}(L) \\prod_{p\\in L} \\left( \\prod p \\right).$$\nThe second statement of the lemma follows from the fact that for two routings $L_1,L_2\\in R$,\nwe have $L_1 = L_2$ if and only if $\\bigcup_{p\\in L_1} \\left| p \\right|_A = \\bigcup_{p\\in L_2} \\left| p \\right|_A$. For the non-trivial direction: assume we have a set of arcs $L_A$ that are traversed by the paths of a linking, and let $V_A=\\SET{u,v~\\middle|~(u,v)\\in L_A}$.\n Then the initial vertices of that linking are the elements of the set\n$S_A = \\SET{u\\in V_A\\mid \\forall (v,w)\\in L_A\\colon\\,u\\not= w}$. The terminal vertices are the elements of the set\n$T_A = \\SET{w\\in V_A\\mid \\forall (u,v)\\in L_A\\colon\\,u\\not= w}$, and the paths can be reconstructed from\nthe initial vertices $v\\in S_A$ by following the unique arcs $(v,w),(w,x),\\ldots \\in L_A$ until a vertex $t\\in T_A$ is reached. Clearly, for $L\\in R$, $\\prod_{p\\in L} \\left( \\prod p \\right) \\not= 0$, and since $w$ is an indeterminate weighting, two summands $L,L'\\in R$ can only cancel each other when the corresponding monomials are equal, i.e. $\\prod_{p\\in L} \\left( \\prod p \\right) = \\prod_{p\\in L'} \\left( \\prod p \\right)$; but then $L_A = L'_A$ holds, and so $L = L'$. Thus no summand in the determinant formula\n which belongs to a routing from $R$\ncan be cancelled out by another summand belonging to another routing from $R$. Therefore\n$\\det\\left( \\mu| S\\times T \\right) = 0$ if and only if $R=\\emptyset$, i.e. there is no linking from $S$ to $T$ in $D$.\n\\end{proof}\n\n\\begin{corollary}\\PRFR{Feb 15th}\n\tLet $D=(V,A)$ be an acyclic digraph, $T,E\\subseteq V$, and $w\\colon A\\longrightarrow \\mathbb{R}$ be an indeterminate weighting of $D$. Furthermore,\n\tlet\n\t \t$\\mu\\in \\mathbb{R}^{E\\times T}$ be the matrix with\n \t\\[ \\mu(e,t) = \\sum_{p\\in {\\mathbf{P}}{(D;e,t)}} \\left( \\prod p \\right).\\]\n \n \tThen $\\Gamma(D,T,E) = M(\\mu)$.\n\\end{corollary}\n\\begin{proof}\\PRFR{Feb 15th}\n\tThis is straightforward from the Definition~\\ref{def:Mmu} and the Lindstr\u00f6m Lemma~\\ref{lem:lindstrom}.\n\\end{proof}\n\n\\noindent\n\\PRFR{Feb 15th}\nClearly, for an arbitrary gammoid $M = \\Gamma(D,T,E)$, we cannot assume that $D$ is acyclic (Remark~\\ref{rem:weNeedCycles}). \nThere are several ways to work around this. Either {\\em a)}\\footnote{This is what is implied by the rationale given in \\cite{Ar06}.} \nwe adjust our definition of a routing such that routings with non-path walks are allowed, making the class of routings in $D$ \ninfinite whenever there is a cycle in $D$. Then we could use power series to calculate the entries of $\\mu$ as well as its sub-determinants, \nwhere convergence is sufficiently guaranteed if $\\prod p \\in (0,1)$ holds for every cycle walk $p\\in {\\mathbf{W}}(D)$. \nA sufficient condition would be to use a weighting $w$ where $0 < w(a) < 1$ for all $a\\in A$.\n The construction of $\\phi$ in the proof of the Lindstr\u00f6m Lemma would still go through, but for the second \n statement we would have to choose the indeterminate weights more carefully, since a cycle walk $q\\in {\\mathbf{W}}(D)$ \n gives rise to the formal power\nseries $\\sum_{i=0}^\\infty \\left( \\prod q \\right)^{i}$ which converges to $\\frac{1}{1-\\prod q}$. \nClearly, a similar cardinality-argument as in Lemma~\\ref{lem:enoughZindependents} \nguarantees that we can find a sufficient number of carefully chosen indeterminates in $\\mathbb{R}$. \nOr {\\em b)} we could try to find a construction that removes cycles from $D$, possibly changing the\ngammoid represented by the resulting digraph $D'$, then use the Lindstr\u00f6m Lemma to obtain a \nmatrix $\\nu$, and then revert the constructions in order to obtain $\\mu$ from $\\nu$; which is what we will do now.\n\n\\begin{definition}\\PRFR{Feb 15th}\n\tLet $D=(V,A)$ be a digraph, $x,t\\notin V$ be distinct new elements, and let $c=(c_i)_{i=1}^n\\in {\\mathbf{W}}(D)$ be a cycle walk.\n\tThe \\deftext[lifting of c in D@lifting of $c$ in $D$]{lifting of $\\bm c$ in $\\bm D$ by $\\bm(\\bm x\\bm,\\bm t\\bm)$} is the digraph\n\t$D^{(c)}_{(x,t)} = (V\\mathbin{\\dot{\\cup}}\\SET{x,t}, A')$ where\n\t\\[ A' = A \\BSET{(c_1,c_2)} \\cup \\SET{(c_1,t),(x,c_2),(x,t)}. \\qedhere\\]\n\\end{definition}\n\n\\noindent Observe that the cycle walk $c\\in{\\mathbf{W}}(D)$ is not a walk in the lifting of $c$ in $D$ anymore.\n\n\\begin{example}\\PRFR{Feb 15th}\n\tConsider $D=(\\SET{c_1,c_2,c_3,c_4},\\SET{(c_1,c_2),(c_2,c_3),(c_3,c_4),(c_4,c_1)})$. Then\n\t$c_1c_2c_3c_4c_1\\in{\\mathbf{W}}(D)$ is a cycle. The lifting of $c$ in $D$ by $(x,t)$ is then defined to be\n\tthe digraph $D'=(\\SET{c_1,c_2,c_3,c_4,x,t},\\SET{(c_1,t),(c_2,c_3),(c_3,c_4),(c_4,c_1),(x,c_2),(x,t)})$. \\\\[5mm]\n\t\\hspace*{4cm}\n\t\\includegraphics{digraphlifting2} \\qedhere\n\\end{example}\n\n\\begin{lemma}\\label{lem:liftingNoNewCycles}\\PRFR{Feb 15th}\n\tLet $D=(V,A)$ be a digraph, $x,t\\notin V$, and $c=(c_i)_{i=1}^n\\in {\\mathbf{W}}(D)$ a cycle walk, and let\n\t$D'=D^{(c)}_{(x,t)}$ be the lifting of $c$ in $D$ by $(x,t)$.\n\tIf $c'\\in {\\mathbf{W}}(D')$ is a cycle walk, then\n\t$c'\\in {\\mathbf{W}}(D)$. In other words, the lifting of cycle walks does not introduce new cycle walks.\n\\end{lemma}\n\\begin{proof}\\PRFR{Feb 15th}\n\tLet $D'=(V',A')$.\n\tClearly, $x$ is a source in $D^{(c)}_{(x,t)}$ and $t$ is a sink in $D^{(c)}_{(x,t)}$. Thus $x,t\\notin \\left| c' \\right|$.\n\tBut then $\\left| c' \\right|_A \\subseteq A'\\cap \\left( V\\times V \\right)$ and therefore $c'$ is also a cycle walk in $D$.\n\\end{proof}\n\n\n\\begin{definition}\\label{def:completeLifting}\\PRFR{Feb 15th}\n\tLet $D=(V,A)$ be a digraph. A \\deftext[complete lifting of D@complete lifting of $D$]{complete lifting of $\\bm D$}\n\tis an acyclic digraph $D'=(V',A')$ for which there is a suitable $n\\in \\mathbb{N}$ such that there is a\n\tset $X=\\dSET{x_1,t_1,x_2,t_2,\\ldots,x_n,t_n}$ with $X\\cap V = \\emptyset$,\n\ta family of digraphs $D^{(i)} = (V^{(i)},A^{(i)})$ for $i\\in \\SET{0,1,\\ldots,n}$\n\twhere $D' = D^{(n)}$, $D^{(0)} = D$, and for all $i\\in\\SET{1,2,\\ldots,n}$\n\t $$D^{(i)} = \\left( D^{(i-1)}\\right)^{(c_i)}_{(x_i,t_i)}$$\n\twith respect to a cycle walk $c_i\\in {\\mathbf{W}}\\left( D^{(i-1)}\\right)$.\n\tIn this case, we say that the set $$R = \\SET{(x_i,t_i)\\mid i\\in\\SET{1,2,\\ldots,n}}$$ \\deftextX{realizes} \n\tthe complete lifting $D'$ of $D$.\n\\end{definition}\n\n\\begin{lemma}\\label{lem:completelifting}\\PRFR{Feb 15th}\n\tLet $D=(V,A)$ be a digraph. Then $D$ has a complete lifting.\n\\end{lemma}\n\\begin{proof}\\PRFR{Feb 15th}\n\tBy induction on the number of cycle walks in $D$. If $D$ has no cycle walk, $D$ is a complete lifting of $D$.\n\tNow let $c\\in {\\mathbf{W}}(D)$ be a cycle walk, and let $x,t\\notin V$.\n\tLet $D' = D^{(c)}_{(x,t)}$. By construction $c\\notin {\\mathbf{W}}(D')$,\n\tthus\n\t $D'$ has strictly fewer cycle walks than $D$ (Lemma~\\ref{lem:liftingNoNewCycles}), therefore there is a complete lifting $D''$ of $D'$ by induction hypothesis.\n\t Since $D'$ is a lifting of $D$, $D''$ is also a complete lifting of $D$.\n\\end{proof}\n\n\\needspace{3\\baselineskip}\n\\begin{lemma}\\label{lem:cyclelifting}\\PRFR{Feb 15th}\n\tLet $D=(V,A)$, $E,T\\subseteq V$, $c\\in {\\mathbf{W}}(D)$ a cycle, $x,t\\notin V$, and let $D'=D^{(c)}_{(x,t)}$ be the lifting of $c$ in $D$.\n\tThen $\\Gamma(D,T,E) = \\Gamma(D',T\\cup\\SET{t},E\\cup\\SET{x})|' E$.\n\\end{lemma}\n\\begin{proof}\\PRFR{Feb 15th}\n\tLet $M=\\Gamma(D,T,V)$ be the strict gammoid induced by the representation $(D,T,E)$ \n\tof the not necessarily strict gammoid $\\Gamma(D,T,E)$, and let \\linebreak\n\t$M' = \\Gamma(D',T\\cup\\SET{t},V')$ be the strict gammoid obtained from the lifting of $c$.\n\tThen $M'' = \\left( M' \\right)|'\\left( V\\cup\\SET{t} \\right)$ is a strict gammoid that is represented by $(D'',T,V\\cup\\SET{t})$\n\twhere the digraph\n\t $D'' = (V_0\\BSET{x},A_0\\backslash\\left( V_0\\times\\SET{x} \\right))$\n\tis induced from the $x$-$t$-pivot $D_0$ of $D'$, i.e.\n\t $D_0 = D'_{x\\leftarrow t} = (V_0,A_0)$. This follows from the proof of Lemma~\\ref{lem:contractionStrictGammoid}\n\t along with the single-arc routing $\\SET{xt}\\colon \\SET{x}\\double{\\rightarrow} T\\cup\\SET{t}$ in $D'$.\n\tLet $A''$ denote the arc set of $D''$. \n\tIt is easy to see from the involved constructions (Fig.~\\ref{fig:liftingcycles}), \n\tthat $A'' = \\left( A\\BSET{(c_1,c_2)} \\right) \\cup\\SET{(c_1, t), (t, c_2)}$.\n\t Clearly, a routing $R$ in $D$ can have at most one path $p\\in R$ such that $(c_1,c_2)\\in \\left| p \\right|_A$, \n\t and since $t\\notin V$, we obtain a routing $R'= \\left(R\\BSET{p}\\right)\\cup\\SET{q t r}$ \n\t for $q,r\\in {\\mathbf{P}}(D)$ such that \n\t$p=qr$ with $q_{-1}=c_1$ and $r_1=c_2$.\n\t Clearly, $R'$ routes $X$ to $Y$ in $D''$ whenever $R$ routes $X$ to $Y$ in $D$.\n\tConversely, let $R'\\colon X'\\double{\\rightarrow} Y'$ be a routing in $D''$ with $t\\notin X'$.\n\t Then there is at most one $p\\in R'$ with $t\\in \\left| p \\right|$.\n\t We can invert the construction and let $R''= \\left( R'\\BSET{p} \\right)\\cup\\SET{qr}$\n\t for the appropriate paths $q,r\\in {\\mathbf{P}}(D'')$ with $p=qtr$. \n\tThen $R''$ is a routing from $X'$ to $Y'$ in $D'$. Thus we have shown that $M''| V = M$,\n\tand consequently, with $E\\subseteq V$ and Lemma~\\ref{lem:contractrestrictcommutes}, it follows that\n\t\\begin{align*}\n\t\\Gamma(D,T,E) = M| E = \\left( M'' \\right)| E & = \\left( \\Gamma(D',T\\cup\\SET{t},V') |' \\left( V\\cup\\SET{t} \\right) \\right) | E \\\\& = \\Gamma(D',T\\cup\\SET{t},E\\cup\\SET{x})|' E. \\qedhere\n\\end{align*}\n\\end{proof}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics{liftingtrick}\n\\end{center}\n\\caption{\\label{fig:liftingcycles}Constructions involved in Lemma~\\ref{lem:cyclelifting}.}\n\\end{figure}\n\n\\begin{corollary}\\label{cor:acyclicQuasiRepresentationOfGammoids}\\PRFR{Feb 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a gammoid. Then there is an acyclic digraph $D=(V,A)$ and sets $T,E'\\subseteq V$ such that\n\t$M = \\Gamma \\left( D,T,E' \\right)|' E$\n\tand such that $$\\left| T \\right| = \\mathrm{rk}_M(E) + \\left| E'\\backslash E \\right|.$$\n\\end{corollary}\n\\begin{proof}\\PRFR{Feb 15th}\n\tLet $M=\\Gamma(D',T',E)$ with $\\left| T' \\right|=\\mathrm{rk}_M(E)$.\n\tThen let $D$ be a complete lifting of $D'$ (Lemma~\\ref{lem:completelifting}),\n\tand let $D^{(0)},D^{(1)},\\ldots, D^{(n)}$ be the family of digraphs and $c_1,c_2,\\ldots,c_n$ be the cycle walks that correspond to \n\tthe complete lifting $D$ of $D'$ \n\tas required by Definition~\\ref{def:completeLifting},\n\tand let $\\dSET{x_1,t_1,\\ldots,x_n,t_n}$ denote the new elements such that\n\t\\[ D^{(i)} = \\left( D^{(i-1)} \\right)^{(c_i)}_{(x_i,t_i)} \\]\n\tholds for all $i\\in\\SET{1,2,\\ldots,n}$.\n\tInduction on the index $i$ with Lemma~\\ref{lem:cyclelifting} yields that\n\t\\[ \\Gamma(D',T,E) = \\Gamma(D^{(i)},T\\cup\\SET{t_1,t_2,\\ldots,t_i},E\\cup\\SET{x_1,x_2,\\ldots,x_i})|' E\\]\n\tholds for all $i\\in\\SET{1,2,\\ldots,n}$.\n\tClearly,\n\t\\[ \\left| T\\cup\\SET{t_1,t_2,\\ldots,t_n} \\right| = \\left| T \\right| + n = \\mathrm{rk}_M(E) + n = \\mathrm{rk}_M(E) + \\left| \\SET{x_1,x_2,\\ldots,x_n} \\right|. \\qedhere \\]\n\\end{proof}\n\n\\begin{theorem}\\label{thm:gammoidOverR}\\PRFR{Feb 15th}\n\tLet $M=(E,{\\mathcal{I}})$ be a gammoid, $T=\\dSET{t_1,t_2,\\ldots,t_{\\mathrm{rk}_M(E)}}$. Then there is a matrix $\\mu\\in \\mathbb{R}^{E\\times T}$ such\n\tthat $M= M(\\mu)$.\n\\end{theorem}\n\\begin{proof}\\PRFR{Feb 15th}\n\tBy Corollary~\\ref{cor:acyclicQuasiRepresentationOfGammoids}, there is an acyclic digraph $D=(V,A)$ and\n\tthere are sets $E',T' \\subseteq V$,\n\tsuch that $M = N|' E$ where $N=\\Gamma(D,T',E')$\n\tand $\\left| T' \\right| = \\mathrm{rk}_M(E) + \\left| E' \\backslash E \\right|$.\n\tRemember that $E'\\backslash E$ is independent in $N$,\n\tand every base $B$ of $M$ induces a base $B\\cup \\left( E'\\backslash E \\right)$ of $N$.\n\tThe Lindstr\u00f6m Lemma~\\ref{lem:lindstrom} yields a matrix $\\nu\\in \\mathbb{R}^{E'\\times T'}$ such that \n\t$N = M(\\nu)$. In Lemma~\\ref{lem:contractequalspivot} and Remark~\\ref{rem:contraction} we have seen that we can pivot in the\n\tindependent set $E'\\backslash E$\n\tin $\\nu$, which yields a new matrix $\\nu'\\in \\mathbb{R}^{E'\\times T'}$. Let $T_0 = \\SET{t'\\in T' ~\\middle|~ \\forall e'\\in E'\\backslash E\\colon\\, \\nu'(e',t') = 0 }$ denote the remaining columns of $\\nu'$ that have not been used to pivot in an element of $E'\\backslash E$.\n\tWe set $\\mu = \\nu'| E\\times T_0$. Thus $M(\\mu) = M(\\nu)|' E = N|' E = M$.\n\\end{proof}\n\n\\noindent Let us compare the two methods {\\em a)} and {\\em b)} mentioned above. In our opinion, \nboth methods are connected to aspects of the same underlying phenomenon that cycle paths do not interfere with \nthe existence of linkings between given sets of vertices in a digraph.\n\n\\input{Text\/Ex\/302_repr_circle}\n\n\\PRFR{Mar 7th}\n\\noindent\nIn the paper {\\em A parameterized view on matroid optimization problems} \\cite{Marx09}, D.~Marx shows that there is a randomized polynomial time algorithm with respect to the size of the ground set of a gammoid, \nthat constructs a matrix $\\mu$ from $(D,T,E)$ such that $M(\\mu) = \\Gamma(D,T,E)$.\nThe method of D.~Marx starts with the construction of the dual $N^\\ast$ of the underlying strict gammoid $N=\\Gamma(D,T,V)$ \nfor a given representation $(D,T,E)$\nwith $D=(V,A)$ through the linkage system of $D$ to $T$ (Definition~\\ref{def:linkageSystem} and Lemma~\\ref{lem:linkage}). \nThen a matrix $\\nu$ with $M(\\nu) = N^\\ast$ is constructed with a small probability of failure (see Proposition~\\ref{prop:randompolytimeTransversalMatroid} below), \nwhich in turn is converted into a standard representation (Remark~\\ref{rem:stdRep})\nof the form $(I_r\\,\\, A^\\top)^\\top$ using Gaussian Elimination. Then $(-A \\,\\, I_{n-r})^\\top$ is the desired representation of $M$.\nBefore we present the main proposition that leads to this result, we need the following lemma.\n\n\\begin{lemma}[\\cite{Sch80}, Corollary 1]\\PRFR{Mar 7th}\\label{lem:SchwartzNumberZeros}\\PRFR{Mar 7th}\n\tLet ${\\mathbb{F}}$ be a field, $X = \\dSET{x_1,x_2,\\ldots,x_n}$,\n\t let $p \\in {\\mathbb{F}}[X]$ be a polynomial\n\twith $p \\not= 0$.\n\tFurthermore, let $F\\subseteq {\\mathbb{F}}$ be a finite subset of elements of the coefficient field with $\\left| F \\right| \\geq c\\cdot \\deg(p)$\n\tfor some $c\\in {\\mathbb{Q}}$ with $c > 0$. Then\n\t \\[\t\\left| \\SET{\\xi \\in F^X ~\\middle|~ p[X=\\xi] = 0} \\right| \\leq \\frac{\\left| F \\right|^n}{c}.\\]\n\\end{lemma}\n\n\\PRFR{Mar 7th}\n\\noindent For a formal proof, we refer the reader to J.T.~Schwartz's \n{\\em Fast Probabilistic Algorithms for Verification of Polynomial Identities} \\cite{Sch80}. \nThe proof idea is to do induction on the number of variables involved. The base case is the fact that a polynomial\nin a single variable of degree $d$ can have at most $d$ different roots. In the induction step, we fix the values of all but one variable,\nif the resulting polynomial in a single variable is the zero polynomial, we may choose any value from $F$ for that variable. Otherwise,\nthere are at most the degree of the resulting polynomial many choices for the last variable such that the polynomial evaluates to zero.\n\n\\begin{lemma}[\\cite{Marx09}, Lemma 1, \\cite{Sch80}, \\cite{Zi79}]\\label{lem:probOfZero}\n\tLet ${\\mathbb{F}}$ be a field, \\linebreak\n\tlet $X = \\dSET{x_1,x_2,\\ldots,x_n}$ be a finite set,\n\t let $p \\in {\\mathbb{F}}[X]$ be a polynomial\n\twith $p \\not= 0$, and let $F\\subseteq {\\mathbb{F}}$ be a finite set.\n\tLet $\\xi$ be a random variable sampled from a uniform distribution on the set $F^X$.\n\tThen the probability that $\\xi$ is a zero of $p$ may be estimated by\n\t\\[ \\Pr\\left(\\vphantom{A^A}p[X=\\xi] = 0\\right) \\leq \\frac{\\deg(p)}{\\left| F \\right|} .\\]\n\\end{lemma}\n\n\\begin{proof}\\footnote{D.~Marx omits the proof and instead cites \\cite{Sch80} and \\cite{Zi79}.} \\PRFR{Mar 7th}\n\tIn Lemma~\\ref{lem:SchwartzNumberZeros} we set $c = \\frac{\\left| F \\right|}{\\deg(p)}$ and get\n\t\\[ \\frac{\\left| \\SET{\\xi \\in F^X ~\\middle|~ p[X=\\xi] = 0} \\right|}{ \\left| F^X \\right|} \\leq \\frac{\\left| F \\right|^n}{c\\cdot \\left| F \\right|^n} = \\frac{1}{c} = \\frac{\\deg(p)}{\\left| F \\right|}. \\qedhere\\]\n\\end{proof}\n\n\\begin{proposition}[\\cite{Marx09}, Proposition 3.11]\\label{prop:randompolytimeTransversalMatroid}\\PRFR{Mar 7th}\n\tLet $E$ be a finite set, $r\\in \\mathbb{N}$, and ${\\mathcal{A}} = (A_i)_{i=1}^r \\subseteq E$ be a family of subsets of $E$.\n\tThen a matrix $\\mu\\in \\mathbb{R}^{E\\times \\SET{1,2,\\ldots,r}}$ with $M(\\mu) = M({\\mathcal{A}})$ can be constructed in randomized polynomial time.\n\\end{proposition}\n\\begin{proof}\\PRFR{Mar 7th}\n\tFor all $k\\in \\mathbb{N}$ with $k>1$, we write $\\mathrm{unif}(k)$\n\tin order to denote an integer that has been randomly sampled from a uniform distribution on $\\SET{1,2,\\ldots,k}$. Several instances\n\tof $\\mathrm{unif}(k)$ shall denote independently sampled random variables.\n\t%\n\n\tLet $p\\in \\mathbb{N}$ be an arbitrary parameter.\n\tWe define the random matrix $\\mu\\in \\mathbb{R}^{E\\times \\SET{1,2,\\ldots,r}}$ by\\footnote{D.~Marx uses samples \n\tfrom $\\mathrm{unif}\\left( 2^p\\cdot \\left| E \\right|\\cdot 2^{\\left| E \\right|} \\right)$ and uses the argument that there are at most\n\t$2^{\\left| E \\right| }$ independent sets. This line of arguments is valid, \n\tyet it does not use the fact that if $X$ is independent in $M(\\mu)$, then all subsets of $X$ are independent in $M(\\mu)$, too; \n\tconsequently, the probability of failure is overestimated.}\n\t\\[ \\mu(e,i) = \\begin{cases}[r]\n\t\t\t\t\t\\mathrm{unif} \\left( 2^p\\cdot r\\cdot Q \\right) & \\quad\\text{if~} e\\in A_i, \\\\\n\t\t\t\t\t0 & \\quad\\text{otherwise,}\n\t\\end{cases} \n\t\\]\n\twhere $$Q = \\binom{\\left| E \\right|}{\\left\\lceil \\frac{\\left| E \\right|}{2}\\right\\rceil}.$$\n\tClearly, $Q \\leq 2^{\\left| E \\right|}$ with equality if $\\left| E \\right| = 1$.\n\tObserve that sampling $\\mathrm{unif}(2^k)$ can be done by sampling $k$ bits from a uniform distribution. Thus $\\mu$ can be obtained\n\tby sampling at most\n\t $\\left| E \\right|\\cdot r \\cdot \\left(p + \\lceil \\log_2\\left( Q + r \\right) \\rceil \\right)$ uniform random bits.\n\tWe show that \\mbox{$\\Pr\\left( M(\\mu) \\not= M({\\mathcal{A}}) \\vphantom{A^A}\\right) \\leq \\frac{1}{2^p}$.}\n\tLet $X\\subseteq E$ be an independent set with respect to $M(\\mu)$. Then ${\\mathrm{idet}~}(M| X\\times \\SET{1,2,\\ldots,r}) = 1$,\n\tso there is an injective map $\\phi\\colon X\\longrightarrow \\SET{1,2,\\ldots,r}$ such that $\\mu(x,\\phi(x)) \\not= 0$\n\tfor all $x\\in X$. By construction of $\\mu$ we obtain that in this case $x\\in A_{\\phi(x)}$. Therefore $X$ is a partial transversal\n\tof ${\\mathcal{A}}$, and so $X$ is independent in $M({\\mathcal{A}})$, too.\n\n\t\\PRFR{Mar 7th}\n\t\\noindent\n\tNow let $X\\subseteq E$ be a base of $M({\\mathcal{A}})$. Thus $X$ is a maximal partial transversal of ${\\mathcal{A}}$ and\n\tthere is an injective map $\\phi\\colon X\\longrightarrow \\SET{1,2,\\ldots,r}$ such that $x\\in A_{\\phi(x)}$ for all $x\\in X$.\n\tLet $X=\\dSET{x_1,x_2,\\ldots,x_k}$, then we may define the matrix $\\nu \\in \\mathbb{R}[X]^{X\\times \\phi[X]}$\n\twhere\n\t\t\\[ \\nu(x,i) = \\begin{cases}[r]\n\t\t\t\t\t\tx_i & \\quad\\text{if~} i = \\phi(x),\\\\\n\t\t\t\t\t\t\\mu(x,i) & \\quad\\text{otherwise.}\n\t\t\\end{cases}\\]\n\tThen $\\det(\\nu)$ is a polynomial of degree $\\left| X \\right| = k \\leq r$ with leading monomial $x_1x_2\\cdots x_k$ in $\\mathbb{R}[X]$,\n\tand if $\\xi\\in \\mathbb{R}^X$ is the vector\n\twhere $\\xi(x) = \\mu(x,\\phi(x))$ for all $x\\in X$, we have the equality\n\t\\[ \\det(\\mu| X\\times \\phi[X]) = \\left( \\det(\\nu) \\right)[X=\\xi] .\\]\n\tRemember that each value $\\xi(x)$ has been uniformly sampled from a set with cardinality $ 2^p\\cdot r \\cdot Q$,\n\tthus Lemma~\\ref{lem:probOfZero} yields\n\t\\[ \\Pr\\left( \\vphantom{A^A}\\det(\\mu| X\\times\\phi[X]) = 0 \\right) \\leq \\frac{\\left| X \\right|}{2^p\\cdot r \\cdot Q} \\leq \\frac{1}{2^p\\cdot Q}.\\]\n\tThere are at most $\\binom{\\left| E \\right|}{\\mathrm{rk}_{M({\\mathcal{A}})}(E)}$ different bases in $M({\\mathcal{A}})$,\n\tand the family of all subsets of $E$ with cardinality $\\left\\lceil \\frac{\\left| E \\right|}{2} \\right\\rceil$\n\tis a maximal-cardinality anti-chain in the power set lattice of $E$.\n\tTherefore, there are at most $Q$ different bases\n\tin $M({\\mathcal{A}})$\n\tneeded to detect failure of $M({\\mathcal{A}})=M(\\mu)$.\n\tThus we obtain\n\t\\[ \\Pr\\left( M(\\mu) \\not= M({\\mathcal{A}}) \\vphantom{A^A}\\right) \\leq \\sum_{B\\in {\\mathcal{B}}(M({\\mathcal{A}}))} \\frac{1}{2^p\\cdot Q} \\leq \\frac{1}{2^p} . \\qedhere\\]\n\\end{proof}\n\n\\noindent Clearly, if the rank of $M({\\mathcal{A}})$ is known, we may use the better factor\n$Q = \\binom{\\left| E \\right|}{\\mathrm{rk}_M(E)}$ in the probabilistic construction of $\\mu$\ngiven in the proof of Proposition~\\ref{prop:randompolytimeTransversalMatroid}. \n If we also know the number of bases, we may even use $Q= \\left| {\\mathcal{B}}(M({\\mathcal{A}})) \\right|$.\n\n\\chapter{Oriented Matroids}\\label{ch:OMs}\n\\remblue{\n\\remred{TODO} Oriented matroids have been studied by....\n\n\\noindent Historically, oriented matroids have been defined by J.~Folkman and J.~Lawrence \\cite{FoLa78} and by R.G.~Bland and M.~Las~Vergnas \\cite{BlV78}\nas structures that arise when the circuits of a matroid are turned into signed subsets of the ground set such that certain properties mimicking those of the\nsign patterns associated with the coefficients of non-trivial linear combinations of the zero vector are fulfilled. Both papers show that\n many aspects of ordinary matroid theory carry over to oriented matroids, most notably the concepts of duality and minors are reflected by oriented matroids\n in a way that is compatible with the structure of the underlying matroids. Therefore we are at liberty to merge aspects of the theory of oriented matroids\n into our definition of oriented matroids.\n}\n\n\\section{Quick Introduction to Oriented Matroids}\n\n\\marginpar{Jan 8th}\nLet us consider a matroid $M(\\mu)$ where $\\mu\\in \\mathbb{R}^{E\\times \\SET{1,2,\\ldots,r}}$ is a finite matrix over the reals\nwith full column rank, i.e. such that $\\mathrm{rk}_{M(\\mu)}(E) = r$.\nWhenever $C\\in {\\mathcal{C}}(M(\\mu))$ is a circuit, there are coefficients $\\alpha\\colon C\\longrightarrow \\mathbb{R}$ such that\n$\\alpha(c)\\not= 0$ for all $c\\in C$ and such that\n\\[ \\sum_{c\\in C} \\alpha(c) \\cdot \\mu_c = 0 \\]\nholds in the vector space $\\mathbb{R}^r$. Furthermore, $\\alpha$ is uniquely determined by $\\SET{\\mu_c\\mid c\\in C}$\nup to a homogeneous factor $\\lambda \\in \\mathbb{R}\\BSET{0}$, i.e. whenever the\nequality $\\sum_{c\\in C} \\beta(c) \\cdot \\mu_c = 0$ holds for $\\beta\\colon C\\longrightarrow \\mathbb{R}$ with $\\beta$ not\nconstantly zero on $C$, then there is some $\\lambda \\in \\mathbb{R}\\BSET{0}$ with\n$\\alpha(c) = \\lambda \\beta(c)$ for all $c\\in C$. Therefore, the signs of the coefficients are determined \nup to a possible negation of all signs by\nthe circuit $C$ and the matrix $\\mu$. \n\n\\begin{definition}\\marginpar{Jan 8th}\n\tLet $E$ be a set. A \\deftext[signed subset]{signed subset of $\\bm E$} is a map\\label{n:signedsubset}\n\t\\[ X\\colon E\\longrightarrow \\SET{-1,0,1}.\\]\n\tWe denote the \\deftext[positive elements of a signed subset]{positive elements of $\\bm X$} by\n\t\\( X_+ = \\SET{x\\in E\\mid X(x) = 1}, \\)\\label{n:xplus}\n\tthe \\deftext[negative elements of a signed subset]{negative elements of $\\bm X$} shall be denoted by\n\t\\( X_- = \\SET{x\\in E\\mid X(x)=-1},\\)\\label{n:xminus}\n\tthe \\deftext[support of a signed subset]{support of $\\bm X$} is defined as\n\t\\( X_\\pm = \\SET{x\\in E\\mid X(x)\\not= 0},\\)\\label{n:xpm}\n\tand the \\deftext[zero-set of a signed subset]{zero-set of $\\bm X$} is \n\tdenoted by $X_0 = E\\backslash X_\\pm$.\\label{n:xzero} The \\deftext[negation of a signed subset]{negation of $\\bm X$}\n\tis the signed subset \\(-X\\)\\label{n:minusx} where $-X\\colon E\\longrightarrow \\SET{-1,0,1},$ $e\\mapsto -X(e)$. \n\tLet $C\\subseteq E$ and $\\alpha \\colon C\\longrightarrow \\mathbb{R}$ be a vector of coefficients. \n\tThe \\deftext[signs of a over E@signs of $\\alpha$ over $E$]{signs of $\\bm \\alpha$ over $\\bm E$} shall be denoted by $E_\\alpha$, which is defined\n\tto be the map\n\t\\[ E_\\alpha \\colon E\\longrightarrow \\SET{-1,0,1},\\,e\\mapsto \\begin{cases} \\hphantom{-}0 & \\quad \\text{if }e\\notin C {\\mathop{\\text{~or~}}} \\alpha(e) = 0,\\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t-1 & \\quad \\text{if }\\alpha(e) < 0,\\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\hphantom{-}1 & \\quad \\text{if }\\alpha(e) > 0.\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\end{cases}\\]\n\tThe \\deftextX{class of all signed subsets of $\\bm E$}\\label{n:signedsubsets} shall be denoted by $\\sigma E$.\n\tLet $X,Y\\in \\sigma E$ be signed subsets of $E$. We say that \\deftext[signed subset]{$\\bm X$ is a signed subset of $\\bm Y$},\n\tif $X_+\\subseteq Y_+$ and $X_- \\subseteq Y_-$. We denote this fact by writing $X\\subseteq_\\sigma Y$.\\label{n:subsetsigma}\n\tFurthermore, we write $X\\subsetneq_\\sigma Y$ whenever $X\\subseteq_\\sigma Y$ and $X_\\pm \\subsetneq Y_\\pm$ holds.\n\tThe \\deftext{empty signed subset} of $E$ is the map\\label{n:emptysigma}\n\\( \\emptyset_{\\sigma E} \\colon E\\longrightarrow \\SET{-1,0,1},\\,e\\mapsto 0 .\\)\n\\end{definition}\n\n\\begin{notation}\\PRFR{Apr 5th}\n\tLet $E$ be a finite set, and let $C\\in \\sigma E$ such that $C_+ = \\dSET{p_1,p_2,\\ldots,p_m}$ and $C_- = \\dSET{n_1,n_2,\\ldots,n_k}$.\n\tWe shall denote $C$ by both\n\t\\[ \\SET{p_1,p_2,p_3,\\ldots,p_m, -n_1, -n_2,\\ldots,-n_k} \\]\n\tand\n\t\\[ \\SET{+p_1,+p_2,+p_3,\\ldots,+p_m, -n_1, -n_2,\\ldots,-n_k},\\]\n\ti.e. we write a list of $C_\\pm$ where every element from $C_+$ has either no prefix or a $+$-sign, and where every element of $C_-$\n\thas a $-$-sign as prefix. The elements of $C_0$ are not listed. As with normal sets, we disregard the order in which the elements of $C_\\pm$ are listed.\n\\end{notation}\n\n\\begin{example}\\marginpar{Jan 8th}\nWith regard to a fixed representation $\\mu\\in \\mathbb{R}^{E\\times\\SET{1,2,\\ldots,r}}$,\nevery circuit $C\\in {\\mathcal{C}}(M(\\mu))$ gives rise to two different signed subsets of $E$:\nLet $\\alpha\\colon C\\longrightarrow \\mathbb{R}$ be not constantly zero on $C$ with $\\sum_{c\\in C}\\alpha(c)\\cdot\\mu_c=0$,\nthen $E_\\alpha$ and $-E_\\alpha$ are the signed subsets of $E$ \nthat correspond to the signs of non-trivial coefficients $\\alpha\\colon C\\longrightarrow \\mathbb{R}$ with $\\sum_{c\\in C}\\alpha(c)\\cdot \\mu_c = 0$.\n\\end{example}\n\n\n\\begin{definition}\\marginpar{Jan 8th}\n\tLet $E$ be a finite set, $C,D\\in \\sigma E$ be signed subsets of $E$. We define the \\deftext[separator of signed subsets]{separator of $\\bm C$ and $\\bm D$}\n\tto be the set\\label{n:sep} \\[ {\\mathrm{sep}}(C,D) = \\left( C_+ \\cap D_- \\right) \\cup \\left( C_- \\cap D_+ \\right).\n\t\\qedhere \\]\n\\end{definition}\n\n\n\\noindent There is a notion of orthogonality for signed subsets which generalizes the ordinary orthogonality in vector spaces\n(see \\cite{BlVSWZ99}, p.115; \\cite{Ni12}, p.27).\n\n\\begin{definition}\\label{def:XorthoY}\\marginpar{Jan 8th}\n\tLet $E$ be a finite set, $C,D\\in\\sigma E$ be signed subsets of $E$\n\tThen $C$ and $D$ shall be called \\deftext{orthogonal signed subsets},\n\tif either\n\t\\begin{enumerate}\\renewcommand{\\theenumi}{(\\roman{enumi})}\\renewcommand{\\labelenumi}{\\theenumi}\n\t\\item there are $e,f\\in E$, such that\n\t\\[ C(e)\\cdot D(e) = - C(f)\\cdot D(f) \\not= 0 \\]\n\tholds; or\n\t\\item for all $e\\in E$, the equation\n\t\\[ C(e)\\cdot D(e) = 0 \\]\n\tholds.\n\\end{enumerate}\n\tWe write $X\\bot Y$\\label{n:XorthoY} in order to denote that $X$ and $Y$ are orthogonal,\n\tand $X\\!\\!\\not\\!\\!\\bot\\, Y$ to denote that $X$ and $Y$ are not orthogonal. In the latter case $X_\\pm\\cap Y_\\pm \\not=\\emptyset$ and \n\tthe common elements of the supports of $X$ and $Y$ \n\tall have the same relative sign with respect to $X$ and $Y$, i.e. $X(e) = \\alpha\\cdot Y(e)$ for all $e\\in X_\\pm \\cap Y_\\pm$\n\tand some $\\alpha\\in \\SET{-1,1}$ that does not depend on the choice of $e$.\n\\end{definition}\n\n\\needspace{3\\baselineskip}\n\n\\begin{lemma}\\label{lem:MinusCOrthoD}\\marginpar{Jan 8th}\n\tLet $E$ be a finite set, $C,D\\in \\sigma E$. Then\n\t$C\\bot D$ if and only if $\\left( -C \\right)\\bot D$ if and only if $C\\bot\\left( -D \\right)$ if and only if $\\left( -C \\right)\\bot \\left( -D \\right)$.\n\\end{lemma}\n\\begin{proof}\\marginpar{Jan 8th}\n\tSince $\\bot$ is obviously a symmetric relation, it suffices to show that $C\\bot D$ implies $\\left( -C \\right)\\bot D$.\n\tBut for every $e\\in E$, $\\left( -C \\right)(e) = -C(e)$, \n\ttherefore both properties {\\em (i)} and {\\em (ii)} of Definition~\\ref{def:XorthoY} carry over from $C$ to $-C$.\n\\end{proof}\n\n\n\\begin{lemma}\\marginpar{Jan 8th}\n\tLet $E$ be a finite set, $\\alpha,\\beta \\in \\mathbb{R}^E$ with $\\langle \\alpha,\\beta \\rangle = 0$.\n\tThen $E_\\alpha \\bot E_\\beta$.\n\\end{lemma}\n\\begin{proof}\\marginpar{Jan 8th}\n\tIf $\\left( E_\\alpha \\right)_\\pm \\cap \\left( E_\\beta \\right)_\\pm = \\emptyset$, then\n\t{\\em (ii)} of Definition~\\ref{def:XorthoY} holds, thus $E_\\alpha \\bot E_\\beta$.\n\tOtherwise, there is some $e\\in E$ with $\\alpha(e)\\cdot \\beta(e) \\not= 0$.\n\tLet $$E_e = \\SET{e'\\in E ~\\middle|~ \\mathrm{sgn}\\left(\\alpha(e)\\cdot \\beta(e)\\right) = \\mathrm{sgn}\\left(\\alpha(e')\\cdot \\beta(e')\\right)}.$$\n\tSince $\\langle a,b \\rangle=0$, we have\n\t\\[ -\\sum_{e'\\in E_e} \\alpha(e') \\cdot \\beta(e')\\,\\,\\, =\\,\\,\\, \\langle \\alpha,\\beta\\rangle -\\sum_{e'\\in E_e} \\alpha(e') \\cdot \\beta(e')\n\t\\,\\,\\, =\\,\\,\\, \\sum_{f\\in E\\backslash E_{e}} \\alpha(f)\\cdot \\beta(f).\\]\n\tWe give an indirect argument and assume that {\\em (i)} does not hold. Then for all $f\\in E\\backslash{E_e}$,\n\twe have $\\alpha(f)\\cdot \\beta(f) = 0$. Thus $-\\sum_{e'\\in E_e} \\alpha(e') \\cdot \\beta(e') = 0$, but the sign of $\\alpha(e')\\cdot \\beta(e')$\n\tis the same for every $e'\\in E_e$. Therefore $\\alpha(e')\\cdot \\beta(e')=0$ for all $e\\in E_e$, \n\tcontradicting the assumption that $\\alpha(e)\\cdot \\beta(e) \\not= 0$, so {\\em (i)} must hold. Thus $E_\\alpha\\bot E_\\beta$.\n\\end{proof}\n\n\n\n\\needspace{6\\baselineskip}\n\\begin{definition}\\label{def:OrientedMatroid}\\marginpar{Jan 8th}\n\tLet $E$ be a finite set, ${\\mathcal{C}}\\subseteq \\sigma E$ and ${\\mathcal{C}}^\\ast \\subseteq \\sigma E$.\n\tThe triple ${\\mathcal{O}} = (E,{\\mathcal{C}},{\\mathcal{C}}^\\ast)$\\label{n:Ocal} is called \\deftext{oriented matroid},\n\tif the following properties hold:\n\t\\begin{enumerate}\\label{n:Cx}\n\t\t\\item[(${\\mathcal{C}}$1)] $\\emptyset_{\\sigma E}\\notin {\\mathcal{C}}$,\n\t\t\\item[(${\\mathcal{C}}$2)] for all $C\\in\\sigma E$, $C\\in {\\mathcal{C}}$ if and only if $-C \\in {\\mathcal{C}}$,\n\t\t\\item[(${\\mathcal{C}}$3)] for all $X,Y\\in {\\mathcal{C}}$, $X_\\pm \\subseteq Y_\\pm$ implies $X = Y$ or $X = -Y$,\n\t\t\\item[(${\\mathcal{C}}$4)] for all $X,Y\\in {\\mathcal{C}}$ with $X\\not= -Y$ and all $e\\in X_+\\cap Y_-$ and $f\\in X_\\pm \\backslash {\\mathrm{sep}}(X,Y)$, there is some $Z\\in{\\mathcal{C}}$ such that\n\t\t\t\t$e\\notin Z_\\pm$, $Z(f) = X(f)$, $Z_+ \\subseteq X_+\\cup Y_+$ and $Z_- \\subseteq X_-\\cup Y_-$;\n\n\t\t\\item[(${\\mathcal{C}}^\\ast$1)] $\\emptyset_{\\sigma E}\\notin {\\mathcal{C}}^\\ast$,\n\t\t\\item[(${\\mathcal{C}}^\\ast$2)] for all $C'\\in\\sigma E$, $C'\\in {\\mathcal{C}}^\\ast$ if and only if $-C' \\in {\\mathcal{C}}^\\ast$,\n\t\t\\item[(${\\mathcal{C}}^\\ast$3)] for all $X',Y'\\in {\\mathcal{C}}^\\ast$, $X'_\\pm \\subseteq Y'_\\pm$ implies $X' = Y'$ or $X' = -Y'$,\n\t\t\\item[(${\\mathcal{C}}^\\ast$4)] for all $X',Y'\\in {\\mathcal{C}}^\\ast$ with $X'\\not= -Y'$ and all $e\\in X'_+\\cap Y'_-$ and $f\\in X'_\\pm \\backslash {\\mathrm{sep}}(X',Y')$, there some is $Z'\\in{\\mathcal{C}}^\\ast$ such that\n\t\t\t\t$e\\notin Z'_\\pm$, $Z'(f) = X'(f)$, $Z'_+ \\subseteq X'_+\\cup Y'_+$ and $Z'_- \\subseteq X'_-\\cup Y'_-$;\n\n\t\t\\item[(${\\mathcal{O}}$1)] for all $C\\in {\\mathcal{C}}$ and $C'\\in{\\mathcal{C}}^\\ast$, we have $C\\bot C'$,\n\t\t\\item[(${\\mathcal{O}}$2)] there is a matroid $M=(E,{\\mathcal{I}})$, such that $$\\SET{C_\\pm ~\\middle|~\\vphantom{C'} C\\in {\\mathcal{C}}} = {\\mathcal{C}}(M) \\,\\,{\\mathop{\\text{~and~}}}\\,\\,\n\t\t\\SET{C'_\\pm ~\\middle|~ C'\\in {\\mathcal{C}}^\\ast} = {\\mathcal{C}}(M^\\ast). $$\n\n\t\n\t\\end{enumerate}\n\tIn this case, the elements $C \\in {\\mathcal{C}}$ shall be called \\deftext[signed circuits of O@signed circuits of ${\\mathcal{O}}$]{signed circuits of $\\bm {\\mathcal{O}}$},\n\tand ${\\mathcal{C}}$ shall be the \\deftextX{family of signed circuits of $\\bm {\\mathcal{O}}$}. \n\tLikewise, the elements $C' \\in {\\mathcal{C}}^\\ast$ shall be called \\deftext[signed cocircuits of O@signed cocircuits of ${\\mathcal{O}}$]{signed cocircuits of $\\bm {\\mathcal{O}}$},\n\tand ${\\mathcal{C}}^\\ast$ shall be the \\deftextX{family of signed cocircuits of $\\bm {\\mathcal{O}}$}.\n\tFurthermore, $M({\\mathcal{O}})$\\label{n:MOcal} shall denote the \\deftext[underlying matroid of O@underlying matroid of ${\\mathcal{O}}$]{underlying matroid of ${\\mathcal{O}}$}, whose existence and uniqueness is guaranteed by {\\em (${\\mathcal{O}}$2)}.\n\\end{definition}\n\n\\begin{remark}\\label{rem:sufficientConditionOM}\\marginpar{Jan 8th}\nThe above definition of an oriented matroid is redundant in the sense that some of the properties follow from other properties easily. For\ninstance, {\\em (${\\mathcal{O}}$1)} and {\\em(${\\mathcal{O}}$2)} imply all other properties from Definition~\\ref{def:OrientedMatroid} \n(see \\cite{Ox11}, p.401).\n We give a quick overview over the most common cryptomorphic ways to define oriented matroids via signed circuits.\nFor full disclosure on these\nless redundant definitions of oriented matroids, we refer the reader to \\cite{BlVSWZ99}, \\cite{BlV78}, and \\cite{FoLa78}. \n\nIn \\cite{BlV78}, R.G.~Bland and M.~Las~Vergnas define oriented matroids to be pairs $(E,{\\mathcal{C}})$ such that ${\\mathcal{C}}\\subseteq \\sigma E$\n has the properties {\\em (${\\mathcal{C}}$1)}, {\\em (${\\mathcal{C}}$2)}, {\\em (${\\mathcal{C}}$3)}, and the property\n\\begin{enumerate}\\label{n:Ccal4p}\n\t\t\\item[(${\\mathcal{C}}$4')] for all $X,Y\\in {\\mathcal{C}}$ with $X\\not= -Y$ and all $e\\in X_+\\cap Y_-$, there is some $Z\\in{\\mathcal{C}}$ such that\n\t\t\t\t$e\\notin Z_\\pm$, $Z_+ \\subseteq X_+\\cup Y_+$ and $Z_- \\subseteq X_-\\cup Y_-$.\n\\end{enumerate}\n\tIn Theorem~2.1 \\cite{BlV78}, R.G.~Bland and M.~Las~Vergnas prove that if we assume {\\em (${\\mathcal{C}}$1)}, {\\em (${\\mathcal{C}}$2)}, {\\em (${\\mathcal{C}}$3)}, then \n\tthe properties {\\em (${\\mathcal{C}}$4)} and {\\em (${\\mathcal{C}}$4')} are equivalent. In Theorem~2.2 \\cite{BlV78}, they prove\n\tthat if $(E,{\\mathcal{C}},{\\mathcal{C}}^\\ast)$ is an oriented matroid as in our Definition~\\ref{def:OrientedMatroid}, then ${\\mathcal{C}}$ uniquely\n\tdetermines ${\\mathcal{C}}^\\ast$ and vice versa.\n\tTherefore, in order to define an oriented matroid on the ground set $E$, it suffices to determine ${\\mathcal{C}}$, and show that \n\t{\\em (${\\mathcal{C}}$1)}, {\\em (${\\mathcal{C}}$2)}, {\\em (${\\mathcal{C}}$3)}, and {\\em (${\\mathcal{C}}$4')} hold. Since the underlying matroid $M({\\mathcal{O}})$\n\tis already uniquely determined by the supports of the elements of ${\\mathcal{C}}$, we can reconstruct the supports of ${\\mathcal{C}}^\\ast$\n\tby examining the cocircuits of $M({\\mathcal{O}})$.\n\tIn order to find the correct signatures of $D\\in{\\mathcal{C}}^\\ast$,\n\twe can set the sign $D(d)$ for an arbitrarily chosen $d\\in D_\\pm$ to $+1$, or to $-1$ in order to generate the corresponding \n\tnegation $-D$. If $D_\\pm =\\SET{d}$, we are done. \n\tIf $\\left| D_\\pm \\right| > 1$, \n\tthen Lemma~\\ref{lem:CircuitCocircuitIntersectInTwo}\n\twith respect to the dual matroid $M({\\mathcal{O}})^\\ast$ \n\tindicates that for every $c\\in D_\\pm\\BSET{d}$, there is a circuit $C\\in {\\mathcal{C}}$ such that $C_\\pm \\cap D_\\pm = \\SET{c,d}$.\n\tWe let $D(c) = D(d)$ if $C(c) \\not= C(d)$, and $D(c) = -D(d)$ if $C(c) = C(d)$. Clearly, this is the only possibility which\n\tyields $D\\bot C$.\n\n\nIn \\cite{FoLa78}, J.~Folkman and J.~Lawrence independently defined\n their version of oriented matroids to be triples $(E_\\sigma,{\\mathcal{C}},-)$ subject to basically \nthe same properties as R.G.~Bland's and M.~Las~Vergnas's version of oriented matroids, but where the latter used explicit signs $+$ and $-$, the former used a fix-point free involution on $E_\\sigma$ that maps $+e$ to $-e$ and vice versa. Thus the oriented matroids of\n J.~Folkman and J.~Lawrence correspond to {\\em reorientation classes} of oriented matroids in this work.\n\\end{remark}\n\n\n\\begin{example}\\PRFR{Apr 5th}\n\tLet $E$ be a finite set and $M=(E,2^E)$ be the free matroid on $E$. Then\n\t$${\\mathcal{O}} = \\left(E,\\emptyset,\\SET{\\vphantom{E^E}\\SET{e},\\SET{-e}~\\middle|~ e\\in E}\\right)$$ is the only possible oriented matroid with $M({\\mathcal{O}}) = M$.\n\\end{example}\n\n\\noindent It is straightforward, that applying an $M({\\mathcal{O}})$-automorphism to the signed circuits ${\\mathcal{C}}$ and cocircuits ${\\mathcal{C}}^\\ast$\nof an oriented matroid ${\\mathcal{O}}$ yields another oriented matroid.\n\n\\needspace{4\\baselineskip}\n\\begin{definition}\\PRFR{Apr 5th}\n\tLet ${\\mathcal{O}}=(E,{\\mathcal{C}},{\\mathcal{C}}^\\ast)$ be an oriented matroid, and let $\\phi\\colon E\\longrightarrow E$ be an $M({\\mathcal{O}})$-automorphism.\n\tThe \\deftext[relabeling of O by p@relabeling of ${\\mathcal{O}}$ by $\\phi$]{relabeling of $\\bm {\\mathcal{O}}$ by $\\bm \\phi$}\n\tshall be the triple \\[ \\phi[{\\mathcal{O}}] = \\left(E,{\\mathcal{C}}_\\phi, {\\mathcal{C}}_\\phi^\\ast\\right) \\]\n\twhere \\(\n\t{\\mathcal{C}}_\\phi = \\SET{C\\circ \\phi \\in \\sigma E ~\\middle|~ C \\in {\\mathcal{C}}}\n\t\\) and \\( {\\mathcal{C}}_\\phi^\\ast = \\SET{C'\\circ \\phi \\in \\sigma E ~\\middle|~ C'\\in {\\mathcal{C}}^\\ast} \\).\n\\end{definition}\n\n\\begin{lemma}\n\tLet ${\\mathcal{O}}$ be an oriented matroid and $\\phi$ and $M({\\mathcal{O}})$-automorphism. Then $\\phi[{\\mathcal{O}}]$ is an oriented matroid.\n\\end{lemma}\n\\begin{proof}\n\tFor $X\\in\\sigma E$, we have $(X\\circ\\phi)_+ = \\phi[X_+]$, $(X\\circ\\phi)_- = \\phi[X_-]$,\n\t$(X\\circ\\phi)_\\pm = \\phi[X_\\pm]$, and $-(X\\circ\\phi) = (-X)\\circ \\phi$. As a consequence, for $C,D\\in \\sigma E$,\n\twe have that $C\\bot D$ if and only if $C\\circ \\phi \\bot D\\circ \\phi$ as well as ${\\mathrm{sep}}(C\\circ \\phi,D\\circ \\phi) = \\phi[{\\mathrm{sep}}(C,D)]$.\n\tFurthermore, for $Y\\subseteq E$,\n\twe have $Y\\in {\\mathcal{C}}(M({\\mathcal{O}}))$ if and only if $\\phi[Y] \\in {\\mathcal{C}}(M({\\mathcal{O}}))$, as well as $Y\\in {\\mathcal{C}}(M({\\mathcal{O}})^\\ast)$\n\tif and only if $\\phi[Y]\\in {\\mathcal{C}}(M({\\mathcal{O}})^\\ast)$.\n\tWith these properties, it is straightforward yet tiresome to verify using the definition of $\\phi[{\\mathcal{O}}]$,\n\tthat the axioms for ${\\mathcal{O}}$ carry over to $\\phi[{\\mathcal{O}}]$.\n\\end{proof}\n\n\n \n\\begin{definition}\n\tLet ${\\mathcal{O}}=(E,{\\mathcal{C}},{\\mathcal{C}}^\\ast)$ be an oriented matroid. The \\deftext[dual oriented matroid]{dual oriented matroid of $\\bm {\\mathcal{O}}$}\n\tis the triple ${\\mathcal{O}}^\\ast = (E,{\\mathcal{C}}^\\ast,{\\mathcal{C}})$.\\label{n:OcalBot}\n\\end{definition}\n\n\\needspace{3\\baselineskip}\n\\begin{lemma}\n\tLet ${\\mathcal{O}}$ be an oriented matroid. Then ${\\mathcal{O}}^\\ast$ is an oriented matroid, too.\n\\end{lemma}\n\\begin{proof}\n\tObserve that the axioms {\\em (${\\mathcal{C}} i$)} for ${\\mathcal{O}}$ are equivalent to the axioms {\\em (${\\mathcal{C}}^\\ast i$)} for ${\\mathcal{O}}^\\ast$,\n\tanalogously {\\em (${\\mathcal{C}}^\\ast i$)} for ${\\mathcal{O}}$ are equivalent to {\\em (${\\mathcal{C}} i$)} for ${\\mathcal{O}}^\\ast$; where $i\\in \\SET{1,2,3,4}$. Furthermore,\n\t{\\em (${\\mathcal{O}}$1)} is symmetric in itself, so it holds for ${\\mathcal{O}}$ if and only if it holds for ${\\mathcal{O}}^\\ast$. Moreover,\n\tevery witness $M({\\mathcal{O}})$, that certifies {\\em (${\\mathcal{O}}$2)} for ${\\mathcal{O}},$ yields a witness $\\left( M({\\mathcal{O}}) \\right)^\\ast$,\n\tthat certifies {\\em (${\\mathcal{O}}$2)} for ${\\mathcal{O}}^\\ast$, and vice versa. Therefore, the triple ${\\mathcal{O}}$ is an oriented matroid\n\tif and only if the triple ${\\mathcal{O}}^\\ast$ is an oriented matroid, so we obtain that ${\\mathcal{O}}^\\ast$ is an oriented matroid from the premises of this lemma.\n\\end{proof}\n\n\n\n\n\\needspace{6\\baselineskip}\n\\begin{definition}\\label{def:Omu}\\marginpar{Jan 8th}\n\tLet $E$ be a finite set and $\\mu\\in \\mathbb{R}^{E\\times \\SET{1,2,\\ldots,r}}$ be a matrix.\n\tThe \\deftext[oriented matroid represented by mu@oriented matroid represented by $\\mu$]{oriented matroid represented by $\\bm \\mu$}\n\tis the uniquely determined oriented matroid \\label{n:OcalMu}${\\mathcal{O}}(\\mu) = (E,{\\mathcal{C}}_\\mu,{\\mathcal{C}}_\\mu^\\ast)$ where\n\t\\[ {\\mathcal{C}}_\\mu = \\SET{C\\in {\\mathcal{D}}_\\mu ~\\middle|~ \\forall C'\\in{\\mathcal{D}}_\\mu \\colon \\,C'_\\pm \\subseteq C_\\pm \\Rightarrow C'_\\pm = C_\\pm} \\]\n\tand \n\t\\[ {\\mathcal{D}}_\\mu = \\SET{E_\\alpha \\in \\sigma E\\BSET{\\emptyset_{\\sigma E}} ~~\\middle|~~ \\alpha\\in \\mathbb{R}^E,\\,\\alpha\\not\\equiv 0,\\,\\sum_{e\\in E}\\alpha(e)\\cdot \\mu_e = 0 } .\\]\n\\end{definition}\n\n\\begin{lemma}\\label{lem:OcalMu}\\marginpar{Jan 8th}\n\tLet $E$ be a finite set and $\\mu\\in \\mathbb{R}^{E\\times \\SET{1,2,\\ldots,r}}$ be a matrix. Then ${\\mathcal{O}}(\\mu)$ is indeed an oriented matroid.\n\\end{lemma}\n\\begin{proof}\\marginpar{Jan 8th}\nBy Remark~\\ref{rem:sufficientConditionOM}, it suffices to show that {\\em (${\\mathcal{C}}$1)}, {\\em (${\\mathcal{C}}$2)}, {\\em (${\\mathcal{C}}$3)}, and {\\em (${\\mathcal{C}}$4')} hold for ${\\mathcal{C}}_\\mu$\n in Definition~\\ref{def:Omu}.\n{\\em (${\\mathcal{C}}$1)} is obvious from the construction.\nLet $C\\in {\\mathcal{C}}_\\mu$, then there is a vector $\\alpha\\in \\mathbb{R}^E$ such that\n $\\sum_{e\\in E}\\alpha(e)\\cdot \\mu_e = 0$, thus there is $\\alpha'\\in \\mathbb{R}^E$ with $\\alpha'(e)=-\\alpha(e)$\n such that $$\\sum_{e\\in E}\\alpha'(e)\\cdot \\mu_e = -\\sum_{e\\in E}\\alpha(e)\\cdot \\mu_e = 0.$$ Clearly, $E_{\\alpha'} = -E_\\alpha$ and therefore $\\left( E_\\alpha \\right)_\\pm = \\left( E_{\\alpha'} \\right)_\\pm$. Thus $-C\\in{\\mathcal{C}}_\\mu$, so\n {\\em (${\\mathcal{C}}$2)} holds.\n\n \\par \\vspace*{.6\\baselineskip}\\noindent \\mbox{---\\hspace*{2.5mm}} \\marginpar{Jan 8th}\n We give an indirect argument for {\\em (${\\mathcal{C}}$3)}: Let $X,Y\\in {\\mathcal{C}}_\\mu$ with $X_\\pm = Y_\\pm$ and\n \\linebreak $Y\\notin \\SET{X,-X}$, such that\n $X_\\pm$ is minimal in ${\\mathcal{C}}_\\mu$ with respect to set-inclusion $\\subseteq$. There is an element\n$f\\in \\left( X_+\\cap Y_+ \\right)\\cup \\left( X_-\\cap Y_- \\right)$\nand an element $f'\\in {\\mathrm{sep}}(X,Y)$. Now let $\\alpha,\\alpha'\\in \\mathbb{R}^E$ with $\\sum_{e\\in E}\\alpha(e)\\cdot\\mu_e\n=\\sum_{e\\in E}\\alpha'(e)\\cdot\\mu_e = 0$ such that $E_\\alpha = X$ and $E_{\\alpha'} = Y$. Let $\\beta\\in\\mathbb{R}^E$ where\n $$\\beta(e) = \\alpha(e) - \\frac{\\alpha(f)}{\\alpha'(f)}\\alpha'(e)$$ for all $e\\in E$. Then\n \\begin{align*}\n \t\\sum_{e\\in E}\\beta(e)\\cdot \\mu_e & = \\sum_{e\\in E}\\left(\\alpha(e) - \\frac{\\alpha(f)}{\\alpha'(f)}\\alpha'(e) \\right)\\cdot \\mu_e \\\\\n \t& = \\left( \\sum_{e\\in E}\\alpha(e)\\cdot\\mu_e\\right) - \\frac{\\alpha(f)}{\\alpha'(f)}\\left(\\sum_{e\\in E}\\alpha'(e)\\cdot\\mu_e\\right) \\\\\n \t& = 0,\n \\end{align*}\n with $\\beta(f) = 0$ and $$\\beta(f') = \\alpha(f') - \\frac{\\alpha(f)}{\\alpha'(f)}\\cdot \\alpha'(f) \\not= 0$$\n since $f'\\in {\\mathrm{sep}}(E_\\alpha,E_{\\alpha'})$, therefore\n $\\emptyset \\not= \\left( E_\\beta \\right)_\\pm \\subsetneq \\left( E_\\alpha \\right)_\\pm$ which contradicts the minimality \n of $X_\\pm$. Therefore our assumption must be wrong and $Y\\in \\SET{X,-X}$.\n\n \\par \\vspace*{.6\\baselineskip}\\noindent \\mbox{---\\hspace*{2.5mm}} \\marginpar{Jan 8th}\n The proof of {\\em (${\\mathcal{C}}$4')} is similar: Let $X,Y\\in{\\mathcal{C}}_\\mu$ with $X\\not= -Y$, and let $f\\in X_+\\cap Y_-$. Again\n let $\\alpha,\\alpha'\\in \\mathbb{R}^E$ with $\\sum_{e\\in E}\\alpha(e)\\cdot\\mu_e\n=\\sum_{e\\in E}\\alpha'(e)\\cdot\\mu_e = 0$ such that $E_\\alpha = X$ and $E_{\\alpha'} = Y$. Define $\\beta\\in \\mathbb{R}^E$ as above,\nthen again $\\sum_{e\\in E}\\beta(e)\\cdot\\mu_e = 0$ and $\\beta(f) = 0$. Since $X\\not= -Y$, and obviously $X\\not= Y$,\nwe obtain that $X_\\pm \\not= Y_\\pm$ from {\\em (${\\mathcal{C}}$3)}. Thus there is an element $g\\in \\left( X_\\pm \\cap Y_0 \\right) \\cup \\left( X_0\\cap Y_\\pm \\right)$. Since either $\\alpha(g) = 0$ or $\\alpha'(g) = 0$, we obtain that\n $$\\beta(g) = \\alpha(g) -\\frac{\\alpha(f)}{\\alpha'(f)}\\alpha'(g) \\not= 0.$$\n\n So $\\emptyset_{\\sigma E} \\not= E_\\beta \\in {\\mathcal{D}}_\\mu$. \n Furthermore, for all $g\\in \\left( X_\\pm \\cap Y_0 \\right)$,\n we have $\\beta(g) = \\alpha(g)$, and thus $E_\\beta(g) = X(g)$.\n Also,\n for all $g\\in \\left(X_0\\cap Y_\\pm\\right)$ we have $$\\beta(g) = -\\frac{\\alpha(f)}{\\alpha'(f)}\\alpha'(g)$$ and \n since $\\mathrm{sgn}\\left(-\\frac{\\alpha(f)}{\\alpha'(f)}\\right) = 1$, \n we have $E_\\beta(g) = Y(g)$. Finally, for all $g\\in X_0\\cap Y_0$, we clearly have $\\beta(g) = 0$.\n Thus we found $Z=E_\\beta \\in {\\mathcal{D}}_\\mu$ with $Z(f) = 0$, $Z_+ \\subseteq X_+ \\cup Y_+$, and $Z_- \\subseteq X_- \\cup Y_-$.\n We claim that there is some $Z'\\in{\\mathcal{C}}_\\mu$ with $Z' \\subseteq_\\sigma Z$, yielding the desired signed circuit for {\\em (${\\mathcal{C}}$4')}.\n We give a constructive argument for this claim.\n Assume that $Z\\notin {\\mathcal{C}}_\\mu$, then there is an $Z'\\in {\\mathcal{C}}_\\mu$ such that $Z'_\\pm \\subsetneq Z_\\pm$.\n Let $\\gamma\\in \\mathbb{R}^E$ with $\\sum_{e\\in E}\\gamma(e)\\cdot\\mu_e = 0$ such that $E_\\gamma = Z'$.\n Let $f\\in Z'_\\pm \\cap Z_\\pm$ such that $\\left| \\frac{\\beta(f)}{\\gamma(f)} \\right|$ is minimal, thus for all $f'\\in Z'_\\pm \\cap Z_\\pm$ we have\n \\[ \\left| \\beta(f') \\right| \\geq \\left| \\frac{\\beta(f)}{\\gamma(f)} \\gamma(f') \\right|.\\]\n Therefore if we let $\\delta\\in \\mathbb{R}^E$ such that $\\delta(e) = \\beta(e) - \\frac{\\beta(f)}{\\gamma(f)}\\gamma(e)$,\n we have $\\sum_{e\\in E}\\delta(e)\\cdot \\mu_e = 0$ and $\\emptyset_{\\sigma E} \\not= E_\\delta \\subsetneq_\\sigma E_\\beta$,\n guaranteed by the choice of $f$.\n So we found some $Z'' = E_\\delta \\in {\\mathcal{D}}_\\mu$ with $Z'' \\subsetneq_\\sigma Z$.\nIf $Z''\\in {\\mathcal{C}}_\\mu$ we are done, otherwise we continue the last construction where $Z''$ takes on the role of $Z$.\nSince $E$ is finite and our construction\nstrictly reduces the cardinality of the support of the signed subset in question, we finally \nconstruct a signed subset $Z^{(2n)'}$ with minimal possible support, and therefore\nwe eventually find some $Z^{(2n)'}\\in {\\mathcal{C}}_\\mu$\nwith $Z^{(2n)'} \\subseteq_\\sigma Z$.\n\\end{proof}\n\n\\begin{corollary}\\label{cor:MOmuEQMmu}\\marginpar{Jan 8th}\n\tLet $E,C$ be finite sets, and $\\mu\\in \\mathbb{R}^{E\\times C}$ a real matrix.\n\tThen $$M({\\mathcal{O}}(\\mu)) = M(\\mu),$$ i.e. \n\tthe underlying matroid of the oriented matroid represented by $\\mu$ is the matroid represented by $\\mu$.\n\\end{corollary}\n\\begin{proof}\\marginpar{Jan 8th}\n\tObvious from Definition~\\ref{def:Mmu} and Definition~\\ref{def:Omu}. \n\\end{proof}\n\n\\begin{definition}\\marginpar{Jan 8th}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid. Then $M$ shall be called \\deftext{orientable matroid},\n\tif there is an oriented matroid ${\\mathcal{O}}=(E,{\\mathcal{C}},{\\mathcal{C}}^\\ast)$, such that $M = M({\\mathcal{O}})$,\n\tor equivalently\n\t$\\SET{C_\\pm \\mid C\\in {\\mathcal{C}}} = {\\mathcal{C}}(M)$. Furthermore, every oriented matroid ${\\mathcal{O}}$ with this property\n\tshall be called \\deftext[orientation of M@orientation of $M$]{orientation of $\\bm M$}.\n\\end{definition}\n\n\\begin{corollary}\\label{cor:Rorientable}\\marginpar{Jan 8th}\n\tEvery matroid that can be represented over $\\mathbb{R}$ is orientable.\n\\end{corollary}\n\\begin{proof}\\marginpar{Jan 8th}\n\tFollows from Lemma~\\ref{lem:OcalMu} and Corollary~\\ref{cor:MOmuEQMmu}.\n\\end{proof}\n\n\n\\noindent Thus every gammoid is orientable (Lemma~\\ref{lem:gammoidOrientable}).\n\n\\begin{definition}\\marginpar{Jan 8th}\n\tLet ${\\mathcal{O}}=(E,{\\mathcal{C}},{\\mathcal{C}}^\\ast)$ be an oriented matroid. We say that ${\\mathcal{O}}$ is \\deftext[realizable oriented matroid]{realizable},\n\tif there is a finite set $Q$ and a matrix $\\mu\\in \\mathbb{R}^{E\\times Q}$, such that ${\\mathcal{O}} = {\\mathcal{O}}(\\mu)$.\n\tIf there is no such matrix, we shall call ${\\mathcal{O}}$ \\deftext[non-realizable oriented matroid]{non-realizable}.\n\\end{definition}\n\n\\begin{remark} \\marginpar{Jan 8th}\nOf course, not every oriented matroid arises in this way from a matrix over a linearly ordered field. \n H.~Miyata, S.~Moriyama, and K.~Fukuda published\na listing of all non-realizable oriented matroids\\footnote{See: \\url{http:\/\/www-imai.is.s.u-tokyo.ac.jp\/~hmiyata\/oriented_matroids\/} \\cite{OM2}} of rank $4$ with $\\left| E \\right| = 8$, and of rank $3$ and rank $6$\n with $\\left| E \\right| = 9$; the results are based on the oriented matroid database\\footnote{See: \\url{http:\/\/www.om.math.ethz.ch\/} \\cite{OM1}} \n by L.~Finschi and K.~Fukuda.\n \\end{remark}\n\n\\begin{example}[\\cite{BlVSWZ99}, p.20]\\label{ex:NonRepresentableOrientationOfUniform}\\label{ex:UniformNonRealizable}\\index{RS(8)@$\\mathtt{RS}(8)$}\\PRFR{Apr 5th}\n The oriented matroid we want to present now has been named $\\mathtt{RS}(8)$. \n It is an orientation of the rank $4$ uniform matroid with $8$ elements, and therefore clearly an orientation of a gammoid.\n It has $2\\cdot \\binom{8}{5} = 112$ signed circuits as well as $112$ signed cocircuits.\n Since these signed subsets come in pairs $X$ and $-X$, we only have to list half of them.\n Let $E=\\SET{1,2,\\ldots,8}$. Then $\\mathtt{RS}(8) = (E,{\\mathcal{C}}, {\\mathcal{C}}^\\ast)$ where\n\n \\begin{align*}\n \t{\\mathcal{C}} = \\pm \\{ & \n\\SET{ \\hphantom{-} 1 , - 2 , - 3 , \\hphantom{-} 4 , - 5 },\\,\n\\SET{ \\hphantom{-} 1 , \\hphantom{-} 2 , - 3 , \\hphantom{-} 4 , - 6 },\\,\n\\SET{ \\hphantom{-} 1 , - 2 , \\hphantom{-} 3 , \\hphantom{-} 4 , - 7 },\\,\n \\\\ &\n\\SET{ \\hphantom{-} 1 , - 2 , - 3 , \\hphantom{-} 4 , - 8 },\\,\n\\SET{ - 1 , \\hphantom{-} 2 , - 3 , \\hphantom{-} 5 , - 6 },\\,\n\\SET{ - 1 , \\hphantom{-} 2 , \\hphantom{-} 3 , \\hphantom{-} 5 , - 7 },\\,\n \\\\ &\n\\SET{ - 1 , \\hphantom{-} 2 , \\hphantom{-} 3 , \\hphantom{-} 5 , - 8 },\\,\n\\SET{ \\hphantom{-} 1 , - 2 , \\hphantom{-} 3 , \\hphantom{-} 6 , - 7 },\\,\n\\SET{ - 1 , - 2 , \\hphantom{-} 3 , \\hphantom{-} 6 , - 8 },\\,\n \\\\ &\n\\SET{ - 1 , \\hphantom{-} 2 , - 3 , \\hphantom{-} 7 , - 8 },\\,\n\\SET{ - 1 , \\hphantom{-} 2 , - 4 , \\hphantom{-} 5 , - 6 },\\,\n\\SET{ \\hphantom{-} 1 , - 2 , \\hphantom{-} 4 , - 5 , - 7 },\\,\n \\\\ &\n\\SET{ - 1 , - 2 , \\hphantom{-} 4 , \\hphantom{-} 5 , - 8 },\\,\n\\SET{ \\hphantom{-} 1 , \\hphantom{-} 2 , \\hphantom{-} 4 , - 6 , - 7 },\\,\n\\SET{ - 1 , - 2 , \\hphantom{-} 4 , \\hphantom{-} 6 , - 8 },\\,\n \\\\ &\n\\SET{ - 1 , \\hphantom{-} 2 , - 4 , \\hphantom{-} 7 , \\hphantom{-} 8 },\\,\n\\SET{ - 1 , \\hphantom{-} 2 , \\hphantom{-} 5 , - 6 , - 7 },\\,\n\\SET{ - 1 , \\hphantom{-} 2 , \\hphantom{-} 5 , - 6 , - 8 },\\,\n \\\\ &\n\\SET{ \\hphantom{-} 1 , - 2 , - 5 , - 7 , \\hphantom{-} 8 },\\,\n\\SET{ \\hphantom{-} 1 , \\hphantom{-} 2 , - 6 , - 7 , \\hphantom{-} 8 },\\,\n\\SET{ - 1 , \\hphantom{-} 3 , - 4 , \\hphantom{-} 5 , \\hphantom{-} 6 },\\,\n \\\\ &\n\\SET{ \\hphantom{-} 1 , - 3 , - 4 , - 5 , \\hphantom{-} 7 },\\,\n\\SET{ \\hphantom{-} 1 , - 3 , - 4 , - 5 , \\hphantom{-} 8 },\\,\n\\SET{ \\hphantom{-} 1 , \\hphantom{-} 3 , \\hphantom{-} 4 , - 6 , - 7 },\\,\n \\\\ &\n\\SET{ \\hphantom{-} 1 , - 3 , \\hphantom{-} 4 , - 6 , - 8 },\\,\n\\SET{ - 1 , \\hphantom{-} 3 , - 4 , - 7 , \\hphantom{-} 8 },\\,\n\\SET{ - 1 , \\hphantom{-} 3 , \\hphantom{-} 5 , \\hphantom{-} 6 , - 7 },\\,\n \\\\ &\n\\SET{ - 1 , \\hphantom{-} 3 , \\hphantom{-} 5 , \\hphantom{-} 6 , - 8 },\\,\n\\SET{ \\hphantom{-} 1 , - 3 , - 5 , \\hphantom{-} 7 , - 8 },\\,\n\\SET{ \\hphantom{-} 1 , \\hphantom{-} 3 , - 6 , - 7 , \\hphantom{-} 8 },\\,\n \\\\ &\n\\SET{ \\hphantom{-} 1 , \\hphantom{-} 4 , - 5 , - 6 , - 7 },\\,\n\\SET{ - 1 , \\hphantom{-} 4 , \\hphantom{-} 5 , - 6 , - 8 },\\,\n\\SET{ \\hphantom{-} 1 , - 4 , - 5 , - 7 , \\hphantom{-} 8 },\\,\n \\\\ &\n\\SET{ \\hphantom{-} 1 , \\hphantom{-} 4 , - 6 , - 7 , - 8 },\\,\n\\SET{ - 1 , \\hphantom{-} 5 , \\hphantom{-} 6 , \\hphantom{-} 7 , - 8 },\\,\n\\SET{ - 2 , \\hphantom{-} 3 , - 4 , - 5 , \\hphantom{-} 6 },\\,\n \\\\ &\n\\SET{ \\hphantom{-} 2 , - 3 , - 4 , - 5 , \\hphantom{-} 7 },\\,\n\\SET{ \\hphantom{-} 2 , \\hphantom{-} 3 , - 4 , - 5 , \\hphantom{-} 8 },\\,\n\\SET{ \\hphantom{-} 2 , - 3 , \\hphantom{-} 4 , - 6 , \\hphantom{-} 7 },\\,\n \\\\ &\n\\SET{ \\hphantom{-} 2 , \\hphantom{-} 3 , - 4 , - 6 , \\hphantom{-} 8 },\\,\n\\SET{ - 2 , \\hphantom{-} 3 , - 4 , - 7 , \\hphantom{-} 8 },\\,\n\\SET{ - 2 , \\hphantom{-} 3 , \\hphantom{-} 5 , \\hphantom{-} 6 , - 7 },\\,\n \\\\ &\n\\SET{ - 2 , \\hphantom{-} 3 , - 5 , \\hphantom{-} 6 , - 8 },\\,\n\\SET{ \\hphantom{-} 2 , - 3 , - 5 , \\hphantom{-} 7 , - 8 },\\,\n\\SET{ \\hphantom{-} 2 , - 3 , - 6 , \\hphantom{-} 7 , \\hphantom{-} 8 },\\,\n \\\\ &\n\\SET{ \\hphantom{-} 2 , \\hphantom{-} 4 , \\hphantom{-} 5 , - 6 , - 7 },\\,\n\\SET{ - 2 , \\hphantom{-} 4 , - 5 , \\hphantom{-} 6 , - 8 },\\,\n\\SET{ \\hphantom{-} 2 , - 4 , - 5 , \\hphantom{-} 7 , \\hphantom{-} 8 },\\,\n \\\\ &\n\\SET{ \\hphantom{-} 2 , - 4 , - 6 , \\hphantom{-} 7 , \\hphantom{-} 8 },\\,\n\\SET{ - 2 , - 5 , \\hphantom{-} 6 , \\hphantom{-} 7 , - 8 },\\,\n\\SET{ \\hphantom{-} 3 , \\hphantom{-} 4 , \\hphantom{-} 5 , - 6 , - 7 },\\,\n \\\\ &\n\\SET{ - 3 , \\hphantom{-} 4 , \\hphantom{-} 5 , - 6 , - 8 },\\,\n\\SET{ \\hphantom{-} 3 , - 4 , - 5 , - 7 , \\hphantom{-} 8 },\\,\n\\SET{ \\hphantom{-} 3 , - 4 , - 6 , - 7 , \\hphantom{-} 8 },\\,\n \\\\ &\n\\SET{ - 3 , - 5 , \\hphantom{-} 6 , \\hphantom{-} 7 , - 8 },\\,\n\\SET{ - 4 , - 5 , \\hphantom{-} 6 , \\hphantom{-} 7 , \\hphantom{-} 8 }\n \t\t\\}\n \\end{align*}\nand where ${\\mathcal{C}}^\\ast$ is uniquely determined by ${\\mathcal{C}}$.\\footnote\n{There has not been a complete example of an oriented matroid in this work so far, thus we present ${\\mathcal{C}}^\\ast$\nin order for the reader to check their understanding of the signed cocircuits of an oriented matroid with a non-trivial example.\n \\begin{align*}\n \t{\\mathcal{C}}^\\ast = \\pm \\{ &\n\\SET{ - 1 , \\hphantom{-} 2 , \\hphantom{-} 3 , \\hphantom{-} 4 , - 5 },\\,\n\\SET{ - 1 , - 2 , - 3 , - 4 , - 6 },\\,\n\\SET{ \\hphantom{-} 1 , \\hphantom{-} 2 , \\hphantom{-} 3 , \\hphantom{-} 4 , \\hphantom{-} 7 },\\,\n\\SET{ - 1 , - 2 , - 3 , \\hphantom{-} 4 , \\hphantom{-} 8 },\\,\n \\\\ &\n\\SET{ - 1 , - 2 , - 3 , - 5 , - 6 },\\,\n\\SET{ - 1 , \\hphantom{-} 2 , - 3 , - 5 , - 7 },\\,\n\\SET{ - 1 , \\hphantom{-} 2 , \\hphantom{-} 3 , - 5 , - 8 },\\,\n\\SET{ \\hphantom{-} 1 , - 2 , \\hphantom{-} 3 , - 6 , \\hphantom{-} 7 },\\,\n \\\\ &\n\\SET{ - 1 , - 2 , - 3 , - 6 , \\hphantom{-} 8 },\\,\n\\SET{ \\hphantom{-} 1 , \\hphantom{-} 2 , \\hphantom{-} 3 , \\hphantom{-} 7 , - 8 },\\,\n\\SET{ \\hphantom{-} 1 , \\hphantom{-} 2 , - 4 , \\hphantom{-} 5 , \\hphantom{-} 6 },\\,\n\\SET{ \\hphantom{-} 1 , - 2 , - 4 , \\hphantom{-} 5 , \\hphantom{-} 7 },\\,\n \\\\ &\n\\SET{ \\hphantom{-} 1 , \\hphantom{-} 2 , - 4 , \\hphantom{-} 5 , - 8 },\\,\n\\SET{ - 1 , - 2 , - 4 , - 6 , \\hphantom{-} 7 },\\,\n\\SET{ - 1 , - 2 , - 4 , - 6 , - 8 },\\,\n\\SET{ \\hphantom{-} 1 , \\hphantom{-} 2 , - 4 , - 7 , - 8 },\\,\n \\\\ &\n\\SET{ - 1 , - 2 , - 5 , - 6 , \\hphantom{-} 7 },\\,\n\\SET{ \\hphantom{-} 1 , \\hphantom{-} 2 , \\hphantom{-} 5 , \\hphantom{-} 6 , \\hphantom{-} 8 },\\,\n\\SET{ \\hphantom{-} 1 , - 2 , \\hphantom{-} 5 , \\hphantom{-} 7 , \\hphantom{-} 8 },\\,\n\\SET{ \\hphantom{-} 1 , \\hphantom{-} 2 , \\hphantom{-} 6 , - 7 , - 8 },\\,\n \\\\ &\n\\SET{ - 1 , \\hphantom{-} 3 , \\hphantom{-} 4 , - 5 , - 6 },\\,\n\\SET{ \\hphantom{-} 1 , \\hphantom{-} 3 , \\hphantom{-} 4 , \\hphantom{-} 5 , \\hphantom{-} 7 },\\,\n\\SET{ - 1 , \\hphantom{-} 3 , \\hphantom{-} 4 , - 5 , \\hphantom{-} 8 },\\,\n\\SET{ \\hphantom{-} 1 , \\hphantom{-} 3 , \\hphantom{-} 4 , - 6 , \\hphantom{-} 7 },\\,\n \\\\ &\n\\SET{ - 1 , - 3 , \\hphantom{-} 4 , \\hphantom{-} 6 , \\hphantom{-} 8 },\\,\n\\SET{ \\hphantom{-} 1 , - 3 , - 4 , - 7 , - 8 },\\,\n\\SET{ - 1 , - 3 , - 5 , - 6 , - 7 },\\,\n\\SET{ - 1 , \\hphantom{-} 3 , - 5 , - 6 , - 8 },\\,\n \\\\ &\n\\SET{ \\hphantom{-} 1 , \\hphantom{-} 3 , \\hphantom{-} 5 , \\hphantom{-} 7 , - 8 },\\,\n\\SET{ - 1 , - 3 , \\hphantom{-} 6 , - 7 , \\hphantom{-} 8 },\\,\n\\SET{ \\hphantom{-} 1 , - 4 , \\hphantom{-} 5 , \\hphantom{-} 6 , \\hphantom{-} 7 },\\,\n\\SET{ - 1 , \\hphantom{-} 4 , - 5 , \\hphantom{-} 6 , \\hphantom{-} 8 },\\,\n \\\\ &\n\\SET{ \\hphantom{-} 1 , - 4 , \\hphantom{-} 5 , \\hphantom{-} 7 , - 8 },\\,\n\\SET{ \\hphantom{-} 1 , - 4 , - 6 , - 7 , - 8 },\\,\n\\SET{ \\hphantom{-} 1 , \\hphantom{-} 5 , \\hphantom{-} 6 , \\hphantom{-} 7 , \\hphantom{-} 8 },\\,\n\\SET{ - 2 , - 3 , - 4 , \\hphantom{-} 5 , - 6 },\\,\n \\\\ &\n\\SET{ \\hphantom{-} 2 , \\hphantom{-} 3 , \\hphantom{-} 4 , - 5 , \\hphantom{-} 7 },\\,\n\\SET{ - 2 , - 3 , \\hphantom{-} 4 , \\hphantom{-} 5 , \\hphantom{-} 8 },\\,\n\\SET{ - 2 , \\hphantom{-} 3 , - 4 , - 6 , \\hphantom{-} 7 },\\,\n\\SET{ \\hphantom{-} 2 , - 3 , \\hphantom{-} 4 , \\hphantom{-} 6 , \\hphantom{-} 8 },\\,\n \\\\ &\n\\SET{ \\hphantom{-} 2 , \\hphantom{-} 3 , \\hphantom{-} 4 , \\hphantom{-} 7 , \\hphantom{-} 8 },\\,\n\\SET{ - 2 , \\hphantom{-} 3 , - 5 , - 6 , \\hphantom{-} 7 },\\,\n\\SET{ - 2 , - 3 , \\hphantom{-} 5 , - 6 , \\hphantom{-} 8 },\\,\n\\SET{ - 2 , - 3 , \\hphantom{-} 5 , - 7 , \\hphantom{-} 8 },\\,\n \\\\ &\n\\SET{ \\hphantom{-} 2 , - 3 , \\hphantom{-} 6 , - 7 , - 8 },\\,\n\\SET{ - 2 , - 4 , \\hphantom{-} 5 , - 6 , \\hphantom{-} 7 },\\,\n\\SET{ \\hphantom{-} 2 , \\hphantom{-} 4 , - 5 , \\hphantom{-} 6 , \\hphantom{-} 8 },\\,\n\\SET{ \\hphantom{-} 2 , - 4 , - 5 , - 7 , - 8 },\\,\n \\\\ &\n\\SET{ \\hphantom{-} 2 , \\hphantom{-} 4 , \\hphantom{-} 6 , \\hphantom{-} 7 , \\hphantom{-} 8 },\\,\n\\SET{ - 2 , \\hphantom{-} 5 , - 6 , \\hphantom{-} 7 , \\hphantom{-} 8 },\\,\n\\SET{ - 3 , - 4 , \\hphantom{-} 5 , \\hphantom{-} 6 , - 7 },\\,\n\\SET{ - 3 , \\hphantom{-} 4 , \\hphantom{-} 5 , \\hphantom{-} 6 , \\hphantom{-} 8 },\\,\n \\\\ &\n\\SET{ - 3 , - 4 , - 5 , - 7 , - 8 },\\,\n\\SET{ - 3 , - 4 , \\hphantom{-} 6 , - 7 , - 8 },\\,\n\\SET{ \\hphantom{-} 3 , - 5 , - 6 , \\hphantom{-} 7 , - 8 },\\,\n\\SET{ - 4 , - 5 , - 6 , - 7 , - 8 } \\}.\n \\end{align*}}\n\tProposition~1.5.1 in \\cite{BlVSWZ99} states that the oriented matroid $\\mathtt{RS}(8)$ \n\tis a non-realizable orientation of the \n\tunderlying uniform matroid, thus we may expect gammoids to have non-realizable orientations. \n\tFor the full proof, we refer the reader to p.~23 in \\cite{BlVSWZ99}. \n\tThe idea of the proof is the following: Assume that $\\mathtt{RS}(8)$ is realizable, \n\tthen there is a matrix $\\mu\\in \\mathbb{R}^{\\SET{1,2,\\ldots, 8}\\times\\SET{1,2,3,4}}$ \n\tsuch that $\\mu|\\SET{1,2,3,4}\\times\\SET{1,2,3,4}$ is an identity matrix. This leaves us with a variable matrix\n\t$\\mu| \\SET{5,6,7,8}\\times \\SET{1,2,3,4}$, for which we would have to find values that yield the \n\tcorrect signed circuits of $\\mathtt{RS}(8)$. The signed circuits of $\\mathtt{RS}(8)$ can be translated to\n\tstrict inequalities that $\\mu$ must obey. For instance, the signed circuit $\\SET{1,-2,-3,4,-5}$ states that\n\t$\\mu_5 = \\alpha \\mu_1 - \\beta \\mu_2 - \\gamma \\mu_3 + \\delta \\mu_4$ must have a solution with $\\alpha,\\beta,\\gamma,\\delta > 0$. \n\tIn this particularly easy case, we obtain the inequalities $\\mu(5,1) > 0$, $\\mu(5,2) < 0$, $\\mu(5,3) < 0$, $\\mu(5,4) > 0$.\\footnote{In general, the strict inequalities derived are not linear, as Cramer's rule yields polynomial terms eliminating the coefficients of the non-trivial linear combinations of the zero that correspond to signed circuits of ${\\mathcal{O}}$.}\n\t The proof of non-realizability is completed by the observation that the constructed\n\tsystem of inequalities has no solutions, therefore there is no matrix $\\mu$ with ${\\mathcal{O}}(\\mu)=\\mathtt{RS}(8)$, and $\\mathtt{RS}(8)$ is non-realizable.\n\\end{example}\n\n\\marginpar{Jan 8th}\n\\noindent For every realizable oriented matroid of the form ${\\mathcal{O}}(\\mu)$, we obtain another realizable oriented matroid ${\\mathcal{O}}(\\mu')$ \nwhere $\\mu'$ is obtained from $\\mu$\nby multiplying an arbitrary set of rows with $-1$.\n Clearly, for the underlying matroids we have $M(\\mu) = M(\\mu')$. \n It is easy to see that the next definition carries this operation over to all oriented matroids (\\cite{BlVSWZ99}, p.3).\n\n\\begin{definition}\\label{def:Cflip}\\marginpar{Jan 8th}\n \tLet $E$ be a set, $X\\subseteq E$, and $C\\in \\sigma E$ a signed subset of $E$.\n \tThe \\deftext[flip of a signed subset]{$\\bm X$-flip of $\\bm C$} is defined to be\n \tthe signed subset\\label{n:Xflip}\n \t\\[ C_{-X} \\colon E\\longrightarrow \\SET{-1, 0, 1},\\quad e\\mapsto \\begin{cases} -C(e)&\\quad \\text{if~} e\\in X,\\\\\n \t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\hphantom{-}C(e)&\\quad\\text{otherwise.} \\end{cases} \\]\n\\end{definition}\n\n\\begin{definition}\\marginpar{Jan 8th}\n\tLet ${\\mathcal{O}}=(E,{\\mathcal{C}},{\\mathcal{C}}^\\ast)$ be an oriented matroid, and let $X\\subseteq E$.\n\tThe \\deftext[flip reorientation of O@flip reorientation of ${\\mathcal{O}}$]{$\\bm X$-flip reorientation of $\\bm {\\mathcal{O}}$}\n\tis the triple ${\\mathcal{O}}_{-X} = (E,{\\mathcal{C}}_{-X},{\\mathcal{C}}_{-X}^\\ast)$ where\n\t $${\\mathcal{C}}_{-X} = \\SET{C_{-X}~\\middle|~\\vphantom{C'} C\\in{\\mathcal{C}}} \\quad\\text{and}\\quad\n\t{\\mathcal{C}}_{-X}^\\ast = \\SET{C'_{-X}~\\middle|~ C'\\in{\\mathcal{C}}^\\ast}.$$\n\tLet ${\\mathcal{O}}'$ also be an oriented matroid. We say that \n\t${\\mathcal{O}}'$ is a \\deftext[reorientation of O@reorientation of ${\\mathcal{O}}$]{reorientation of $\\bm {\\mathcal{O}}$}, if\n\tthere is a subset $X\\subseteq E$ with ${\\mathcal{O}}' = {\\mathcal{O}}_{-X}$.\n\\end{definition}\n\n\n\\begin{lemma}\\label{lem:reorientationIsOM}\\marginpar{Jan 8th}\n\tLet ${\\mathcal{O}}=(E,{\\mathcal{C}})$ be an oriented matroid, and let $X\\subseteq E$. Then ${\\mathcal{O}}_{-X}$ is an oriented matroid.\n\\end{lemma}\n\\begin{proof}\\marginpar{Jan 8th}\n\tFor every $X\\subseteq E$ and $C\\in \\sigma E$, it is clear from Definition~\\ref{def:Cflip}, that\n\t\\linebreak $\\left( C_{-X} \\right)_{-X} = C$,\n\ttherefore the map $\\phi_X \\colon \\sigma E\\longrightarrow \\sigma E, C\\mapsto C_{-X}$ is an involution on $\\sigma E$.\n\tSince $\\left( \\emptyset_{\\sigma E} \\right)_{-X} = \\emptyset_{\\sigma E}$, we obtain that $\\emptyset_{\\sigma E} \\notin {\\mathcal{C}}_{-X}$ from $\\emptyset_{\\sigma E} \\notin {\\mathcal{C}}$, thus {\\em (${\\mathcal{C}}$1)} holds. Furthermore, for any $C\\in \\sigma E$\n\twe have\n\t$\\left( C_{-X} \\right)_+ = \\left( C_+ \\backslash X \\right) \\cup \\left( C_- \\cap X \\right)$ and\n\t$\\left( C_{-X} \\right)_- = \\left( C_- \\backslash X \\right) \\cup \\left( C_+ \\cap X \\right)$. In particular we have for $C,D\\in \\sigma E$ that\n\t$C = -D$ if and only if $C_{-X} = -D_{-X}$. We also have $C\\bot D$ if and only $C_{-X}\\bot D_{-X}$: \n\tCase {\\em (ii)} of Definition~\\ref{def:XorthoY} is oblivious of any sign flips in $C$ and $D$ since $C_0 = \\left( C_{-X} \\right)_0$\n\tand $D_0 = \\left( D_{-X} \\right)_0$; whereas in case {\\em (i)} the passage from $C$ and $D$ to $C_{-X}$ and $D_{-X}$, respectively,\n\tintroduces an even amount of sign flips. \n\t So $C(e)D(e) = C_{-X}(e)D_{-X}(e)$ and $C(f)D(f) = C_{-X}(f)D_{-X}(f)$.\n\t Therefore {\\em (${\\mathcal{C}}$2)}, {\\em (${\\mathcal{C}}$3)}, and {\\em (${\\mathcal{C}}$4)} carry over from ${\\mathcal{C}}$ to ${\\mathcal{C}}_{-X}$.\n\t Since $\\SET{C_\\pm ~\\middle|~ C\\in {\\mathcal{C}}} = \\SET{C_\\pm~\\middle|~ C\\in {\\mathcal{C}}_{-X}}$,\n\t $\\SET{C'_\\pm ~\\middle|~ C'\\in {\\mathcal{C}}^\\ast} = \\SET{C_\\pm'~\\middle|~ C'\\in {\\mathcal{C}}_{-X}^\\ast}$,\n\t and for all $C\\in{\\mathcal{C}}_{-X}$ and $D\\in {\\mathcal{C}}_{-X}^\\ast$, we have $C\\bot D$; \n\t we obtain that ${\\mathcal{C}}_{-X}^\\ast$ is\n\t indeed the unique family of signed cocircuits of the oriented matroid on $E$ with the family of signed circuits ${\\mathcal{C}}_{-X}$,\n\t thus ${\\mathcal{O}}_{-X}$ is an oriented matroid (Remark~\\ref{rem:sufficientConditionOM}).\n\\end{proof}\n\n\\begin{corollary}\\marginpar{Jan 8th}\n\tLet ${\\mathcal{O}}=(E,{\\mathcal{C}},{\\mathcal{C}}^\\ast)$ be an oriented matroid, and let $X\\subseteq E$. Then $M({\\mathcal{O}}_{-X}) = M({\\mathcal{O}})$.\n\\end{corollary}\n\\begin{proof}\\marginpar{Jan 8th}\n\tFor $C\\in \\sigma E$, we have $C_\\pm = \\left( C_{-X} \\right)_\\pm$.\n\\end{proof}\n\n\\begin{definition}\\marginpar{Jan 8th}\n\tLet ${\\mathcal{O}}=(E,{\\mathcal{C}},{\\mathcal{C}}^\\ast)$ be an oriented matroid.\n\tThe \\deftext[reorientation class of O@reorientation class of ${\\mathcal{O}}$]{reorientation class of $\\bm {\\mathcal{O}}$}\n\tis defined to be\\label{n:reorientationclass}\n\t\\( [{\\mathcal{O}}] = \\SET{{\\mathcal{O}}_{-X}\\mid X\\subseteq E} .\\)\n\\end{definition}\n\n\\begin{example}\\label{ex:nonStrictGammoidOrientations}\\PRFR{Apr 5th}\n\n\tAccording to L.~Finschi's database \\cite{OM1}, the gammoid from Example~\\ref{ex:nonStrictGammoid} has exactly one equivalence class\n\twith respect to relabeling and reorientation of oriented matroids.\n\t Thus it is easy to check that it has two reorientation classes, ${\\mathcal{O}}_1 = (E,{\\mathcal{C}}_1,{\\mathcal{C}}_1^\\ast)$ and ${\\mathcal{O}}_2 = (E,{\\mathcal{C}}_2,{\\mathcal{C}}_2^\\ast)$\n\t where\n\\begin{align*}\n {\\mathcal{C}}_1 = \\pm \\{ &\n\\SET{ a , b , - c , e } ,\n\\SET{ a , b , - d , - f } ,\n\\SET{ b , - c , - d , g } ,\n\\SET{ d , e , f , - g } ,\n \\\\ &\n\\SET{ - a , \\hphantom{-} b , - c , \\hphantom{-} f , \\hphantom{-} g } ,\n\\SET{ - a , - c , \\hphantom{-} d , \\hphantom{-} e , \\hphantom{-} f } ,\n\\SET{ - a , - c , \\hphantom{-} d , \\hphantom{-} f , \\hphantom{-} g } ,\n \\\\ &\n\\SET{ \\hphantom{-} a , \\hphantom{-} b , \\hphantom{-} d , \\hphantom{-} e , - g } ,\n\\SET{ \\hphantom{-} a , \\hphantom{-} b , \\hphantom{-} e , - f , - g } ,\n\\SET{ \\hphantom{-} a , \\hphantom{-} c , \\hphantom{-} d , \\hphantom{-} e , - g } ,\n \\\\ &\n\\SET{ \\hphantom{-} a , \\hphantom{-} c , \\hphantom{-} e , - f , - g } ,\n\\SET{ \\hphantom{-} b , - c , \\hphantom{-} d , \\hphantom{-} e , \\hphantom{-} f } ,\n\\SET{ \\hphantom{-} b , - c , \\hphantom{-} e , \\hphantom{-} f , \\hphantom{-} g } \\}\n\\end{align*}\nand\n\\begin{align*}\n {\\mathcal{C}}_2 = \\pm \\{ &\n\\SET{ - a , b , c , e } ,\n\\SET{ a , - b , d , - f } ,\n\\SET{ b , c , - d , - g } ,\n\\SET{ d , e , - f , g } ,\n \\\\ &\n\\SET{ - a , \\hphantom{-} b , - c , \\hphantom{-} f , \\hphantom{-} g } ,\n\\SET{ - a , \\hphantom{-} b , \\hphantom{-} d , \\hphantom{-} e , \\hphantom{-} g } ,\n\\SET{ - a , \\hphantom{-} b , \\hphantom{-} e , \\hphantom{-} f , \\hphantom{-} g } ,\n \\\\ &\n\\SET{ - a , - c , \\hphantom{-} d , \\hphantom{-} e , \\hphantom{-} g } ,\n\\SET{ - a , - c , \\hphantom{-} d , \\hphantom{-} f , \\hphantom{-} g } ,\n\\SET{ \\hphantom{-} a , \\hphantom{-} c , \\hphantom{-} d , \\hphantom{-} e , - f } ,\n \\\\ &\n\\SET{ \\hphantom{-} a , \\hphantom{-} c , \\hphantom{-} e , - f , - g } ,\n\\SET{ \\hphantom{-} b , \\hphantom{-} c , \\hphantom{-} d , \\hphantom{-} e , - f } ,\n\\SET{ \\hphantom{-} b , \\hphantom{-} c , \\hphantom{-} e , - f , - g } \\}\n\\end{align*}\nHere, $[{\\mathcal{O}}_2] = [\\phi[{\\mathcal{O}}_1]]$ where $\\phi = (ac)(fg)$ is a corresponding relabeling.\n\\end{example}\n\n\\begin{lemma}\\marginpar{Jan 8th}\n\tLet $E,Y$ be finite sets, $\\mu\\in \\mathbb{R}^{E\\times Y}$, and $X\\subseteq E$.\n\tLet $\\nu\\in \\mathbb{R}^{E\\times Y}$ be the matrix where for every $e\\in E$ and $y\\in Y$,\n\t\\[ \\nu(e,y) = \\begin{cases} -\\mu(e,y)& \\quad \\text{if~} e\\in X,\\\\\n\t\t\t\t\t\t\t\t\\hphantom{-}\\mu(e,y)&\\quad\\text{otherwise.}\n\t\\end{cases} \\]\n\tThen $\\left( {\\mathcal{O}}(\\mu) \\right)_{-X} = {\\mathcal{O}}(\\nu)$.\n\\end{lemma}\n\n\\begin{proof}\\marginpar{Jan 8th}\n\tLet $\\alpha\\in \\mathbb{R}^E$. Let $\\beta\\in \\mathbb{R}^E$ be defined such that\n\t\\[ \\beta(e) = \\begin{cases} -\\alpha(e)&\\quad \\text{if~}e \\in X,\\\\\n\t\t\t\t\t\t\\hphantom{-}\\alpha(e)& \\quad \\text{otherwise.}\n\t\t\t\t\t\\end{cases} \\]\n\tClearly,\n\t$(E_\\alpha)_{-X} = E_\\beta$ and \\( \\sum_{e\\in E} \\beta(e)\\cdot \\nu(e) = \\sum_{e\\in E}\\alpha(e)\\cdot \\mu(e)\\).\n\tThus $\\sum_{e\\in E}\\alpha(e)\\cdot \\mu(e) = 0$ if and only if $\\sum_{e\\in E}\\beta(e)\\cdot \\nu(e) = 0$,\n\tand consequently $E_\\alpha \\in {\\mathcal{C}}_\\mu$ if and only if $E_\\beta \\in {\\mathcal{C}}_\\nu$.\n\tTherefore $\\left( {\\mathcal{O}}(\\mu) \\right)_{-X} = {\\mathcal{O}}(\\nu)$.\n\\end{proof}\n\n\\begin{corollary}\\marginpar{Jan 8th}\n\tLet ${\\mathcal{O}}=(E,{\\mathcal{C}},{\\mathcal{C}}^\\ast)$ be an oriented matroid, and let $X\\subseteq E$. Then ${\\mathcal{O}}$ is realizable if and only if ${\\mathcal{O}}_{-X}$ is realizable.\n\\end{corollary}\n\n\\subsection{Minors}\n\n\\begin{definition}\\label{def:OMrestriction}\\marginpar{Jan 8th}\n\tLet ${\\mathcal{O}}=(E,{\\mathcal{C}},{\\mathcal{C}}^\\ast)$ be a oriented matroid, $R\\subseteq E$.\n\tThe \\deftext[restriction of O to R@restriction of ${\\mathcal{O}}$ to $R$]{restriction of $\\bm {\\mathcal{O}}$ to $\\bm R$}\n\tis the triple\\label{n:OrestrictR}\n\t\\( {\\mathcal{O}} | R = \\left(R,{\\mathcal{C}}_R,{\\mathcal{C}}_R^\\ast\\right) \\)\n\twhere\n\t\\[ {\\mathcal{C}}_R = \\SET{C\\in \\sigma R ~\\middle|~ \\exists D\\in {\\mathcal{C}}\\colon\\, D_\\pm \\subseteq R \\text{~s.t.~} D|_R = C}\\]\n\tand\n\t\\[ {\\mathcal{C}}_R^\\ast = \\SET{C'\\in {\\mathcal{D}}_R^\\ast ~\\middle|~ \\nexists D'\\in {\\mathcal{D}}_R^\\ast\\colon\\, D'_\\pm \\subsetneq C'_\\pm},\\]\n\tand where\n\t\\[ {\\mathcal{D}}_R^\\ast = \\SET{C'\\in \\sigma R \\BSET{\\emptyset_{\\sigma R}} ~\\middle|~ \\exists D'\\in {\\mathcal{C}}^\\ast \\colon \\, D'|_R = C'} .\\]\n\t\\label{def:OMcontraction}\n\tLet $Q\\subseteq E$.\n\tThe \\deftext[contraction of O to Q@contraction of ${\\mathcal{O}}$ to $Q$]{contraction of $\\bm {\\mathcal{O}}$ to $\\bm Q$}\n\tis the triple\\label{n:OcontractQ}\n\t\\( {\\mathcal{O}} |' Q = \\left(Q,{\\mathcal{C}}_{'Q},{\\mathcal{C}}^\\ast_{'Q}\\right) \\)\n\twhere \n\t\\[ {\\mathcal{C}}^\\ast_{'Q} = \\SET{C'\\in \\sigma R ~\\middle|~ \\exists D'\\in {\\mathcal{C}}^\\ast\\colon\\, D'_\\pm \\subseteq R \\text{~s.t.~} D'|_R = C'}\\]\n\tand\n\t\\[ {\\mathcal{C}}_{'Q} = \\SET{C\\in {\\mathcal{D}}_{'Q} ~\\middle|~ \\nexists D\\in {\\mathcal{D}}_{'Q}\\colon\\, D_\\pm \\subsetneq C_\\pm},\\]\n\tand where\n\t\\[ {\\mathcal{D}}_{'Q} = \\SET{C\\in \\sigma R \\BSET{\\emptyset_{\\sigma R}} ~\\middle|~ \\exists D\\in {\\mathcal{C}} \\colon \\, D|_R = C} . \\qedhere\\]\n\\end{definition}\n\n\\begin{lemma}\\label{lem:OMminors}\\marginpar{Jan 8th}\nLet ${\\mathcal{O}}=(E,{\\mathcal{C}},{\\mathcal{C}}^\\ast)$ be a oriented matroid, $X\\subseteq E$.\nThen ${\\mathcal{O}}| X$ and ${\\mathcal{O}}|' X$ are oriented matroids,\nand further \\[\n\t\\left( {\\mathcal{O}}^\\ast | X \\right)^\\ast = {\\mathcal{O}}|' X\n\t\\quad\\text{as well as}\\quad\n\t\\left( {\\mathcal{O}}^\\ast |' X \\right)^\\ast = {\\mathcal{O}}| X\n\\]\nholds.\n\\end{lemma}\n\n\\noindent For a proof, please refer to Propositions 3.3.1 and 3.3.2 (\\cite{BlVSWZ99}, p.110) in {\\em Oriented Matroids} by A.~Bj\u00f6rner, M.~Las~Vergnas, B.~Sturmfels, N.~White,~and G.~Ziegler.\n\n\n\\section{Colorings}\n\n\\PRFR{Mar 7th}\nThe notion of colorings used in this work originates from {\\em Antisymmetric Flows in Matroids} \nby J.~Ne\u0161et\u0159il and W.~Hochst\u00e4ttler \\cite{HN06}. A recommended source for the properties and bearings of this notion\nis R.~Nickel's thesis {\\em Flows and Colorings in Oriented Matroids} \\cite{Ni12}.\n\n\\begin{remark}\\PRFR{Mar 7th}\nThe first appearance of a notion of a coloring of general oriented matroids\ncan be tracked down to the paper {\\em On $(k,d)$-Colorings and Fractional Nowhere-Zero Flows} \nby L.A.~Goddyn, M.~Tarsi, and C.-Q.~Zhang \\cite{GTZ98},\nwho define the {\\em star flow index} of an reorientation class $[{\\mathcal{O}}']$ of oriented matroids\nto be \\[\n\t\\xi^\\ast\\left( \\left[{\\mathcal{O}}'\\right] \\right) = \\min_{{\\mathcal{O}} \\in [{\\mathcal{O}}']} \\max_{D\\in {\\mathcal{C}}^\\ast_{\\mathcal{O}}} \\frac{\\left| D_\\pm \\right|}{\\left| D_+ \\right|}\n\\]\nwhere ${\\mathcal{C}}^\\ast_{\\mathcal{O}}$ denotes the family of signed cocircuits of ${\\mathcal{O}}$. The star flow index is closely related to\nthe chromatic number of a graph $G$ through a result of J.G.~Minty \\cite{Mi62}:\n If $G=(V,E)$ is a graph, and $\\mu\\in \\mathbb{R}^{E\\times V}$ is the signed edge-vertex-incidence matrix of an orientation of the edges of $G$, then for ${\\mathcal{O}}' = \\left( {\\mathcal{O}}(\\mu) \\right)^\\ast$ we have $\\lceil \\xi^\\ast([{\\mathcal{O}}'])\\rceil = \\chi(G)$ where $\\chi(G)$ is the \n well-known\n chromatic number of $G$.\n {\\em{We would like to point out that this is \\textbf{not the chromatic number} of oriented matroids that \\textbf{we are concerned with} in this work.}}\\footnote{For further details on the differences between the oriented flow number of L.A.~Goddyn, M.~Tarsi, and C.-Q.~Zhang \\cite{GTZ98} and\n the chromatic number of J.~Ne\u0161et\u0159il and W.~Hochst\u00e4ttler \\cite{HN06}, see \\cite{Ni12}, pp.~98f.}\n \\end{remark}\n\n\n\\begin{definition}\\marginpar{Jan 8th}\n\tLet $E$ be a set.\n\tA \\deftext[signed multiset]{signed multi-subset of $\\bm E$} -- or shorter: \\deftextX{signed multiset} --\n\tis a map $S\\colon E\\longrightarrow \\mathbb{Z}$. The \\deftextX{family of signed multi-subsets of E}\n\tshall be denoted by $\\mathbb{Z} .E$.\\label{n:ZE}\n\tSince $\\SET{-1,0,1}\\subseteq \\mathbb{Z}$, we shall identify the signed subsets with the corresponding signed multisets,\n\ti.e. $\\sigma E \\equiv \\SET{F\\in \\mathbb{Z}. E\\mid \\forall e\\in E\\colon F(e) \\in \\SET{-1,0,1}}$.\n\tThe \\deftext[empty signed multiset]{empty signed multi-subset of $\\bm E$}\\label{n:emptysetZE} is the map\n\t\\[ \\emptyset_{\\mathbb{Z}. E} \\colon E\\longrightarrow \\mathbb{Z},\\, e\\mapsto 0. \\qedhere\\]\n\\end{definition}\n\n\n\n\\needspace{5\\baselineskip}\n\\begin{definition}[Dual of Definition 1, \\cite{HN06}]\\marginpar{Jan 8th}\n\tLet ${\\mathcal{O}}=(E,{\\mathcal{C}},{\\mathcal{C}}^\\ast)$ be an oriented matroid. The \\deftext[coflow lattice of O@coflow lattice of ${\\mathcal{O}}$]{coflow lattice of $\\bm {\\mathcal{O}}$}\n\tshall consist of all integral linear combinations of cocircuits of ${\\mathcal{O}}$, i.e.\n\t\\[ \\mathbb{Z} .{\\mathcal{C}}^\\ast = \\SET{F\\in \\mathbb{Z}.E ~~\\middle|~~ \\exists \\alpha\\in \\mathbb{Z}^{{\\mathcal{C}}^\\ast}\\colon\\,\\forall e\\in E\\colon\\, F(e) = \\sum_{C'\\in{\\mathcal{C}}^\\ast} \\alpha(C')\\cdot C'(e)}. \\]\n\tEach element $F\\in \\mathbb{Z}.{\\mathcal{C}}^\\ast$ shall be called a \\deftextX{coflow of $\\bm {\\mathcal{O}}$}.\n\tA \\deftext[nowhere-zero coflow of O@nowhere-zero coflow of ${\\mathcal{O}}$]{nowhere-zero coflow of $\\bm {\\mathcal{O}}$} is a coflow $F\\in \\mathbb{Z}.{\\mathcal{C}}^\\ast$\n\twhere $F(e)\\not= 0$ for all $e\\in E$.\n\\end{definition}\n\n\n\n\n\\begin{definition}\\label{def:chiO}\\PRFR{Mar 7th}\n\tLet ${\\mathcal{O}}=(E,{\\mathcal{C}},{\\mathcal{C}}^\\ast)$ be an oriented matroid. We define the \\deftext[chromatic number of ${\\mathcal{O}}$]{chromatic number of $\\bm {\\mathcal{O}}$}\n\tto be\\label{n:chiO} \\[ \\chi({\\mathcal{O}}) = \\min \\SET{\\vphantom{A^{A^A}}\\max \\SET{\\vphantom{A^A} \\left| F( e ) \\right| + 1~\\middle|~ e\\in E} ~~ \\middle| ~~ \n\tF\\in \\mathbb{Z}.{\\mathcal{C}}^\\ast,\\,\\forall e\\in E\\colon F( e )\\not= 0}. \\]\n\tBy convention, we set $\\chi({\\mathcal{O}}) = \\infty$ if there is no nowhere-zero coflow in $\\mathbb{Z}.{\\mathcal{C}}^\\ast$.\n\\end{definition}\n\n\\noindent\nThe only oriented matroid ${\\mathcal{O}}$ with $\\chi({\\mathcal{O}}) = 1$ is the trivial oriented matroid,\\footnote{That is, if we set $\\min \\SET{\\max\\SET{\\left| F(e) \\right| + 1 ~\\middle|~ e\\in \\emptyset} ~\\middle|~ F \\in \\mathbb{Z}.\\emptyset } = 1$. The rationale behind this is that a matroid $M=(E,{\\mathcal{I}})$ with $E=\\emptyset$ is the graphical matroid of every edge-less graph; and the trivial oriented matroid is an orientation of $M$.}\ni.e. the oriented matroid ${\\mathcal{O}} = (E,{\\mathcal{C}},{\\mathcal{C}}^\\ast)$ where $E = {\\mathcal{C}} = {\\mathcal{C}}^\\ast =\\emptyset$.\n\n\\begin{remark}\\PRFR{Mar 7th}\n\tLet ${\\mathcal{O}}=(E,{\\mathcal{C}},{\\mathcal{C}}^\\ast)$ be an oriented matroid. \n\tWe have $\\chi({\\mathcal{O}}) = \\infty$ if and only if there is an element $e\\in E$ such that $D(e) = 0$ for all $D\\in {\\mathcal{C}}^\\ast$:\n\tLet ${\\mathcal{C}}^\\ast = \\pm \\dSET{D_1, D_2,\\ldots, D_k}$, i.e. for each pair $D\\in {\\mathcal{C}}^\\ast$ we choose precisely one element from $\\SET{D,-D}$.\n\tThen there is no cancellation of non-zero summands in \n\t\t\\[ F(e) = \\sum_{i=1}^{k} 2^{i-1} \\cdot D_i(e)\\]\n\tfor $e\\in E$, thus $F$ is a nowhere-zero coflow of ${\\mathcal{O}}$ if and only if for every $e\\in E$ there is some $D\\in {\\mathcal{C}}^\\ast$ with $e\\in D_\\pm$.\n\tThis is the case if and only if $M({\\mathcal{O}})$ has no loop.\n\\end{remark}\n\n\\noindent We give a quick tour justifying why the name {\\em chromatic number} is appropriate in this context. For a more detailed introduction, we refer the reader to Chapter~4 in \\cite{Ni12}.\n\n\\needspace{5\\baselineskip}\n\n\\vspace*{-\\baselineskip}\n\\begin{wrapfigure}{r}{3cm}\n\\vspace{\\baselineskip}\n\\begin{centering}~\n\\includegraphics{pentagonG}\n\\end{centering}%\n\\vspace*{-1\\baselineskip}\n\\end{wrapfigure}\n~\n \n\n\n\\begin{example}\\PRFR{Mar 7th}\n\tConsider the undirected graph $G=(V,E)$ where $V = \\SET{1,2,3,4,5}$ and $E = \\{\\SET{1,2},$ $\\SET{1,5},$ $\\SET{2,3},$ $\\SET{3,4},$ $\\SET{4,5}\\}$.\n\tA proper coloring of $G$ is a map $\\phi\\colon V \\longrightarrow \\mathbb{Z}$ such that $\\phi(v)\\not=\\phi(w)$ whenever $\\SET{v,w}\\in E$.\n\tThe chromatic number of $G$ is defined as $\\chi(G) = \\min\\SET{\\left| \\phi[V] \\right| \\vphantom{A^A}~\\middle|~ \\phi \\text{~proper coloring of~}G}$,\n\tin this case $\\chi(G) = 3$ and $\\phi(1)=\\phi(3)=1$, $\\phi(2)=\\phi(4)=2$, $\\phi(5) = 3$ is a corresponding proper coloring.\n\tAn orientation of $G$ is a digraph $D=(V,A)$ such that for every $\\SET{u,v}\\in E$ we have the equivalency $(u,v)\\in A\\Leftrightarrow (v,u)\\notin A$,\n\tand such that $E = \\SET{\\SET{u,v}~\\middle|~ (u,v)\\in A}$. \n\tEvery orientation $D$ of $G$ gives rise to a map $\\sigma\\colon V\\times V\\longrightarrow \\SET{-1,0,1}$ \n\twhere $\\sigma(u,v) = +1$ if $(u,v)\\in A$, $\\sigma(u,v) = -1$ if $(v,u)\\in A$, and $\\sigma(u,v) = 0$ if $\\SET{u,v}\\notin E$. A nowhere-zero-coflow on $G$\n\twith respect to the orientation $D$\tis a map $f\\colon E\\longrightarrow \\mathbb{Z}$, such that $f(e)\\not= 0$ for all $e\\in E$, and such that for every closed \n\twalk $v_1 v_2 \\ldots v_k$ in $G$, i.e. $v_1=v_k$, we have $$\\sum_{i=1}^{k-1}\\left( \\sigma(v_i,v_{i+1}) \\cdot f\\left( \\SET{v_i,v_{i+1}} \\right) \\right) = 0.$$\n\tEvery coloring $\\phi\\colon V\\longrightarrow \\mathbb{Z}$ induces a coflow $\\hat{\\phi}\\colon E\\longrightarrow \\mathbb{Z}$ on $G$ with respect to an orientation $D$ by setting\n\t$\\hat \\phi(\\SET{u,v}) = \\sigma(u,v)\\cdot \\left( \\phi(v) - \\phi(u) \\right)$. Furthermore, $\\hat \\phi$ is a nowhere-zero-coflow if and only if $\\phi$ is a\n\tproper coloring of $G$. Conversely, if $f$ is a nowhere-zero-coflow on $G$ with respect to $D$,\n\t then we may reconstruct a proper coloring $\\tilde f \\colon V\\longrightarrow \\mathbb{Z}$ from it by choosing a vertex $v$ per component of $G$ and setting $\\tilde f(v) = 0$. For every other vertex $x$,\n\t let $w_1 w_2 \\ldots w_k$ be a walk from the chosen vertex $v=w_1$ of the component containing $x$ to $x=w_k$ in $G$. \n\t We set $$\\tilde f(x) = \\left( \\sum_{i=1}^{k-1} \\left( \\sigma(w_i,w_{i+1})\\cdot f(\\SET{w_i,w_{i+1}}) \\right) \\right)\\mathrm{mod}_{\\max\\SET{\\left| f(e) \\right|+1~\\middle|~\\vphantom{A^A} e\\in E}.}$$\n\t Then $\\tilde f(x)$ is a proper coloring of $G$ which uses at most $\\max\\SET{\\vphantom{A^A}\\left| f(e) \\right|+1~\\middle|~ e\\in E}$ colors.\n\t Furthermore, every graph $G=(V,E)$ gives rise to a cycle matroid $M(G) = (E,{\\mathcal{I}})$ where a set of edges $X\\subseteq E$ is independent, if and only if\n\t $(V,X)$ does not contain a cycle walk\\footnote{Remember that the trivial walk $v$ is not a cycle walk.} (see \\cite{We76}, p.10). The cycle matroid associated with the above graph $G$ is\n\t $M(G) = \\left( E,\\SET{X\\subseteq E~\\middle|~\\vphantom{A^A} \\left| X \\right| \\leq 4} \\right)$, the uniform matroid of rank $4$ on $E$. The cocircuits of cycle matroids\n\t are the $\\subseteq$-minimal subsets $D\\subseteq E$, such that the graph $G\\backslash X = (V,E\\backslash X)$ has more components than $G=(V,E)$. Furthermore, every\n\t orientation $D$ of $G$ yields an oriented matroid ${\\mathcal{O}}=(E,{\\mathcal{C}},{\\mathcal{C}}^\\ast)$ with $M({\\mathcal{O}}) = M(G)$ by the following construction: $C\\in {\\mathcal{C}}$ if\n\t and only if there is a cycle walk $x_1x_2\\ldots x_k$ in $G$ with $C_\\pm = \\SET{x_1,x_2,\\ldots,x_k}$ and\n\t $C(\\SET{x_i,x_{i+1}}) = \\sigma(x_i,x_{i+1})$ for all $i\\in\\SET{1,2,\\ldots,k-1}$. In other words, an edge in the support of a signed circuit\n\t $C$ of ${\\mathcal{O}}$ is assigned $+1$ if its orientation agrees with the corresponding arc in the cycle walk, and $-1$ otherwise. Dually, we have $D\\in {\\mathcal{C}}^\\ast$\n\t if and only if there is a minimal edge-cut $X\\subseteq E$ and a partition $L,R$ of $V$ such that every edge $e\\in X$ has\n\t the property $\\left| e\\cap L \\right| = 1$; with $D_\\pm = X$ and\n\t $D(\\SET{l,r}) = \\sigma(l,r)$ for all $l\\in L$ and $r\\in R$ with $\\SET{l,r}\\in X$.\n\t Let ${\\mathcal{O}}=(E,{\\mathcal{C}},{\\mathcal{C}}^\\ast)$ be the oriented matroid that corresponds to the orientation $D$ of $G$ as above. We have\n\t ${\\mathcal{C}} = \\pm \\SET{ \\vphantom{\\sum_A^{A^A}}\\SET{\\vphantom{A^A}\\SET{1,2},-\\SET{1,5},\\SET{2,3},\\SET{3,4},\\SET{4,5}} }$ and\n\t \\begin{align*}\n\t \t{\\mathcal{C}}^\\ast = \\pm \\left\\{\\vphantom{A^{A^A}}\\right. & \n\t \t\\SET{\\vphantom{A^A}\\SET{1,2},\\SET{1,5}}, \\SET{\\vphantom{A^A}\\SET{1,2},-\\SET{2,3}}, \n\t \t \\SET{\\vphantom{A^A}\\SET{1,2},-\\SET{3,4}}, \\SET{\\vphantom{A^A}\\SET{1,2},-\\SET{4,5}},\t \t\\\\ \n\t \t &\n\t \t\\SET{\\vphantom{A^A}\\SET{2,3},\\SET{1,5}}, \n\t \t \\SET{\\vphantom{A^A}\\SET{2,3},-\\SET{3,4}}, \\SET{\\vphantom{A^A}\\SET{2,3},-\\SET{4,5}},\t \t\\\\ \n\t \t &\n\t \t\\SET{\\vphantom{A^A}\\SET{3,4},\\SET{1,5}}, \\SET{\\vphantom{A^A}\\SET{3,4},-\\SET{4,5}},\t\\SET{\\vphantom{A^A}\\SET{4,5},\\SET{1,5}}\n\t \t \\left.\\vphantom{A^{A^A}}\\right\\} .\n\t \\end{align*}\n\t Furthermore, all coflows of $G$ with respect to $D$ are integral linear combinations of the signed cocircuits of the oriented matroid\n\t ${\\mathcal{O}}$ corresponding to the orientation $D$ of $G$; the coflow $\\hat \\phi$ may be written as the a linear combination of cocircuits of ${\\mathcal{O}}$\n\t \\( \\hat \\phi = \\SET{\\vphantom{A^A}\\SET{1,2},-\\SET{2,3}} + \\SET{\\vphantom{A^A}\\SET{3,4},-\\SET{4,5}} + 2\\cdot\\SET{\\vphantom{A^A}\\SET{4,5},\\SET{1,5}}.\\)\n\t For all graphs $G=(V,A)$, the equation $\\chi(G) = \\chi({\\mathcal{O}})$ holds, \n\t where ${\\mathcal{O}}$ corresponds to an orientation $D$ of the cycle matroid of $G$. Thus the chromatic number of \n\t oriented matroids is a generalization of the chromatic number of graphs.\n\\end{example}\n\n\\begin{example}\\PRFR{Mar 7th}\n\tConsider the oriented matroids ${\\mathcal{O}}_1$ and ${\\mathcal{O}}_2$ given in Example~\\ref{ex:nonStrictGammoidOrientations}.\n\t$M({\\mathcal{O}}_1)=M({\\mathcal{O}}_2)=(E,{\\mathcal{I}})$ is the matroid given in Example~\\ref{ex:nonStrictGammoid}.\n\tFor both orientations, the corresponding coflow lattice is the free integer module $\\mathbb{Z}^E$. Therefore we have\n\t$\\chi({\\mathcal{O}}_1) = \\chi({\\mathcal{O}}_2) = 2$.\n\\end{example}\n\n\\PRFR{Mar 7th}\n\\noindent\nLet ${\\mathcal{O}}$ and ${\\mathcal{O}}'$ be two oriented matroids such that $M({\\mathcal{O}}) = M({\\mathcal{O}}')$. We would like to mention that it is still an open problem whether\nin this case\nthe equation $\\chi({\\mathcal{O}}) = \\chi({\\mathcal{O}}')$ holds in general (\\cite{Ni12} (Q4), p.69). This question clearly is beyond the scope of this work.\nHowever, if ${\\mathcal{O}}' = {\\mathcal{O}}_{-X}$ is the reorientation of ${\\mathcal{O}}$ with respect to some set $X\\subseteq E$, then \n$$\\mathbb{Z}.{\\mathcal{C}}^\\ast_{-X} = \\SET{ F\\in \\mathbb{Z}^{E} ~\\middle|~ \\exists F'\\in \\mathbb{Z}.{\\mathcal{C}}^\\ast \\colon \\, \\forall e\\in E\\colon\\, F(e)=(-1)^{\\chi_X(e)} F'(e) }$$\nwhere $\\chi_X$ is the characteristic function of $X\\subseteq E$, i.e. $\\chi_X(e) = 1$ if $e\\in X$ and $\\chi_X(e) = 0$ if $e\\notin X$.\nTherefore the nowhere-zero coflows of ${\\mathcal{O}}$ are in a $\\left| \\cdot \\right|$-preserving one-to-one correspondence with the nowhere-zero coflows of ${\\mathcal{O}}'$,\nthus $\\chi({\\mathcal{O}}) =\\chi({\\mathcal{O}}')$ whenever ${\\mathcal{O}}'$ is a reorientation of ${\\mathcal{O}}$.\n\n\\begin{theorem}[\\cite{HN06}, Theorem 1]\\label{thm:chiOuniform}\\PRFR{Mar 7th}\n\tLet $r\\in \\mathbb{N}$ and ${\\mathcal{O}} = (E,{\\mathcal{C}},{\\mathcal{C}}^\\ast)$ be an oriented matroid \n\tsuch that $M({\\mathcal{O}}) = \\left( E,\\SET{\\vphantom{A^A}X\\subseteq E~\\middle|~\\left| X \\right| \\leq r }\\right)$ is the uniform matroid of rank $r$ on $E$.\n\tThen \n\t\\[ \\chi({\\mathcal{O}}) = \\begin{cases}[r] 2 & \\quad\\text{if~} n\\cdot (n-r)\\text{~is even},\\\\\n\t 3 & \\quad\\text{if~} n\\cdot (n-r)\\text{~is odd}.\\\\\n\t\\end{cases} \\]\n\\end{theorem}\n\n\\noindent\nSee \\cite{HN06} for the proof. Theorem~\\ref{thm:chiOuniform} is the generalization of the fact that the chromatic number of a cycle graph -- with at least $3$ vertices -- is $2$ if the cycle graph consists of an even number of vertices, and $3$ if the cycle graph consists of an odd number of vertices.\n\n\\needspace{6\\baselineskip}\n\n\\begin{theorem}[\\cite{HN08}, Theorem 3]\\PRFR{Mar 7th}\n\tLet ${\\mathcal{O}}= (E,{\\mathcal{C}},{\\mathcal{C}}^\\ast)$ be an oriented matroid such that $M({\\mathcal{O}})$ has no loops, no parallel edges, and $\\mathrm{rk}_{M({\\mathcal{O}})}(E) \\geq 3$.\n\n\n\tThen \\[ \\chi({\\mathcal{O}}) \\leq \\mathrm{rk}_{M({\\mathcal{O}})}(E) + 1 \\]\n\twhere equality holds if and only if $M({\\mathcal{O}})$ is isomorphic \n\tto the cycle matroid $M(K)$ of the complete graph $K=\\left(V,\\binom{V}{2}\\right)$ with $\\left| V \\right| =\\mathrm{rk}_{M({\\mathcal{O}})}(E)+1$ vertices.\n\\end{theorem}\n\n\\PRFR{Mar 7th}\n\\noindent\nSee \\cite{HN08} for the proof. If $M({\\mathcal{O}})$ has no loops but it has two parallel edges, i.e. some $e,f\\in E$ with $e\\not=f$ and\n $\\mathrm{rk}_{M({\\mathcal{O}})}\\left( \\SET{e,f} \\right) = 1$,\nthen $\\SET{e,f}$ is a circuit of $M({\\mathcal{O}})$. By Lemma~\\ref{lem:CircuitCocircuitOrthogonality} we obtain that\nthe equality $e\\in D_\\pm \\Leftrightarrow f\\in D_\\pm$ holds for every $D\\in {\\mathcal{C}}^\\ast$.\nLet $C\\in {\\mathcal{C}}$ be the signed circuit with $C_\\pm = \\SET{e,f}$, and let $D_1,D_2\\in {\\mathcal{C}}^\\ast$ with $e\\in \\left( {D_1}_\\pm \\cap {D_2}_\\pm \\right)$.\nSince $C\\bot D_1$ and $C\\bot D_2$, we obtain that $D_1(e)D_2(e) = D_1(f) D_2(f)$, i.e. the sign of $e$ uniquely determines the sign of $f$ \nin any cocircuit of ${\\mathcal{O}}$.\n In the proof of Lemma~\\ref{lem:CircuitCocircuitOrthogonality} we\nestablished that cocircuits are the complements of hyperplanes, therefore every signed cocircuits $D'$ of the restriction\n${\\mathcal{O}} | \\left( E\\BSET{f} \\right)$\ncorresponds to a signed cocircuit $D$ of ${\\mathcal{O}}$ \nwhere $\\SET{e,f}\\subseteq D_\\pm$ if and only if $e\\in D'_\\pm$. \nThus a nowhere-zero coflow $\\phi'$ of ${\\mathcal{O}}| \\left( E\\BSET{f} \\right)$ \nextends naturally to a nowhere-zero coflow $\\phi$ of ${\\mathcal{O}}$ with $\\phi'(f) \\in \\SET{-\\phi(e),\\phi(e)}$ by taking any integer linear combination of cocircuits \nof the restriction ${\\mathcal{O}}| \\left( E\\BSET{f} \\right)$ with respect to the corresponding cocircuits of ${\\mathcal{O}}$.\nConsequently, $\\chi({\\mathcal{O}}) = \\mathrm{rk}_{M({\\mathcal{O}})}(E) + 1$\nif and only if $M({\\mathcal{O}})$ is isomorphic to the cycle matroid of a multi-graph on $\\mathrm{rk}_{M({\\mathcal{O}})}(E) + 1$ vertices that has at least one edge between every pair\nof distinct vertices.\n\n\\section{Lattice Path Matroids are 3-Colorable}\n\n\n\\PRFR{Mar 7th}\nThe results presented in this section\nhave been presented in the technical report {\\em Lattice Path Matroids are 3-Colorable}\nby I.~Albrecht and W.~Hochst\u00e4ttler \\cite{Al15}.\n\n\n\n\n\\begin{definition}[\\cite{GoHoNe15}, Definition~4]\\PRFR{Mar 7th}\n Let $M=(E,{\\mathcal{I}})$ be\na matroid. A flat $X\\in{\\mathcal{F}}(M)$ is called \\deftext[coline of M@coline of $M$]{coline of $\\bm M$},\n if $\\mathrm{rk}_M(X)=\\mathrm{rk}_M(E)-2$.\n A flat $Y\\in{\\mathcal{F}}(M)$ is called\n\\deftext[copoint of M on X@copoint of $M$ on $X$]{copoint of $\\bm M$ on $\\bm X$},\n if $X\\subseteq Y$ and\n$\\mathrm{rk}_M(Y)=\\mathrm{rk}_M(E)-1$. \nIf further $\\left| Y\\backslash X \\right|=1$,\nwe say that $Y$ is a \\deftext[simple copoint on X@simple copoint on $X$]{simple copoint on $\\bm X$}. \nIf otherwise $\\left| Y\\backslash X \\right|>1$, we\nsay that $Y$ is a \\deftext[multiple copoint on X@multiple copoint on $X$]{multiple copoint on $\\bm X$}\\footnote{In \\cite{GoHoNe15} and \\cite{Al15}, multiple copoints are called {\\em fat copoints}.\\index{fat copoint}}.\n A \\deftext{quite simple coline}\\footnote{In \\cite{GoHoNe15} and \\cite{Al15}, quite simple colines are called {\\em positive colines}.\\index{positive coline}} is a coline $X\\in {\\mathcal{F}}(M)$,\n such that there are more simple copoints on $X$ than there are multiple copoints on $X$.\n\\end{definition}\n\n\\noindent The following definitions are basically those found in J.E.~Bonin and A.~deMier's paper {\\em Lattice path matroids: Structural properties} \\cite{Bonin2006701}.\n\\begin{definition}\\PRFR{Mar 7th}\nLet $n\\in\\mathbb{N}$. A \\deftext{lattice path} of length $n$ is a tuple\\label{n:latticePath}\n$(p_{i})_{i=1}^{n}\\in\\{\\mathrm{N},\\mathrm{E}\\}^{n}$. \nWe say\nthat the \\deftext[i-th step of a lattice path@$i$-th step of a lattice path]{$\\bm i$-th step} of $(p_{i})_{i=1}^{n}$ is towards the North if $p_{i}=\\mathrm{N}$,\nand towards the East if $p_{i}=\\mathrm{E}$.\n\\end{definition}\n\n\n\\begin{definition}\\PRFR{Mar 7th}\nLet $n\\in\\mathbb{N}$, and let $p = (p_{i})_{i=1}^{n}$ and $q = (q_{i})_{i=1}^{n}$\nbe lattice paths of length $n$. We say that $p$\nis \\deftext[p south of q@$p$ south of $q$]{south of $\\bm q$}\nif for all $k\\in\\SET{1,2,\\ldots,n}$, \n\\[\n\\left|\\left\\{ i\\in \\mathbb{N} \\BSET {0} \\vphantom{A^A}~\\middle|~ i\\leq k {\\mathop{\\text{~and~}}} p_{i}=\\mathrm{N}\\right\\} \\right|\\leq\\left|\\left\\{ i\\in \\mathbb{N} \\BSET {0} \\vphantom{A^A}~\\middle|~ i\\leq k {\\mathop{\\text{~and~}}} q_{i}=\\mathrm{N}\\right\\} \\right|.\n\\]\nWe say that $p$ and $q$ have \\deftext[lattice paths with common endpoints]{common endpoints}, if\n $$\\left|\\left\\{ i\\in \\mathbb{N} \\BSET {0} \\vphantom{A^A}~\\middle|~ i\\leq n {\\mathop{\\text{~and~}}} p_{i}=\\mathrm{N}\\right\\} \\right| = \\left|\\left\\{ i\\in \\mathbb{N} \\BSET {0} \\vphantom{A^A}~\\middle|~ i\\leq n {\\mathop{\\text{~and~}}} q_{i}=\\mathrm{N}\\right\\} \\right|$$ holds. \n We say that the \\deftextX{lattice path $\\bm p$\nis south of $\\bm q$ with common endpoints},\n if $p$ and $q$ have common endpoints and $p$ is south of $q$.\n In this case, we write $p \\preceq q$.\\label{n:neverabove}\n\\end{definition}\n\n\n\\begin{definition}\\PRFR{Mar 7th}\nLet $n\\in \\mathbb{N}$, and let $p,q\\in \\SET{\\mathrm{E},\\mathrm{N}}^n$ be lattice paths such\nthat $p\\preceq q$. We define the set\nof \\deftext[lattice paths between p and q@lattice paths between $p$ and $q$]{lattice paths between\n $\\bm p$ and $\\bm q$}\nto be\\label{n:LPbetweenPQ}\n\\[\n\\mathrm{P}\\left[p,q\\right]=\\left\\{ r \\in\\{\\mathrm{N},\\mathrm{E}\\}^{n}\n\\vphantom{A^A}~\\middle|~\np \\preceq r \\preceq q\\right\\} . \\qedhere\n\\]\n\\end{definition}\n\n\\needspace{6\\baselineskip}\n\\begin{definition}\\label{def:LPmatroid}\\PRFR{Mar 7th}\nA matroid $M=(E,{\\mathcal{I}})$ is called \\deftext{strong lattice path matroid}, if\nits ground set has the property \n$E = \\SET{1,2,\\ldots,\\left| E \\right|}$ and if\nthere are lattice paths $p,q\\in \\SET{\\mathrm{E},\\mathrm{N}}^{\\left| E \\right|}$\nwith $p\\preceq q$,\nsuch that $M = M[p,q],$\nwhere $M[p,q]$ denotes the transversal matroid presented by\nthe family\n${\\mathcal{A}}_{[p,q]}=(A_i)_{i=1}^{\\mathrm{rk}_M(E)} \\subseteq E$\nwith\n\\[\nA_i = \\SET{j\\in E~\\middle|~\\exists (r_j)_{j=1}^{\\left| E \\right|}\\in \\mathrm{P}[p,q]\\colon\\,\n r_j = \\mathrm{N}{\\mathop{\\text{~and~}}} \\left| \\SET{k\\in E\\mid k\\leq j,\\,r_k=\\mathrm{N} }\\right| = i },\n\\]\ni.e. each $A_i$ consists of those $j\\in E$, such that there is a \nlattice path $r$ between $p$ and $q$ such that the $j$-th step of $r$ is towards the North for the $i$-th time in total.\nFurthermore,\na matroid $M=(E,{\\mathcal{I}})$ is called \\deftext{lattice path matroid}, \nif there is a bijection $\\phi \\colon E\\longrightarrow \\SET{1,2,\\ldots,\\left| E \\right|}$ \nsuch that $\\phi[M] = \\left(\\phi[E],\\SET{\\phi[X]\\vphantom{A^A}~\\middle|~ X\\in{\\mathcal{I}}}\\right)$ is a strong lattice path matroid.\n\\end{definition}\n\n\n\\vspace*{-\\baselineskip}\n\\begin{wrapfigure}{r}{5cm}\n\\vspace{1\\baselineskip}\n\\begin{centering\n~\n\\begin{tikzpicture}\n\\node[inner sep=0pt,circle,minimum size=4pt,fill] at (0,0) {};\n\n\\node[inner sep=0pt,circle,minimum size=4pt,fill] at (1,0) {};\n\n\\node[inner sep=0pt,circle,minimum size=4pt,fill] at (2,0) {};\n\n\\node[inner sep=0pt,circle,minimum size=4pt,fill] at (0,1) {};\n\n\\node[inner sep=0pt,circle,minimum size=4pt,fill] at (1,1) {};\n\n\\node[inner sep=0pt,circle,minimum size=4pt,fill] at (2,1) {};\n\n\\node[inner sep=0pt,circle,minimum size=4pt,fill] at (3,1) {};\n\\node[inner sep=0pt,circle,minimum size=4pt,fill] at (0,2) {};\n\n\\node[inner sep=0pt,circle,minimum size=4pt,fill] at (1,2) {};\n\n\\node[inner sep=0pt,circle,minimum size=4pt,fill] at (2,2) {};\n\n\\node[inner sep=0pt,circle,minimum size=4pt,fill] at (3,2) {};\n\\node[inner sep=0pt,circle,minimum size=4pt,fill] at (1,3) {};\n\n\\node[inner sep=0pt,circle,minimum size=4pt,fill] at (2,3) {};\n\n\\node[inner sep=0pt,circle,minimum size=4pt,fill] at (3,3) {};\n\n\\draw[thin] (1,0) -- (1,3);\n\\draw[thin] (2,0) -- (2,3);\n\\draw[thin] (0,1) -- (3,1);\n\\draw[thin] (0,2) -- (3,2);\n\\draw[very thick] (0,0) -- (2,0) -- (2,1) -- (3,1) -- (3,3);\n\\node at (3.5,.5) {$p$};\n\\node at (-.5,.5) {$q$};\n\n\\draw[very thick] (0,0) -- (0,2) -- (1,2) -- (1,3) -- (3,3);\n\\begin{scope}[shift={(-.1,-.15)}]\n\\node at (.3,.5) {$1$};\n\\node at (1.3,.5) {$2$};\n\\node at (2.3,.5) {$3$};\n\\node at (.3,1.5) {$2$};\n\\node at (1.3,1.5) {$3$};\n\\node at (2.3,1.5) {$4$};\n\\node at (3.3,1.5) {$5$};\n\\node at (1.3,2.5) {$4$};\n\\node at (2.3,2.5) {$5$};\n\\node at (3.3,2.5) {$6$};\n\\end{scope}\n\\end{tikzpicture}\n\\end{centering}%\n\\vspace*{-1\\baselineskip}\n\\end{wrapfigure}\n~\n \n\n\n\\begin{example}\\PRFR{Mar 7th}\nLet us consider the two lattice paths $p=(\\mathrm{E},\\mathrm{E},\\mathrm{N},\\mathrm{E},\\mathrm{N},\\mathrm{N})$\nand $q=(\\mathrm{N},\\mathrm{N},\\mathrm{E},\\mathrm{N},\\mathrm{E},\\mathrm{E})$. \nWe have $p\\preceq q$ and the strong lattice path matroid $M[p,q]$ is the transversal matroid $M({\\mathcal{A}})$ presented\nby the family \\linebreak ${\\mathcal{A}}=(A_i)_{i=1}^3$ of subsets of $\\SET{1,2,\\ldots,6}$ where $A_{1}=\\{1,2,3\\}$, $A_{2}=\\{2,3,4,5\\}$,\nand \n$A_{3}=\\{4,5,6\\}$.%\n\\end{example}\n\n\n\\begin{theorem}[\\cite{Bonin2006701}, Theorem~2.1]\\PRFR{Mar 7th}\n\\label{LPMthm:P} Let $p$, $q$ be lattice\npaths of length $n$, such that $p\\preceq q$.\nLet ${\\mathcal{B}} \\subseteq 2^{\\SET{1,2,\\ldots,n}}$ consist of the bases of the\nstrong lattice path matroid $M=M[p,q]$ on the ground set $\\SET{1,2,\\ldots,n}$.\nLet\n$$\\phi \\colon \\mathrm{P}[p,q] \\longrightarrow {\\mathcal{B}},\\quad (r_i)_{i=1}^n \\mapsto \\SET{j\\in \\mathbb{N} ~\\middle|~ 1\\leq j\\leq n,\\,r_j = \\mathrm{N}} .$$\nThen $\\phi$ is a bijection between \nthe family of lattice paths $\\mathrm{P}[p,q]$ between $p$ and $q$ and the family of bases of $M$.\n\\end{theorem}\n\\begin{proof}\\PRFR{Mar 7th}\n\tClearly, $\\phi$ is well-defined: let $r=(r_i)_{i=1}^n\\in \\mathrm{P}[p,q]$, and let $m = \\mathrm{rk}_M(\\SET{1,2,\\ldots,n})$,\n\tthen there are $j_1 < j_2 < \\ldots < j_m$ such that $r_i = \\mathrm{N}$ if and only if $i\\in \\SET{j_1,j_2,\\ldots,j_m}$.\n\tThus the map $$\\iota_r\\colon \\phi(r) \\longrightarrow \\SET{1,2,\\ldots,m},$$ where\n\t$\\iota_r(i) = k$ for $k$ such that $i = j_k$, witnesses that the set $\\phi(r) \\subseteq \\SET{1,2,\\ldots,n}$ is indeed a transversal of ${\\mathcal{A}}_{[p,q]}$,\n\tand therefore a base of $M[p,q]$. It is clear from Definition~\\ref{def:LPmatroid} that $\\phi$ is surjective.\n\tIt is obvious that if we consider only lattice paths of a fixed given length $n$, then the indexes of the steps towards the North\n\t uniquely determine such a lattice path. Thus $\\phi$ is also injective.\n\\end{proof}\n\n\\needspace{6\\baselineskip}\n\\begin{proposition}\\PRFR{Mar 7th}\n\\label{LPMprop:X}Let $p=(p_{i})_{i=1}^{n}$, $q=(q_{i})_{i=1}^{n}$\nbe lattice paths of length $n$ such that $p\\preceq q$. Let $j\\in E=\\SET{1,2,\\ldots,n}$ and $M=M[p,q]$. Then\n\\begin{enumerate} \\renewcommand{\\theenumi}{(\\roman{enumi})}\\renewcommand{\\labelenumi}{\\theenumi}\n\\item $\\mathrm{rk}_M\\left( \\SET{1,2,\\ldots,j} \\right)=\\left|\\left\\{ i\\in \\SET{1,2,\\ldots,j} \\mid q_{i}=\\mathrm{N}\\right\\} \\right|$.\n\\item \\begin{minipage}[t]{12cm} The element $j$ is a loop in $M$ if and only if \n\\[\n\\left|\\left\\{ i\\in \\SET{1,2,\\ldots,j-1} \\vphantom{A^A}~\\middle|~ p_{i}=\\mathrm{N}\\right\\} \\right|=\\left|\\left\\{ i\\in \\SET{1,2,\\ldots,j}\\vphantom{A^A}~\\middle|~ q_{i}=\\mathrm{N}\\right\\} \\right|,\n\\]\ni.e. the $j$-th step is forced to go towards East for all $r\\in \\mathrm{P}[p,q]$.\\end{minipage}\n\\vtop{%\n \\vskip0pt\n \\vspace*{.8cm}\n \\hbox{%\n \\includegraphics[scale=.75]{loop}%\n }%\n}\n\\item For all $k\\in E$ with $j \\phi(j)}\\right|} . \\qedhere\\]\n\\end{definition}\n\n\\begin{definition}\\label{def:Csigmac}\\PRFR{Mar 7th}\n\tLet $D=(V,A)$ be a digraph such that $V=\\dSET{v_1,v_2,\\ldots,v_n}$ is implicitly ordered,\n\t $(\\sigma,\\ll)$ be a heavy arc signature of $D$, and let $T,E\\subseteq V$\n\tbe subsets that inherit the implicit order of $V$.\n\tFurthermore, let $M=\\Gamma(D,T,E)$ be the corresponding gammoid, and let $C\\in {\\mathcal{C}}(M)$\n\tbe a circuit of $M$ \n\t such that $C = \\dSET{c_1,c_2,\\ldots,c_m}$ inherits its implicit order from $V$;\n\t and let $i\\in\\SET{1,2,\\ldots,m}$. \n\tThe \\deftext[heavy arc circuit signature]{signature of $\\bm C$ with respect to $\\bm M$, $\\bm i$, and $\\bm (\\bm \\sigma \\bm, \\bm \\ll \\bm)$}\n\tshall be the signed subset $C_{(\\sigma,\\ll)}^{(i)}$ of $E$ where\n\t\\[ C_{(\\sigma,\\ll)}^{(i)}(e) = \\begin{cases}[r]\n\t\t\t\t\t\t\t0 &\\quad \\text{if~}e\\notin C,\\\\\n\t\t\t\t\t\t\t- \\mathrm{sgn}_{\\sigma}(R_{i}) &\\quad \\text{if~}e = c_i,\\\\\n\t\t\t\t\t\t\t(-1)^{i-j+1} \\cdot \\mathrm{sgn}_{\\sigma}(R_{j}) &\\quad \\text{if~}e = c_j\\not= c_i,\n\t\t\t\t\t\t\\end{cases} \\]\n\tand where for all $k\\in \\SET{1,2,\\ldots,m}$\n\t\\[ R_k = \\max_{\\llless} \\SET{R \\mid R\\colon C\\BSET{c_k}\\double{\\rightarrow} T\\text{~in~}D} \\]\n\tdenotes the unique $\\llless$-maximal routing from $C\\BSET{c_k}$ to $T$ in $D$.\n\\end{definition}\n\n\\begin{remark}\\PRFR{Mar 7th}\n\tThe factors $(-1)^{i-j+1}$ in Definition~\\ref{def:Csigmac} do not appear explicitly in Lemma~\\ref{lem:CRAMERsrule},\n\twhere\n\t$\\nu_e$ is obtained from the restriction $\\mu| (C\\BSET{c})\\times T_0$ \n\tby replacing the values in row $e$ with the values of $\\mu_c$.\n\tWe have to account for the number of row transpositions that are needed to turn $\\nu_e$ into the restriction $\\mu|(C\\BSET{e})\\times T_0$, which depends on the position of $e=c_j$ relative to $c=c_i$ with respect to the implicit order of $V$.\n\\end{remark}\n\n\\begin{definition}\\label{def:heavyArcWeighting}\\PRFR{Mar 7th}\n\tLet $D=(V,A)$ be a digraph and $(\\sigma, \\ll)$ a heavy arc signature of $D$, and let $w\\colon A\\longrightarrow \\mathbb{R}$ be an indeterminate weighting of $D$.\n\tWe say that $w$ is a \\deftext[heavy arc weighting]{$\\bm( \\bm\\sigma\\bm,\\bm\\ll\\bm)$-weighting of $\\bm D$} if, for all $a\\in A$,\n\tthe inequality $\\left| w(a) \\right| \\geq 1$,\n\tthe strict inequality\n\t\\[\\sum_{L\\subseteq \\SET{x\\in A~\\middle|~ x \\ll a,\\,x\\not=a}} \\left( \\prod_{x\\in L} \\left| w(x) \\right| \\right) < \\left| w(a) \\right|, \\]\n\tand the equality \n\t\\( \\mathrm{sgn}(w(a)) = \\sigma(a) \\)\n\thold.\n\\end{definition}\n\n\\needspace{4\\baselineskip}\n\\begin{lemma}\\label{lem:ExistenceOfHeavyArcWeighting}\\PRFR{Mar 7th}\n\tLet $D=(V,A)$ be a digraph and $(\\sigma, \\ll)$ be a heavy arc signature of $D$.\n\tThere is a $(\\sigma,\\ll)$-weighting of $D$.\n\\end{lemma}\n\\begin{proof}\\PRFR{Mar 7th}\n\tLet $w\\colon A\\longrightarrow \\mathbb{R}$ be an indeterminate weighting of $D$, which exists due to Lemma~\\ref{lem:enoughZindependents}.\n\tIt is clear from Definition~\\ref{def:Zindependent} that for every $\\zeta \\in \\mathbb{Z}^A$ and every $\\tau \\in \\SET{-1,1}^A$, the map\n\t$w_{\\zeta,\\tau}\\colon A\\longrightarrow \\mathbb{R}$, which has $$w_{\\zeta,\\tau}(a) = \\tau(a)\\cdot \\frac{w(a)}{\\mathrm{sgn}(w(a))} + \\tau(a) \\cdot \\zeta(a)$$ for all $a\\in A$, is an indeterminate weighting of $D$, too.\n\tNow, let $\\zeta\\in \\mathbb{Z}^A$, such that for all $a\\in A$ we have the following recurrence relation\n\t\\[ \\zeta(a) = \\left\\lceil \\sum_{L\\subseteq \\SET{x\\in A ~\\middle|~ x \\ll a,\\,x\\not=a}} \\left( \\prod_{x\\in L} \\left(\\vphantom{A^1} \\left| w(x) \\right| + \\zeta(x) \\right) \\right) \\right\\rceil.\\]\n\tThe map $\\zeta$ is well-defined by this recurrence relation because $\\left| A \\right| < \\infty$ and therefore there is a $\\ll$-minimal element $a_0$ in $A$, and we have $\\zeta(a_0) = \\prod_{x\\in \\emptyset} \\left(\\vphantom{A^1} \\left| w(x) \\right| + \\zeta(x) \\right) = 1$.\n\tThen $w_{\\zeta,\\sigma}$ is a $(\\sigma,\\ll)$-weighting of $D$. Clearly,\n\t\\begin{align*}\n\t\t \\mathrm{sgn}\\left( w_{\\zeta,\\sigma}(a) \\right) & = \\mathrm{sgn}\\left( \\sigma(a) \\cdot \\frac{w(a)}{\\mathrm{sgn}(w(a))} + \\sigma(a) \\cdot \\zeta(a) \\right) \n\t\t \\\\ & \n\t\t = \\mathrm{sgn}\\left( \\vphantom{a^1}\\sigma(a) \\right)\\cdot\\mathrm{sgn}\\left( \\frac{w(a)}{\\mathrm{sgn}(w(a))} + \\zeta(a) \\right)\\\\& \n\t\t = \\sigma(a) \\cdot 1 = \\sigma(a)\n\t\t \\end{align*}\n\tholds for all $a\\in A$. Furthermore, we have\n\t\\begin{align*}\n\t\t\\left| w_{\\zeta,\\sigma}(a) \\right| & = \\left| \\sigma(a) \\cdot \\frac{w(a)}{\\mathrm{sgn}(w(a))} + \\sigma(a) \\cdot \\zeta(a) \\right| \\\\\n\t\t& > \\left| \\zeta(a) \\right|\n\t\t = \\left\\lceil \\sum_{L\\subseteq \\SET{x\\in A ~\\middle|~ x \\ll a,\\,x\\not=a}} \\left( \\prod_{x\\in L} \\left(\\vphantom{A^1} \\left| w(x) \\right| + \\zeta(x) \\right) \\right) \\right\\rceil \\\\\n\t\t& \\geq \\sum_{L\\subseteq \\SET{x\\in A~\\middle|~ x \\ll a,\\,x\\not=a}} \\left( \\prod_{x\\in L} \\left| w_{\\zeta,\\sigma}(x) \\right| \\right). \\qedhere\n\t\\end{align*}\n\\end{proof}\n\n\\needspace{5\\baselineskip}\n\n\\begin{lemma}\\label{lem:lllMaximalRoutingsHaveCommonEnd}\\PRFR{Mar 7th}\n\tLet $D=(V,A)$ be a digraph, $(\\sigma,\\ll)$ be a heavy arc weighting of $D$, $E,T\\subseteq V$,\n\t$C\\in {\\mathcal{C}}(\\Gamma(D,T,E))$ be a circuit in the corresponding gammoid, and let $c,d\\in C$.\n\tFurthermore, let $R_c \\colon C\\BSET{c} \\double{\\rightarrow} T$ and $R_d \\colon C\\BSET{d} \\double{\\rightarrow} T$ be \n\tthe $\\lll$-maximal routings in $D$.\n\tThen \\[ \\SET{p_{-1} ~\\middle|~ p\\in R_c} = \\SET{p_{-1} ~\\middle|~ p\\in R_d} \\]\n\tholds.\n\\end{lemma}\n\\begin{proof}\\PRFR{Mar 7th}\n\tLet $S$ be a $C$-$T$-separator of minimal cardinality in $D$, i.e. a $C$-$T$-separator with\n\t$\\left| S \\right| = \\left| C \\right| -1$. Since $R_c$ and $R_d$ are both $C$-$T$-connectors with maximal cardinality,\n\twe obtain that for every $s\\in S$ there is $p_c^s \\in R_c$ and a $p_d^s \\in R_d$ such that $s\\in \\left| p_c^s \\right|$ and\n\t$s\\in \\left| p_d^s \\right|$ (Corollary~\\ref{cor:Menger}), thus there are paths $l_c^s,l_d^s,r_c^s,r_d^s\\in {\\mathbf{P}}(D)$ such that\n\t$p_c^s = l_c^s . r_c^s$ and\n\t\\linebreak\n\t $p_d^s = l_d^s . r_d^s$ with $\\left( {r_c^s} \\right)_1 = \\left( {r_d^s} \\right)_1 = s$. Now let $R_c^S = \\SET{r_c^s ~\\middle|~ s\\in S}$\n\tand $R_d^S = \\SET{r_d^s ~\\middle|~ s\\in S}$, clearly both $R_c^S$ and $R_d^S$ are routings from $S$ to $T$ in $D$.\n\tAssume that $R_c^S \\not= R_d^S$, then we have $R_c^S \\lll R_d^S$ --- without loss of generality, by possibly\n\tswitching names for $c$ and $d$. Then $Q = \\SET{l_c^s . r_d^s ~\\middle|~ s\\in S}$ is a routing from $C\\BSET{c}$ to $T$ in $D$.\n\tBut for the symmetric differences we have the equality \n\t\\[ \\left( \\bigcup_{p\\in Q} \\left| p \\right|_A \\right) \\bigtriangleup \\left( \\bigcup_{p\\in R_c} \\left| p \\right|_A \\right) \n\t= \\left( \\bigcup_{p\\in R_d^S} \\left| p \\right|_A \\right) \\bigtriangleup \\left( \\bigcup_{p\\in R_c^S} \\left| p \\right|_A \\right),\n\t\\]\n\twhich implies $R_c \\lll Q$, a contradiction to the assumption that $R_c$ is the $\\lll$-maximal routing from $C\\BSET{c}$ to $T$.\n\tThus $R_c^S = R_d^S$ and the claim of the lemma follows.\n\\end{proof}\n\n\\PRFR{Mar 7th}\n\\noindent Now we have amassed all ingredients that we need in order to show that every heavy arc signature of $D$ corresponds to an orientation of a gammoid\nwhenever $D$ is an acyclic digraph. Thus heavy arc signatures yield orientations of cascade matroids.\n\n\\needspace{4\\baselineskip}\n\\begin{lemma}\\label{lem:acyclicOrientation}\\PRFR{Mar 7th}\n\tLet $D=(V,A)$ be an acyclic digraph where $V$ is implicitly ordered, $(\\sigma,\\ll)$ be a heavy arc signature of $D$, and $T,E\\subseteq V$.\n\tThen there is a unique oriented matroid ${\\mathcal{O}}=(E,{\\mathcal{C}},{\\mathcal{C}}^\\ast)$ where\n\t \\[ {\\mathcal{C}} = \\SET{ \\pm C_{(\\sigma,\\ll)}^{(1)} ~\\middle|~ C\\in {\\mathcal{C}}(\\Gamma(D,T,E))}.\\]\n\\end{lemma}\n\\begin{proof}\\PRFR{Mar 7th}\n\tLet $M=\\Gamma(D,T,E)$, and let $w\\colon A\\longrightarrow \\mathbb{R}$ be a $(\\sigma,\\ll)$-weighting of $D$ which exists due to Lemma~\\ref{lem:ExistenceOfHeavyArcWeighting}.\n\tFurthermore, let $\\mu\\in \\mathbb{R}^{E\\times T}$ be the matrix defined as in the Lindstr\u00f6m Lemma~\\ref{lem:lindstrom}, with respect to the\n\t$(\\sigma,\\ll)$-weighting $w$ and the implicit order on $V$. Theorem~\\ref{thm:gammoidOverR} along with its proof yields that we have $M = M(\\mu)$.\n\tLet ${\\mathcal{O}} = {\\mathcal{O}}(\\mu) = (E,{\\mathcal{C}}_\\mu,{\\mathcal{C}}_\\mu^\\ast)$ be the oriented matroid that arises from $\\mu$, so $M({\\mathcal{O}}) = M(\\mu)$ holds (Corollary~\\ref{cor:MOmuEQMmu}). We show that ${\\mathcal{C}}_\\mu = {\\mathcal{C}}$. It suffices to prove\n\t that for all $C\\in {\\mathcal{C}}(M)$, all $D \\in {\\mathcal{C}}_\\mu$ with $D_\\pm = C$, and all $D'\\in {\\mathcal{C}}$\n\twith $D_\\pm = C$ we have $D \\in \\SET{D',-D'}$. \n\tNow, let $C\\in {\\mathcal{C}}(M)$ and let $C=\\dSET{c_1,c_2,\\ldots,c_k}$ implicitly ordered respecting the implicit order of $V$.\n\tThe claim follows if $D(c_1)D(c_j) = D'(c_1)D'(c_j)$ holds for all $j\\in \\SET{2,3,\\ldots,k}$.\n\tLet $T_0 \\subseteq T$ be the target vertices onto which the $\\lll$-maximal \n\tand $\\left| \\cdot \\right|$-maximal $C$-$T$-connectors link in $D$ \n\t(Lemma~\\ref{lem:lllMaximalRoutingsHaveCommonEnd}).\n\tFrom Lemma~\\ref{lem:CRAMERsrule} we obtain that\n\t\\begin{align*}\n\t\tD(c_1)D(c_j) & = -1\\cdot \\mathrm{sgn}\\left( \\frac{\\det (\\nu_j)}{\\det(\\mu | (C\\BSET{c_1})\\times T_0)}\\right)\\\\\n\t\t& = -\\mathrm{sgn}(\\det(\\nu_j))\\cdot \\mathrm{sgn}(\\mu | (C\\BSET{c_1})\\times T_0)) \n\t\\end{align*}\n\twhere \\[ \\nu_j \\colon C\\BSET{c_1}\\times T_0 \\longrightarrow \\mathbb{R},\\quad (x,t)\\mapsto \\begin{cases}[r]\n\t \t\t\\mu(c_1,t) & \\quad \\text{if~} x = c_j,\\\\\n\t \t\t\\mu(x,t) & \\quad \\text{otherwise.}\n\t \\end{cases} \n\t \\]\n\t Observe that $\\nu_j$ arises from the restriction $\\mu| C\\BSET{c_j}\\times T_0$ by a \n\t row-permutation, which has at most one non-trivial cycle,\n\t and this cycle then has the length $j-1$, therefore $$\\det(\\nu_j) = (-1)^{j-2} \\det \\left( \\mu| C\\BSET{c_j}\\times T_0 \\right)$$ holds,\n\t so\n\t \\begin{align*}\n\t\tD(c_1)D(c_j) & = (-1)^{1-j}\\mathrm{sgn}\\left( \\det \\left( \\mu| C\\BSET{c_j}\\times T_0 \\right) \\right)\\cdot \\mathrm{sgn}(\\mu | (C\\BSET{c_1})\\times T_0)).\n\t\\end{align*}\n\t We further have\n\t \\begin{align*}\n\t \tD'(c_1)D'(c_j) & = (-1)^{j+1}\\cdot \\mathrm{sgn}_{\\sigma}(R_1)\\cdot \\mathrm{sgn}_{\\sigma}(R_j)\n\t \\end{align*}\n\t where for all $i\\in \\SET{1,2,\\ldots,k}$\n\t\\[ R_i = \\max_{\\llless} \\SET{R \\mid R\\colon C\\BSET{c_i}\\double{\\rightarrow} T\\text{~in~}D} \\]\n\tdenotes the unique $\\llless$-maximal routing from $C\\BSET{c_i}$ to $T$ in $D$.\n\tBy the Lindstr\u00f6m Lemma~\\ref{lem:lindstrom} we obtain that for all $i\\in\\SET{1,2,\\ldots,k}$\n\tthe equation\n\t\\begin{align*}\n\t\t\\det\\left( \\mu| C\\BSET{c_i}\\times T_0 \\right) & = \\sum_{R\\colon C\\BSET{c_i}\\double{\\rightarrow} T_0} \\left( \\mathrm{sgn}(R) \n \t\t\\prod_{p\\in R} \\left( \\prod_{a\\in \\left| p \\right|_A} w(a) \\right) \\right) \n \t\\end{align*}\n \tholds,\n\twhere $\\mathrm{sgn}(R)$ is the sign of the permutation implicitly given by the start and end vertices of the paths in $R$, \n\tboth with respect to the implicit order on $V$.\n\tSince $w$ is a $(\\sigma,\\ll)$-weighting, we have\n\t\\[ \\left| \\sum_{R\\colon C\\BSET{c_i}\\double{\\rightarrow} T_0,\\,R\\not= R_i} \\left( \\mathrm{sgn}(R) \n \t\t\\prod_{p\\in R} \\left( \\prod_{a\\in \\left| p \\right|_A} w(a) \\right) \\right) \\right| < \\left| w(a_i) \\right| \\]\n \twhere $a_i \\in \\bigcup_{p\\in R_i} \\left| p \\right|_A$ is the $\\ll$-maximal arc in the $\\lll$-maximal routing $R_i$ from\n \t$C\\BSET{c_i}$ to $T_0$ in $D$. Therefore the sign of $\\det\\left( \\mu| C\\BSET{c_i}\\times T_0 \\right)$ is determined\n \tby the sign of the summand that contains $w(a_i)$ as a factor, which is the summand that corresponds to $R = R_i$.\n\tTherefore\n\t\\begin{align*}\n\t\t\\mathrm{sgn} \\left( \\det\\left( \\mu| C\\BSET{c_i}\\times T_0 \\right) \\right) & = \n\t\t \\mathrm{sgn} \\left( \\mathrm{sgn}(R_i)\\prod_{p\\in R_i,\\,a\\in \\left| p \\right|_A} w(a) \\right) \\\\\n\t\t & = \\mathrm{sgn}(R_i) \\prod_{p\\in R_i,\\,a\\in \\left| p \\right|_A} \\mathrm{sgn}(w(a)) \\\\\n\t\t & = \\mathrm{sgn}(R_i) \\prod_{p\\in R_i,\\,a\\in \\left| p \\right|_A} \\sigma(a) \\\\\n\t\t & = \\mathrm{sgn}_\\sigma (R_i).\n\t\\end{align*}\n\tSo we obtain\n\t\\begin{align*}\n\t\tD(c_1)D(c_j) & = (-1)^{1-j} \\mathrm{sgn}_\\sigma(R_1) \\cdot \\mathrm{sgn}_{\\sigma}(R_j) \\\\\n\t\t & = (-1)^{j+1}\\cdot \\mathrm{sgn}_{\\sigma}(R_1)\\cdot \\mathrm{sgn}_{\\sigma}(R_j) = D'(c_1)D'(c_j). \\qedhere\n\t\\end{align*}\n\\end{proof}\n\n\\noindent Unfortunately, we cannot omit the assumption that $D$ is an acyclic digraph.\n\n\\needspace{6\\baselineskip}\n\n\\vspace*{-\\baselineskip}\n\\begin{wrapfigure}{r}{6cm}\n\\vspace{\\baselineskip}\n\\begin{centering}~\n\\includegraphics{NonCascadeOrientCex}\n\\end{centering}%\n\\vspace*{-1\\baselineskip}\n\\end{wrapfigure}\n~\n \n\n\\begin{example}\n\tWe consider the digraph \\linebreak $D=(V,A)$ with the implicitly ordered vertex set $V=\\dSET{a,b,c,d,e,f,g,h,i,x,y}$, \n\tand $A$ as depicted on the right. Let $T=\\SET{a,b,c,d}$.\n\tClearly, ${\\mathbf{W}}(D)$ contains the cycle walk $ghig$. Let $(\\sigma,\\ll)$ be the heavy arc signature of $D$ where\n\t$\\sigma(a) = 1$ for all $a\\in A$, and where $a_1 \\ll a_2$ if the tuple $a_1$ is less than the tuple $a_2$ with respect to the\n\tlexicographic order on $V\\times V$ derived from the implicit order of the vertex set.\n\tLet $C_1 = \\SET{f,g,i}$, $C_2 = \\SET{d,e,f,i}$, $C_{f} = \\SET{d,e,g,i}$. Clearly $C_1,C_2,C_{f}\\in {\\mathcal{C}}(\\Gamma(D,T,E))$.\n\tThe following routings are $\\lll$-maximal among all routings in $D$ with the same set of initial vertices and with targets in $T$.\n\t\\begin{align*}\n\t\tR_{\\SET{f,g}} & = \\SET{fxb, gyc} & \\mathrm{sgn}_{\\sigma}\\left( R_{\\SET{f,g}} \\right) & = +1\\\\\n\t\tR_{\\SET{f,i}} & = \\SET{fxb, igyc} & \\mathrm{sgn}_{\\sigma}\\left( R_{\\SET{f,i}} \\right) & = +1\\\\\n\t\tR_{\\SET{g,i}} & = \\SET{gyc, ifxb} & \\mathrm{sgn}_{\\sigma}\\left( R_{\\SET{g,i}} \\right) & = -1\\\\\n\t\tR_{\\SET{d,e,f}} & = \\SET{d,eyc,fxb} & \\mathrm{sgn}_{\\sigma}\\left( R_{\\SET{d,e,f}} \\right) & = -1\\\\\n\t\tR_{\\SET{d,e,i}} & = \\SET{d,exb,igyc} & \\mathrm{sgn}_{\\sigma}\\left( R_{\\SET{d,e,i}} \\right) & = +1\\\\\n\t\tR_{\\SET{d,f,i}} & = \\SET{d,fxb,igyc} & \\mathrm{sgn}_{\\sigma}\\left( R_{\\SET{d,f,i}} \\right) & = +1\\\\\n\t\tR_{\\SET{e,f,i}} & = \\SET{exb,fd,igyc} & \\mathrm{sgn}_{\\sigma}\\left( R_{\\SET{e,f,i}} \\right) & = -1\\\\\n\t\tR_{\\SET{d,e,g}} & = \\SET{d,eyc,ghifxb} & \\mathrm{sgn}_{\\sigma}\\left( R_{\\SET{d,e,g}} \\right) & = -1\\\\\n\t\tR_{\\SET{d,g,i}} & = \\SET{d,gyc,ifxb} & \\mathrm{sgn}_{\\sigma}\\left( R_{\\SET{d,g,i}} \\right) & = -1\\\\\n\t\tR_{\\SET{e,g,i}} & = \\SET{exb,gyc,ifd} & \\mathrm{sgn}_{\\sigma}\\left( R_{\\SET{e,g,i}} \\right) & = +1\\\\\n\t\\end{align*}\n\tNow let us calculate the signatures of $C_1$, $C_2$, and $C_f$ according to Definition~\\ref{def:Csigmac}.\n\tWe obtain \\[\t\n\t\t\\left( C_{1} \\right)^{(1)}_{(\\sigma,\\ll)} = \\SET{f,g,-i}\n\t\t,\\,\n\t\t\\left( C_{2} \\right)^{(1)}_{(\\sigma,\\ll)} = \\SET{d,e,-f,-i}\n\t\t, {\\mathop{\\text{~and~}}} \n\t\t\\left( C_{f} \\right)^{(1)}_{(\\sigma,\\ll)} = \\SET{-d,-e,-g,-i}.\n\t\t\\]\n\tThis clearly violates axiom {\\em (${\\mathcal{C}}$4)}: if we eliminate $f$ from $\\left( C_{1} \\right)^{(1)}_{(\\sigma,\\ll)}$ \n\tand $\\left( C_{2} \\right)^{(1)}_{(\\sigma,\\ll)}$, then the resulting signed circuit must have opposite signs for $d$ and $i$,\n\tbut $d$ and $i$ have the same sign with respect to $\\left( C_{f} \\right)^{(1)}_{(\\sigma,\\ll)}$. Therefore we see that\n\tthe assumption, that $D$ is\n\tacyclic, cannot be dropped from Lemma~\\ref{lem:acyclicOrientation}.\n\\end{example}\n\n\\noindent We can still use the construction involved in Lemma~\\ref{lem:acyclicOrientation} for every representation $(D,T,E)$,\nbut we first have to construct a complete lifting of $D$ (Lemma~\\ref{lem:completelifting}).\nWe then may use Lemma~\\ref{lem:acyclicOrientation}\ntogether with a heavy arc orientation of the lifted digraph in order to obtain an orientation of the lifted representation,\nand then use the contraction formula from Lemma~\\ref{lem:cyclelifting} in order to obtain the orientation of $(D,T,E)$ (Lemma \\ref{lem:OMminors}).\nThus we have found a purely combinatorial way to determine an orientation of a gammoid from its representation,\nand the proof of Lemma~\\ref{lem:acyclicOrientation} yields that every orientation obtained in this way is realizable.\n\n\\section{Orientations of Gammoids}\n\nIn this section, we want to investigate how to obtain oriented matroids from a given gammoid.\n\n\\begin{lemma}\n\tLet $M=(E,{\\mathcal{I}})$ be a gammoid. Then $M$ is orientable.\n\\end{lemma}\n\\begin{proof}\n\tBy Theorem~\\ref{thm:gammoidOverR} there a set $T$ with $\\left| T \\right| = \\mathrm{rk}_M(E)$ and a matrix $\\mu\\in \\mathbb{R}^{E\\times T}$,\n\tsuch that $M=M(\\mu)$. Then ${\\mathcal{O}}(\\mu)$ is an oriented matroid with $M({\\mathcal{O}}(\\mu)) = M(\\mu)$ (Corollary~\\ref{cor:MOmuEQMmu}).\n\\end{proof}\n\n\\noindent Given a gammoid $M=\\Gamma(D,T,E)$, the oriented matroid, whose existence is guaranteed by the previous lemma,\n depends on the actual values of the\n indeterminate weighting $w\\colon A\\longrightarrow \\mathbb{R}$ of $D$, and obtaining the signatures of the circuits from the matrix\n $\\mu$ requires actual computations in the general case.\n Furthermore, Example~\\ref{ex:UniformNonRealizable} indicates that there are orientations of gammoids\n which cannot be represented by a real matrix.\n\n \\subsection{Realizable Combinatorial Orientations}\n\n In this section, we develop a notion of orientations of gammoids which stem from indeterminate weightings with\n a special property that allows us to determine the signed circuit by examining some combinatorial properties of\n a given representation $(D,T,E)$ alone.\n\n \\begin{definition}\\label{def:Xphi-pivot}\n \tLet $R$ be a commutative ring,\n \t$E$, and $T$ be finite sets, and let $\\mu\\in R^{E\\times T}$. Furthermore, let $X=\\dSET{x_1,x_2,\\ldots,x_n}\\subseteq E$\n \tbe a subset of $E$ with implicit order,\n \tand let $\\phi\\colon X\\longrightarrow T$ be an injective map.\n \tThe \n \t\\deftext[(X,f)-pivot of m@$(X,\\phi)$-pivot of $\\mu$]{$\\bm(\\bm X\\bm,\\bm \\phi\\bm)$-pivot of $\\bm \\mu$} shall be the matrix denoted by\\label{n:Xphi-pivot} $\\mu@\\left(x_i,\\phi(x_i)\\right)_{i=1}^n$, which satisfies the following well-founded recursive definition with respect to $\\left| X \\right|$.\n \t If $X=\\emptyset$,\n \tthen we set $$\\mu@\\left( x_i,\\phi(x_i) \\right)_{i=1}^0 = \\mu.$$\n \tOtherwise, $X=\\dSET{x_1,x_2,\\ldots,x_n}$ for some $n\\in \\mathbb{N}\\BSET{0}$. Let $\\nu = \\mu@\\left( x_i,\\phi(x_i) \\right)_{i=1}^{n-1}$\n \tbe the $(X\\BSET{x_n},\\phi|_{X\\BSET{x_n}})$-pivot of $\\mu$. Then we let $\\mu@\\left( x_i,\\phi(x_i) \\right)_{i=1}^{n}\\in \\mathbb{R}^{E\\times T}$\n \twhere \n \t\\[\\mu@\\left( x_i,\\phi(x_i) \\right)_{i=1}^{n} (e,t) = \\begin{cases}[r]\n \t\t\t\\nu(e,t) & \\quad \\text{if~} t = t_n,\\\\\n \t\t\t\\nu(x_n,t_n) \\cdot \\nu(e,t) - \\nu(x_n,t)\\cdot\\nu(e,t_n) & \\quad \\text{otherwise,}\n \t\\end{cases} \\]\n \twith $t_n = \\phi(x_n)$.\n \\end{definition}\n\n \\begin{example}\n \tFirst, observe that the $(X,\\phi)$-pivot of $\\mu$ is derived from $\\mu$ by a sequence of Gau\u00df-Jordan steps without division. Thus the restriction\n \t$$\\mu' = \\mu@(x_i,\\phi(x_i))_{i=1}^n | X\\times \\phi[X]$$ has the property that $\\mu'(x,t) = 0$ for all $x\\in X$,\n \t$t\\in \\phi[X]$ with $t\\not= \\phi(x)$.\n \n \tBut since we cannot make the pivot element values unique without division,\n \t even under the premise that for all $j\\in\\SET{1,2,\\ldots,n}$ the pivot element $\\mu@\\left( x_i,\\phi(x_i) \\right)_{i=1}^{j} (x_i,\\phi(x_i)) \\not= 0$ the $(X,\\phi)$-pivot of $\\mu$ depends on the implicit order of $X=\\dSET{x_1,x_2,\\ldots,x_n}$:\n \tLet $\\mu\\in R^ {\\SET{a,b}\\times\\SET{1,2}}$, let $\\phi(a) = 1$, and $\\phi(b) = 2$,\n \tand let \\[\n \t\t\\mu = \\left( \\begin{array}{cc} a_1 & a_2 \\\\ b_1 & b_2\\end{array} \\right).\n \t\\]\n \tThen\n \t\\[ \\mu@\\left( (a,1),(b,2)\\right) = \\left( \\begin{array}{rr}\n \t\t{a_1}^2 (b_2 - b_1) & 0 \\\\\n \t\t0 & \\hphantom{-}a_1 (b_2 - b_1)\n \t\\end{array} \\right)\\hphantom{.} \\]\n \tbut\n \t\t\\[ \\mu@\\left( (b,2),(a,1)\\right) = \\left( \\begin{array}{rr}\n \t\tb_2(a_1 - a_2) & 0 \\\\\n \t\t0 & \\hphantom{-}{b_2}^2 (a_1 - a_2)\\\\\n \t\t \t\\end{array} \\right).\n\\]\n \\end{example}\n\n \\begin{example}\n \tLet $X = \\mathbb{Z}[\\SET{a_1,a_2,a_3,b_1,b_2,b_3,c_1,c_2,c_3,d_1,d_2,d_3}]$ be a polynomial ring with\n \t variables where the coefficients range\n \t over the integers.\n \tConsider the matrix $\\mu\\in X^{5\\times 4}$ where $\\mu(1,j) = a_j$,\n \t$\\mu(2,j) = b_j$, \n\t$\\mu(3,j) = c_j$, \n\t$\\mu(4,j) = d_j$, and\n\t$\\mu(5,j) = e_j$ for all $j\\in\\SET{1,2,3,4}$.\n\tLet $\\mu_0 = \\mu$ and let $\\mu_i = \\mu_{i-1}@(i,i)$ for $i\\in\\SET{1,2,\\ldots,5}$.\n\tThe following entries of those matrices will be interesting for determining the signs of circuits.\n\n\t\\[\n\t\\begin{array}{rccrcrcrcrcrcrcrc}\n\t \\mu_1(1,1) & = &&a_1\\\\\n\t \\mu_1(2,1) & = &&b_1\\\\\n\t \\end{array}\\]\n\t\\[\n\t\\begin{array}{rccrcrcrcrcrcrcrc}\n\t \\mu_2(1,1) & = &-&a_{1} a_{2} b_{1}& + &a_{1}^{2} b_{2}\\\\\n\t \\mu_2(2,2) & = &-&a_{2} b_{1} &+& a_{1} b_{2}\\\\\n\t \\mu_2(3,1) & = &&a_{1} b_{2} c_{1} &- &a_{1} b_{1} c_{2}\\\\\n\t \\mu_2(3,2) & = &-&a_{2} c_{1} &+ &a_{1} c_{2}\\\\\n\t \\mu_2(3,3) & = &-&a_{1} a_{3} b_{2} c_{1} &+ &a_{1} a_{2} b_{3} c_{1} &+& a_{1} a_{3} b_{1} c_{2} \n\t \\\\ &&- &a_{1}^{2} b_{3} c_{2} &- &a_{1} a_{2} b_{1} c_{3} &+& a_{1}^{2} b_{2} c_{3}\n\t\\end{array}\n\t\\]\n\n\n \\end{example}\n\n \\begin{lemma}\n \tLet $M=\\Gamma(D,T,E)=(E,{\\mathcal{I}})$ be a gammoid, and let $\\mu\\in \\mathbb{R}^{E\\times T}$ be a matrix such that\n \t$M(\\mu) = M$. Furthermore, let $C\\in{\\mathcal{C}}(M)$ be a circuit, let $c\\in C$,\n \tand let $R\\colon C\\BSET{c}\\double{\\rightarrow} T$ be a routing that witnesses $C\\BSET{c}\\in {\\mathcal{I}}$.\n\n \\end{lemma}\n \\begin{proof}\n \\remred{TODO}\n \\end{proof}\n\n \\begin{definition}\n \tA tuple $(D,T,E,S_A,\\ll)$ is called \\deftext{realizable combinatorial orientation of a gammoid},\n \tif\n \t\\begin{enumerate}\\renewcommand{\\theenumi}{(\\roman{enumi})}\\renewcommand{\\labelenumi}{\\theenumi}\n \t\t\\item $D=(V,A)$ is a digraph, $T\\subseteq V$, $E\\subseteq V$,\n \t\t\\item \\( S_A\\colon A\\longrightarrow \\SET{-1, 1} \\) is a signature of $A$, and\n \t\\item $\\ll\\,\\,\\subseteq A\\times A$ is a linear order on $A$.\n\\end{enumerate}\n \\end{definition}\n\n \\begin{definition}\n \tLet $D=(V,A)$ be a digraph, $\\alpha,\\rho \\in\\mathbb{N}$, $w\\colon A\\longrightarrow \\mathbb{R}$ an indeterminate weighting of $D$,\n \t and $\\ll\\,\\,\\subseteq A\\times A$ is a linear order on $A$.\n \tThen $w$ shall be called a \\deftext[ar-compatible indeterminates@$(\\ll,\\alpha,\\rho)$-compatible indeterminates]{$\\bm (\\bm \\ll\\bm,\\bm\\alpha\\bm,\\bm\\rho\\bm)$-compatible indeterminate weighting of $\\bm D$},\n \tif for all $a\\in A$\\remred{TODO: geschicktes absch\u00e4tzen hier; Def. anpassen}\n \t\\[ 2^{2^\\rho \\alpha}\\cdot \\left| \\prod_{a'\\in A,\\,a'\\ll a,\\,a'\\not= a} \\left( w(a') \\right)^\\rho \\right| < w(a) .\\]\n \\end{definition}\n\n\\needspace{4\\baselineskip}\n \\begin{remark}\n \tSince for every $r\\in \\mathbb{R}$ there is some $z\\in \\mathbb{Z}$ such that $r\\cdot z > 1$, and since $X\\subseteq \\mathbb{R}$ is $\\mathbb{Z}$-independent if\n \tand only if for all $x\\in X$ and $z\\in \\mathbb{Z}\\BSET{0}$ the set $(X\\BSET{x})\\cup\\SET{z\\cdot x}$ is $\\mathbb{Z}$-independent\n \tit is clear that we can derive a $(\\ll,\\alpha,\\rho)$-compatible indeterminate weighting $w'$ from any indeterminate weighting\n \t$w$ by setting $$w'(a) = n_a \\cdot w(a)$$ where for all $a\\in A$\n \t$$ n_a = 1 + 2^{2^\\rho \\alpha} \\cdot \\left\\lceil \\left| \\frac{1}{w(a)} \\right| \\cdot \\left| \\prod_{{a'\\in A,\\,a'\\ll a,\\,a'\\not= a}} \\left( w(a')\\cdot \\prod_{a''\\in A,\\,a'' \\ll a'} n_{a''} \\right)^\\rho \\right| \\right\\rceil .$$\n \\end{remark}\n\n \\begin{definition}\n \tLet $A$ be any set, $\\alpha,\\rho\\in \\mathbb{N}$, and $p\\in \\mathbb{Z}[A]$ a polynomial over $\\mathbb{Z}$ with variables in $A$.\n \tThen $p$ shall be called \\deftext[ar-polynomial@$(\\alpha,\\rho)$-polynomial]{$\\bm(\\bm\\alpha\\bm,\\bm\\rho\\bm)$-polynomial},\n \tif \n \t\\( \\sum_{f\\in \\mathbb{N}^{(A)}} \\left| p(f) \\right| \\leq 2^\\alpha\\)\n \tand \\[ \\max\\SET{\\sum_{a\\in A} f(a) ~~\\middle|~~ f\\in \\mathbb{N}^{(A)},\\, p(f) \\not= 0} \\leq \\rho \\]\n \tholds.\n \\end{definition}\n\n \\begin{definition}\n \tLet $A$ be any non-empty set, and $\\ll\\,\\,\\subseteq A\\times A$ is a linear order on $A$. The\n \t\\deftext[monomial order induced by@monomial order induces by $\\ll$]{monomial order induced by $\\bm \\ll$} shall be the order\n \t$\\ll'$ on $\\mathbb{N}^{(A)}$, where for all $f,g\\in \\mathbb{N}^{(A)}$, we have\n \t$f \\ll' g$ if and only if either $f=g$ or there exists $a\\in A$ such that $f(a) < g(a)$ and for all $a'\\in A$ with $f(a') > g(a')$, $a' \\ll a$.\n \\end{definition}\n\n\\begin{remark}\n\tClearly, $\\ll'$ is a linear order on $\\mathbb{N}^{(A)}$, because whenever $f\\not= g$ there is always a $\\ll$-maximal element $a\\in A$ with $f(a) \\not= g(a)$. Then if $f(a) < g(a)$, $f \\ll' g$, otherwise $g \\ll' f$.\n\\end{remark}\n\n\\begin{definition}\n\tLet $A$ be any non-empty set, and $\\ll\\,\\,\\subseteq A\\times A$ is a linear order on $A$, and $p\\in \\mathbb{Z}[A]$ a polynomial over $\\mathbb{Z}$ with variables in $A$ where\n\t$p\\not= 0$.\n\tThe \\deftext[leading monomial of p@$\\ll$-leading monomial of $p$]{$\\bm \\ll$-leading monomial of $\\bm p$} is\n\tdefined to be the $\\ll'$-maximal monomial $f\\in \\mathbb{N}^{(A)}$ with $p(f) \\not = 0$.\n\tBy convention, we say the zero map $z\\colon A\\mapsto \\mathbb{N}, \\,a\\mapsto 0$ is the $\\ll$-leading monomial of the zero polynomial.\n\\end{definition}\n\n \\begin{lemma}\n \tLet $D=(V,A)$ be a digraph, $\\alpha,\\rho\\in \\mathbb{N}$, $\\ll\\,\\,\\subseteq A\\times A$ is a linear order on $A$,\n \t $w\\colon A\\longrightarrow \\mathbb{R}$ an $(\\ll,\\alpha,\\rho)$-compatible indeterminate weighting of $D$, and $p\\in \\mathbb{Z}[A]$\n \t an $(\\alpha,\\rho)$-polynomial over $\\mathbb{Z}$ with variables in $A$ with $\\ll$-leading monomial $f\\in \\mathbb{N}^{(A)}$.\n \t Then\n \t \\[ \\mathrm{sgn} \\left( \\vphantom{x^A_A}{\\mathrm{eval}}_\\mathbb{R}(p)(w) \\right) = \\mathrm{sgn} \\left( p(f) \\cdot \\prod_{a\\in A} w(a)^{f(a)} \\right).\\]\n \\end{lemma}\n\n \\begin{proof}\n \tBy definition \n \t\\[ \t{\\mathrm{eval}}_{{\\mathbb{R}}} (p)(w) = \\sum_{f\\in \\mathbb{N}^{(A)}} p(f)\\cdot \\prod_{a\\in A} w(a)^{f(a)}.\\]\n \tLet $a_f \\in A$ be the $\\ll$-maximal element of $A$ such that $f(a) > 0$, or the minimal element of $A$ if $f$ is the zero map.\n\n \t\\remred{TODO}\n \\end{proof}\n\n\n\n\n\n\\begin{lemma}\n\tLet $(D,T,E,S_A,\\ll)$ be a realizable combinatorial orientation of a gammoid and $I\\subseteq E\\backslash T$,\n\tthen the reorientation \n\t\\[ \\left( {\\mathcal{O}}(D,T,E,S_A,\\ll) \\right)_{-I} \\quad = \\quad {\\mathcal{O}}(D,T,E,S'_A,\\ll) \\]\n\tcorresponds to the combinatorial orientation\n\twhere\n\t\\[ S'_A \\colon A\\longrightarrow \\SET{-1,1},\\quad (u,v)\\mapsto \\begin{cases} -S_A(u,v) & \\quad \\text{if~} \\left| \\SET{u,v}\\cap I \\right| = 1,\\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t \\hphantom{-}S_A(u,v)&\\quad \\text{otherwise.} \\end{cases}\\]\n\\end{lemma}\n\n\\begin{proof}\n\\remred{TODO}\n\\end{proof}\n\\subsection{Duality}\n\n\\remred{TODO: \u00fcberarbeitung def., evtl. k\u00fcrzen}\nIn this section, we introduce the notion of duality for oriented matroids, which is compatible with the notion of duality for\nthe respective underlying ordinary matroids.\n\n\n\\needspace{4\\baselineskip}\n\\begin{definition}\n\tLet ${\\mathcal{O}}=(E,{\\mathcal{C}})$ and ${\\mathcal{O}}'=(E,{\\mathcal{C}}')$ be two oriented matroids on $E$.\n\tThen ${\\mathcal{O}}'$ is the \\deftext[dual of an oriented matroid]{dual oriented matroid of $\\bm {\\mathcal{O}}$},\n\tif $M({\\mathcal{O}})^\\ast = M({\\mathcal{O}})$, i.e. the matroid underlying ${\\mathcal{O}}'$ is the dual of the matroid underlying ${\\mathcal{O}}$, and\n\tif for all $C\\in {\\mathcal{C}}$ and $C'\\in {\\mathcal{C}}'$, $C\\bot C'$ holds.\n\tIn this case, we denote ${\\mathcal{O}}'$ by ${\\mathcal{O}}^\\ast=(E,{\\mathcal{C}}^\\ast)$.\n\\end{definition}\n\n\\noindent The following result justifies that we can speak of the dual oriented matroid of ${\\mathcal{O}}$. \nIt has been published by R.G.~Bland and M.~Las~Vergnas in the 1978 paper {\\em Orientability of Matroids} (\\cite{BlV78}, Theorem 2.2 {(b)}, and Lemmas 2.2.1 and 2.2.2)\\footnote{The results are also presented in \\cite{BlVSWZ99}, p.115.}\n\n\\begin{theorem}\\label{thm:dualOM}\n\tLet ${\\mathcal{O}}=(E,{\\mathcal{C}})$ be an oriented matroid. Then ${\\mathcal{O}}$ has a unique dual oriented matroid ${\\mathcal{O}}^\\ast$.\n\\end{theorem}\n\n\\begin{proof} We have to show existence and uniqueness.\nLet $M=M({\\mathcal{O}})$ denote the underlying matroid of ${\\mathcal{O}}$.\nWe set \n\t$${\\mathcal{C}}' = \\SET{D\\in \\sigma E\\mid D_\\pm \\in {\\mathcal{C}} \\left( M^\\ast \\right),\\,\\forall C\\in {\\mathcal{C}}\\colon\\,C\\bot D}.$$\nNow ${\\mathcal{C}}'$ consists of all signed circuits with support in ${\\mathcal{C}}(M^\\ast)$ that are orthogonal on all signed circuits in ${\\mathcal{C}}$.\nSuppose that ${\\mathcal{O}}$ has no dual oriented matroid, then there are two possibilities: either there is some $C'\\in {\\mathcal{C}}(M^\\ast)$ such that there is no $D\\in {\\mathcal{C}}'$ with\n$D_\\pm = C'$; or every subset of ${\\mathcal{C}}'$ violates at least one axiom from {\\em (${\\mathcal{C}}$1)},{\\em (${\\mathcal{C}}$2)}, {\\em (${\\mathcal{C}}$3)}, {\\em (${\\mathcal{C}}$4)}.\nNow suppose that ${\\mathcal{O}}$ has at least dual oriented matroid, then ${\\mathcal{C}}'$ is the union of all signed circuits of dual oriented matroids of ${\\mathcal{O}}$.\nBut then {\\em (${\\mathcal{C}}$2)} cannot be satisfied for ${\\mathcal{C}}'$ if there is more than one dual oriented matroid of ${\\mathcal{O}}$:\nin that case we would have at least four different signed circuits $D_1,\\ldots,D_4\\in {\\mathcal{C}}'$ with $\\left( D_i \\right)_\\pm = C'$ for some $C'\\in{\\mathcal{C}}(M^\\ast)$.\nThus it suffices to show that ${\\mathcal{O}}' = (E,{\\mathcal{C}}')$ is an oriented matroid with $M({\\mathcal{O}}') = M^\\ast$ in order to proof the lemma.\n\n\\needspace{4\\baselineskip}\n\\par \\vspace*{.6\\baselineskip}\\noindent \\mbox{---\\hspace*{2.5mm}} \n We define the family $$ {\\bf C}^{\\bm \\ast} = \\SET{{\\mathcal{D}} \\subseteq \\sigma E ~~\\middle|~~ \\SET{D_\\pm\\mid D\\in {\\mathcal{D}}} = {\\mathcal{C}}(M^\\ast)}$$\nof freely assigned signatures to the circuits of $M^\\ast$. Let ${\\mathcal{D}}' \\in {\\bf C}^{\\bm \\ast}$ be a choice such that the number of non-orthogonal pairs\n\\[ n_{{\\mathcal{D}}'} = \\left| \\SET{(C,D')\\in {\\mathcal{C}}\\times {\\mathcal{D}}' ~~\\middle|~~ C\\not \\bot D'} \\right| \\]\nis minimal in ${\\bf C}^{\\bm \\ast}$.\nWe give an indirect argument that a choice ${\\mathcal{D}}' \\in {\\bf C}^{\\bm \\ast}$ with $n_{{\\mathcal{D}}'} = 0$ is always possible.\n Assume that $n_{{\\mathcal{D}}'} > 0$, then let $C\\in {\\mathcal{C}}$ and $D'\\in {\\mathcal{D}}'$ be a non-orthogonal, i.e. $C\\not\\bot D$, such that\n $\\left| C_\\pm\\cap D'_\\pm \\right|$ is minimal. Since $C\\not\\bot D'$, for all $x,y\\in C_\\pm\\cap D_\\pm$\n we have $0\\not= C(x)\\cdot D'(x) = C(y)\\cdot D'(y)$, and because $C_\\pm\\in {\\mathcal{C}}(M({\\mathcal{O}}))$ and $D'_\\pm \\in {\\mathcal{C}}(M({\\mathcal{O}})^\\ast)$,\n the Lemma~\\ref{lem:CircuitCocircuitOrthogonality} yields $\\left| C_\\pm\\cap D'_\\pm \\right| \\geq 2$, thus there are elements\n $x,y\\in C_\\pm\\cap D'_\\pm$ with $x\\not=y$\n such that $0\\not= C(x)\\cdot D'(x) = C(y)\\cdot D'(y)$, let us fix two such elements from here on.\n By Lemma~\\ref{lem:CircuitCocircuitIntersectInTwo} there is a circuit $C_{x,y}\\in {\\mathcal{C}}(M({\\mathcal{O}}))$ such that $C_{x,y}\\cap D'_\\pm = \\SET{x,y}$.\n Thus there is a signed circuit $C'\\in {\\mathcal{C}}$ such that $C'_\\pm = C_{x,y}$ where $C'(x) = -C(x)$ due to property {\\em (${\\mathcal{C}}$2)}.\n The circuit elimination property {\\em (${\\mathcal{C}}$4)} of ${\\mathcal{O}}$ guarantees that there is a circuit $Z\\in {\\mathcal{C}}$ such that $x\\notin Z_\\pm \\subsetneq C_\\pm \\cup C'_\\pm$. But then $Z_\\pm \\cap D'_\\pm = \\left( C_\\pm\\cap D'_\\pm \\right) \\cup \\left( C'_\\pm\\cap D'_\\pm \\right) \\subseteq C_\\pm \\BSET{x}$, thus\n $\\left| Z_\\pm \\cap D'_\\pm \\right| < \\left| C_\\pm\\cap D'_\\pm \\right|$ which contradicts the minimality of the choice of $C$ and $D'$. Therefore $n_{{\\mathcal{D}}'} = 0$ must hold. Consequently, for every $D'_1 \\in {\\mathcal{C}}(M({\\mathcal{O}})^\\ast)$ there is a signed subset $D'\\in \\sigma E$ with $D'_1 = D'_\\pm$ such that for all $C\\in {\\mathcal{C}}$ we have $C\\bot D'$.\n Thus $\\SET{C'_\\pm \\mid C'\\in {\\mathcal{C}}'} = {\\mathcal{C}}(M({\\mathcal{O}})^\\ast)$, and therefore $M({\\mathcal{O}}') = M({\\mathcal{O}})^\\ast$ provided that ${\\mathcal{O}}' = (E,{\\mathcal{C}}')$ is indeed a matroid.\n\n \\par \\vspace*{.6\\baselineskip}\\noindent \\mbox{---\\hspace*{2.5mm}} It remains to show that ${\\mathcal{O}}' = (E,{\\mathcal{C}}')$ is an oriented matroid. \n Since $\\emptyset \\notin {\\mathcal{C}}(M({\\mathcal{O}})^\\ast)$, we have $\\emptyset_{\\sigma E} \\notin {\\mathcal{C}}'$, so {\\em (${\\mathcal{C}}$1)} holds.\n It is obvious that {\\em (${\\mathcal{C}}$2)} holds because $C\\bot D \\Leftrightarrow C\\bot \\left( -D \\right)$ (Lemma~\\ref{lem:MinusCOrthoD}).\n Let $C',D'\\in {\\mathcal{C}}'$ such that $C'_\\pm = D'_\\pm$. We give an indirect argument that $C' \\in \\SET{D',-D'}$. If $C'\\notin \\SET{D',-D'}$,\n then there are $e,f\\in E$ with $C'(e) = D'(e) \\not= 0$ and $C'(f) = -D'(f) \\not= 0$. Thus there is a circuit $X\\in {\\mathcal{C}}$ such \n that $X_\\pm \\cap C'_\\pm = \\SET{e,f} = X_\\pm\\cap D'_\\pm$. Since $X\\bot C'$, we have $C'(f) = -C'(e)\\cdot X'(e) \\cdot X'(f)$, but \n then $0\\not= D'(f) = D'(e)\\cdot X'(e)\\cdot X'(f)$ by the above assumption, yet\\linebreak $0\\not= D'(f) = -D'(e)\\cdot X'(e)\\cdot X'(f)$ \n since $X\\bot D'$ --- a contradiction. Thus $C'\\in \\SET{D',-D'}$ and {\\em (${\\mathcal{C}}$3)} holds.\n\n\\par \\vspace*{.6\\baselineskip}\\noindent \\mbox{---\\hspace*{2.5mm}} \n Finally, we have to show that {\\em (${\\mathcal{C}}$4)} holds for ${\\mathcal{C}}'$. \n Let $X,Y\\in {\\mathcal{C}}'$ such that $X\\not= -Y$, and \n let $e\\in E$ \n such that $X(e) = -Y(e) \\not= 0$. \n Then $\\mathrm{rk}_{M({\\mathcal{O}})^\\ast}(X_\\pm) = \\left| X_\\pm \\right| - 1$ and $\\mathrm{rk}_{M({\\mathcal{O}})^\\ast}(Y_\\pm) = \\left| Y_\\pm \\right| - 1$.\n Since $X_\\pm\\cap Y_\\pm \\subsetneq X_\\pm$ and $X_\\pm\\in {\\mathcal{C}}(M({\\mathcal{O}})^\\ast)$, we obtain $\\mathrm{rk}_{M({\\mathcal{O}})^\\ast}(X_\\pm\\cap Y_\\pm) = \\left| X_\\pm\\cap Y_\\pm \\right|$\n and consequently $\\mathrm{rk}_{M({\\mathcal{O}})^\\ast} \\left( \\left( X_\\pm \\cup Y_\\pm \\right) \\BSET{e} \\right) < \\left| \\left( X_\\pm \\cup Y_\\pm \\right) \\BSET{e} \\right|$.\n Thus there is a minimal dependent subset $Z_1\\subseteq \\left( X_\\pm \\cup Y_\\pm \\right) \\BSET{e}$ with respect to set-inclusion and the matroid $M({\\mathcal{O}})^\\ast$, \n i.e. $Z_1\\in{\\mathcal{C}}(M({\\mathcal{O}})^\\ast)$. We already showed that there is some $Z\\in {\\mathcal{C}}'$ such that $Z_\\pm = Z_1$, and that $Z$ is unique in ${\\mathcal{C}}'$, up to \n possible negation. We have to prove that $Z$ is oriented in a way compatible with axiom {\\em (${\\mathcal{C}}$4)}, i.e. that for some $C\\in \\SET{Z,-Z}$\n we have $C_+ \\subseteq X_+ \\cup Y_+$ and $C_- \\subseteq X_- \\cup Y_-$.\n\\remred{... VERTAGT...}\n Due to the symmetry ${\\mathcal{C}}' = -{\\mathcal{C}}'$ it suffices to show that \n either $Z_+\\subseteq X_+\\cup Y_+$ or\n $Z_- \\subseteq X_+\\cup Y_+$ holds for some fixed $Z\\in {\\mathcal{C}}'$ with $Z_\\pm = Z_1$.\n We do this by induction on $\\left| Z_\\pm \\right|$. Clearly, $Z_\\pm$ can neither be empty because $\\emptyset\\notin {\\mathcal{C}}(M({\\mathcal{O}})^\\ast)$ nor a singleton\n because then $X=-Y$ would be implied. \\remred{.... TODO... }\n Thus the base case is $\\left| Z_\\pm \\right| = 2$. \n\n\nLemma~\\ref{lem:CircuitCocircuitOrthogonality}.\n\n \\remred{TODO}\n\\end{proof}\n\n\\begin{definition}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid. We say that $M$ is \\deftext[orientable matroid]{orientable},\n\tif there is an oriented matroid ${\\mathcal{O}}=(E,{\\mathcal{C}})$ such that $M = M({\\mathcal{O}})$.\n\\end{definition}\n\n\\begin{corollary}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid. Then $M$ is orientable if and only if its dual $M^\\ast$ is orientable.\n\\end{corollary}\n\\begin{proof}\n\tClear from Theorem~\\ref{thm:dualOM}.\n\\end{proof}\n\n\n\n\\begin{lemma}\\label{lem:arcCount}\n Let $D=(V,A)$ be a digraph, let $X,T\\subseteq V$ such that $X\\cap T= \\emptyset$,\n such that every $x\\in X$ is a source in $D$,\n and such that every $t\\in T$ is a sink in $D$.\n \n \n Let further $${\\mathcal{P}} \\subseteq \\bigcup_{x\\in X,\\,t\\in T}{\\mathbf{P}}(D;x,t)$$\n be a family of of non-trivial paths in $D$\n\n \n such that for all $p,q\\in {\\mathcal{P}}$ with $p\\not= q$ either $p_1 \\not= q_1$ or $p_{-1}\\not= q_{-1}$ holds.\n Then \\[ \\left| \\bigcup_{p\\in {\\mathcal{P}}} {\\left| p \\right|_A} \\right| \\geq \\left| {\\mathcal{P}} \\right|.\\]\n\\end{lemma}\n\\begin{proof}\n\\remred{BUFF.... TODO}\n We prove the statement by induction on $\\left| T \\right|$.\n The case $\\left| T \\right| = \\emptyset$ is trivial.\n For the induction step, let $t\\in T$. We prove this step by a subsidiary induction on\n $\\left| \\SET{p\\in {\\mathcal{P}}~\\middle|~ p_{-1}=t} \\right|$. The base case of the subsidiary induction is trivial, too.\n Now let $p'\\in {\\mathcal{P}}$ such that $p'_{-1}=t$. Let $X_0 = \\SET{p_1 ~\\middle|~ p\\in {\\mathcal{P}}\\BSET{p'}}$. If $p'_{1} \\notin X_0$,\n then we have $(p'_1,p'_2)\\notin \\bigcup_{p\\in {\\mathcal{P}}\\BSET{p'}} {\\left| p \\right|_A}$ and by induction hypothesis we obtain that in this case\n \\[ \\left| \\bigcup_{p\\in {\\mathcal{P}}} {\\left| p \\right|_A} \\right| \\geq \\left| \\bigcup_{p\\in {\\mathcal{P}}\\BSET{p'}} {\\left| p \\right|_A} \\right| + 1 \\geq \\left| {\\mathcal{P}}\\BSET{p'} \\right| + 1 \\geq \\left| {\\mathcal{P}} \\right|.\\]\n If $\\left| \\bigcup_{p\\in {\\mathcal{P}}\\BSET{p'}} {\\left| p \\right|_A} \\right| > \\left| {\\mathcal{P}}\\BSET{p'} \\right|$, then we have\n \\[ \\left| \\bigcup_{p\\in {\\mathcal{P}}} {\\left| p \\right|_A} \\right| \\geq \\left| \\bigcup_{p\\in {\\mathcal{P}}\\BSET{p'}} {\\left| p \\right|_A} \\right| \\geq \\left| {\\mathcal{P}}\\BSET{p'} \\right| + 1 \\geq \\left| {\\mathcal{P}} \\right|,\\]\n too.\n\n Let $D' = (V,A')$ where $A' = \\bigcup_{p\\in {\\mathcal{P}}} {\\left| p \\right|_A}$.\n and let\n \\[ {\\mathcal{L}} = \\SET{p\\in {\\mathbf{P}}(D') ~\\middle|~ p_{-1}\\in T,\\,\\forall v\\in V\\colon\\, vp \\notin {\\mathbf{P}}(D')} \\]\n consist of all maximal paths in $D'$ that end in $T$.\n \n by induction on $\\left| {\\mathcal{L}} \\right|$.\n\n If ${\\mathcal{P}}= \\emptyset$, the statement is trivial. Now assume that ${\\mathcal{P}}\\not= \\emptyset$.\n Let $E = \\bigcup_{p\\in {\\mathcal{P}}} {\\left| p \\right|_A}$ consists of all arcs that are traversed by some $p\\in {\\mathcal{P}}$.\n Let ${\\mathcal{A}} = (A_p)_{p\\in {\\mathcal{P}}} \\subseteq E$ be the finite family of finite\n subsets of $X$ where $A_p = \\left| p \\right|_A$ consists of the arcs traversed by $p$.\n The statement of the lemma is implied if ${\\mathcal{A}}$ has a transversal (Corollary~\\ref{cor:Hall}). We proof that such a transversal always exists by\n induction on $\\max\\SET{\\left| p \\right|~\\middle|~ p\\in {\\mathcal{P}}}$, i.e. the maximal length of a path in ${\\mathcal{P}}$.\n In the base case we have that all paths of ${\\mathcal{P}}$ have length $2$, i.e. every path $p\\in {\\mathcal{P}}$ traverses the single arc $(p_1,p_{-1})$,\n i.e. $\\left| p \\right|_A = \\SET{(p_1,p_{-1})}$.\n By assumption we have that for all $p,q\\in {\\mathcal{P}}$ with $p\\not= q$ either $p_1 \\not= q_1$ or $p_{-1}\\not= q_{-1}$ holds, thus\n $\\left| p \\right|_A \\cap \\left| q \\right|_A = \\emptyset$ for all $p\\not= q$, thus\n\\end{proof}\n\n\\studyremark{\n\\begin{lemmaX}\n\tLet $D=(V,A)$ be an acyclic digraph. Then $D$ is a cascade digraph if and only if for all $u,v\\in V$ and all $p,q\\in {\\mathbf{P}}(D;u,v)$\n\twe have $\\left| p \\right| = \\left| q \\right|$.\n\t\\remred{Ist glaub ich falsch!}\n\\end{lemmaX}\n\\begin{proof}\n\tAssume that $D$ is a cascade digraph, and let $V_1\\mathbin{\\dot{\\cup}} V_2\\mathbin{\\dot{\\cup}}\\cdots\\mathbin{\\dot{\\cup}} V_k = V$ be a partition such that\n\t$A \\subseteq \\bigcup_{i=1}^{k-1} V_i \\times V_{i+1}$.\n\tLet $u\\in V_i$ and $v\\in V_j$. If $i > j$, then ${\\mathbf{P}}(D; u,v) = \\emptyset$. If $i=j$, then ${\\mathbf{P}}(D;u,v) = \\SET{u}$ if $u=v$ and ${\\mathbf{P}}(D;u,v) = \\emptyset$ if $u\\not= v$.\n\tIf $i < j$, then for every $p\\in {\\mathbf{P}}(D;u,v)$ with $p=(p_l)_{l=1}^{m}$, we may infer\n\t $p_{2}\\in V_{i+1}$, $p_3\\in V_{i+2}$, $\\ldots$, $p_m \\in V_{i+m} = V_j$ from the structure of $A$ with respect to the partition $V_1,\\ldots,V_k$ of $V$. Therefore for all $p\\in {\\mathbf{P}}(D;u,v)$, $\\left| p \\right| = j-i+1$, and thus we have shown that for all $u,v\\in V$ and all $p,q\\in {\\mathbf{P}}(D;u,v)$\n\twe have $\\left| p \\right| = \\left| q \\right|$.\n\n\t\\noindent\n\tIn order to show the converse implication, we construct a suitable partition of $V$.\n\tObserve that if the digraph $D=(V,A)$ has the property that for all $u,v\\in V$ and all $p,q\\in {\\mathbf{P}}(D;u,v)$\n\twe have $\\left| p \\right| = \\left| q \\right|$\n\tthen so does every digraph $D'=(V',A')$ where $V'\\subseteq V$ and $A'\\subseteq A \\cap V'\\times V'$;\n\tso we may prove the implication by induction on $\\left| A \\right|$.\n\tThe base case where $A = \\emptyset$ is clear. For the induction step, let $a=(u,v)\\in A$, such that $u$ is a source in $D$ \n\t--- such a choice is always possible since $\\left| V \\right| < \\infty$ by Definition~\\ref{def:directedGraph} and $D$ is acyclic.\n\tWe give an indirect argument that $D$ is a cascade digraph, too.\n\tAssume that $D$ is not a cascade digraph, then the vertex set $V$ cannot be partitioned into\n\tsubsets $V_1\\mathbin{\\dot{\\cup}} V_2 \\mathbin{\\dot{\\cup}} \\cdots \\mathbin{\\dot{\\cup}} V_k = V$ such that $A\\subseteq \\bigcup_{i=1}^{k-1} V_i\\times V_{i+1}$.\n\t Let $D' = (V, A')$ where\n\t \\[ A' = \\SET{(x,y), (y,x) ~\\middle|~ (x,y)\\in A\\BSET{a}}.\\]\n\t There is a path $p\\in {\\mathbf{P}}(D';u,v)$: Assume the contrary, if we let\n\t \t $$X = \\SET{x\\in V~\\middle|~ {\\mathbf{P}}(D'; u,x) \\not= \\emptyset}$$\n\t and $$Y' = \\SET{y'\\in V~\\middle|~ {\\mathbf{P}}(D'; v,y') \\not= \\emptyset},$$ we then obtain that $X\\cap Y' = \\emptyset$, \n\t i.e. that $u$ and $v$\n\t belong to different weak components of $(V,A\\BSET{a})$.\n\t Let $Y = V\\backslash X \\supseteq Y'$. \n\t The digraphs $D_X = (X,A\\cap X\\times X)$ and $D_Y = (Y,A\\cap Y\\times Y)$ are cascade digraphs by induction hypothesis,\n\t so there are partitions $V^{(X)}_1,\\ldots,V^{(X)}_{k_X}$ and $V^{(Y)}_1,\\ldots,V^{(Y)}_{k_Y}$ of $V$ such that\n\t $$A\\BSET{a} \\subseteq \\left( \\bigcup_{i=1}^{k_X-1} V^{(X)}_i \\times V^{(X)}_{i+1} \\right)\n\t \\cup \\left( \\bigcup_{i=1}^{k_Y-1} V^{(Y)}_i \\times V^{(Y)}_{i+1} \\right)$$\n\t holds, because the only arc of $D$ that is incident with elements from both $X$ and $Y$ is $a$.\n\t Let $i_X \\in\\SET{1,2,\\ldots,k_X}$ such that $u\\in V^{(X)}_{i_X}$, \n\t and let $i_Y \\in\\SET{1,2,\\ldots,k_Y}$ such that $v\\in V^{(Y)}_{i_Y}$. \n\t To ease notation, we let $V^{(X)}_n = \\emptyset$ for $n\\in \\mathbb{Z} \\BSET{1,2,\\ldots,k_X}$ and\n\t $V^{(Y)}_n = \\emptyset$ for $n\\in \\mathbb{Z} \\BSET{1,2,\\ldots,k_Y}$.\n\t Let\n\t \\[ V'_i = V^{(X)}_{i-i_X} \\mathbin{\\dot{\\cup}} V^{(Y)}_{i-i_Y+1} \\]\n\t for $i\\in\\mathbb{Z}$. Then $u\\in V'_0$ and $v\\in V'_1$. Then\n\t $ A \\subseteq \\bigcup_{i= i_0}^{k_0} V'_i $ where $i_0 = \\min\\SET{-i_X,-i_Y+1}$ and $k_0 = \\max\\SET{k_X-i_X,k_Y-i_Y+1}$.\n\t Relabeling the indexes according to the scheme in Definition~\\ref{def:cascadeDigraph} yields\n\t that $D$ is a cascade digraph, contradicting our assumption above.\n\t Therefore, there is a path $p\\in {\\mathbf{P}}(D';u,v)$, that means that $u$ and $v$ belong to the same weak component of $D$.\n\\end{proof}}\n\n\n\\studyremark{Ich bin mir ziemlich sicher, dass man nicht f\u00fcr jede Basis von $M$ auch eine minimum arc-cardinality standard representation findet; da ja $x$ nach dem Pivotschritt auf einem Kreis liegen kann; und sich somit die Anzahl der Sources im Digraphen reduziert.\nWenn man jetzt $x$ zu einer Source machen will, muss man eine zus\u00e4tzliche Kante einf\u00fcgen. Vllt. ein Beispiel hierf\u00fcr? Habe ich jetzt...\nSch\u00e4tzungsweise macht das pivotieren eines anderen Basiselements aber die Eigenschaft der duality respecting representation nicht kaputt; -- doch macht sie!.}\n\n\n\n\n\n\\needspace{5\\baselineskip}\n\n\\begin{lemma}\\label{lem:rCOGaxC3}\n\tLet $\\sigma = (D,T,E,S_A,\\ll)$ be a realizable combinatorial orientation of a gammoid,\n\t $M=\\Gamma(D,T,E)$,\n\t $C\\in {\\mathcal{C}}(M)$\n\twith $C = \\dSET{c_1,c_2,\\ldots,c_r}$ where the implicit order obtained from the indexes of $c_i$ correspond to the implicit\n\torder of $E$ with respect to $\\sigma$. Then for all $i,j\\in \\SET{1,2,\\ldots,r}$\n\twe have\n\t\t\\[ C_{\\sigma {c_i}} = (-1)^{i-j} C_{\\sigma {c_{j}}}. \\]\n\\end{lemma}\n\n\\begin{proof}\n\tBy inspection of Definition~\\ref{def:Csigmac},\n\twe obtain that for all $k\\in \\SET{1,2,\\ldots,r}\\BSET{i,j}$ we have\n\t\\begin{align*} C_{\\sigma c_i}(c_k)& = (-1)^{i-k+1}\\cdot \\mathrm{sgn}_{\\sigma}(R_{c_k})\\\\\n\t\t\t\t\t\t & = (-1)^{i-j+j-k+1}\\cdot\\mathrm{sgn}_{\\sigma}(R_{c_k}) \\\\ &\n\t\t\t\t\t\t = (-1)^{i-j}\\cdot (-1)^{j-k+1}\\cdot\\mathrm{sgn}_{\\sigma}(R_{c_k}) = (-1)^{i-j} \\cdot C_{\\sigma {c_{j}}(k)}.\n\t\\end{align*}\n\tFurthermore,\n\t\\[\n\t\tC_{\\sigma c_i}(c_j) = (-1)^{i-j+1} \\mathrm{sgn}_{\\sigma}(R_{c_j}) =(-1)^{i-j} \\cdot (-\\mathrm{sgn}_{\\sigma}(R_{c_j}))\n\t\t\t\t\t\t= (-1)^{i-j} C_{\\sigma c_j}(c_j) \n\t\\] and by symmetry, $C_{\\sigma c_j}(c_i) = (-1)^{i-j}C_{\\sigma c_i}(c_i)$, thus $C_{\\sigma {c_i}} = (-1)^{i-j} C_{\\sigma {c_{j}}}$.\n\\end{proof}\n\n\n\n\n\\begin{lemma}\n\tLet $\\sigma = (D,T,E,S_A,\\ll)$ be a realizable combinatorial orientation of a gammoid, $M=\\Gamma(D,T,E)$, $C\\in {\\mathcal{C}}(M)$,\n\tand $c_1,c_2\\in C$. Let $R_1\\subseteq {\\mathbf{W}}(D)$ be the unique $\\llless$-maximal routing from $C\\BSET{c_1}$ to $T$ in $D$, and\n\tlet $R_2\\subseteq {\\mathbf{P}}(D)$ be the unique $\\llless$-maximal routing from $C\\BSET{c_2}$ to $T$ in $D$.\n\tThen\n\t\\[ \\SET{p_{-1}\\mid p\\in R_1} = \\SET{p_{-1}\\mid p\\in R_2}.\\]\n\\end{lemma}\n\n\\begin{proof}\n\tWe give an indirect proof and assume that $\\SET{p_{-1}\\mid p\\in R_1} \\not= \\SET{p_{-1}\\mid p\\in R_2}$. \n\n\tBy Lemma~\\ref{lem:rkEqMaxConnector} and Corollary~\\ref{cor:MengerA} there is an $C$-$T$-separator $S\\subseteq V$ in $D$\n\twith $\\left| S \\right| = \\mathrm{rk}(C) = \\left| C \\right| - 1$. By Corollary~\\ref{cor:Menger} \n\tfor all $s\\in S$ there is a path $p_{s,1}\\in R_1$ with $s\\in \\left| p_{s,1} \\right|$\n\tand a path $p_{s,2}\\in R_2$ with $s\\in \\left| p_{s,2} \\right|$. Clearly, we can write $p_{s,i} = x_{s,i} s y_{s,i}$\n\tfor all $s\\in S$ and $i\\in\\SET{1,2}$ where $x_{s,i},y_{s,i}\\in {\\mathbf{P}}(D)$.\n\tThen $Y_1 = \\SET{s y_{s,1}\\mid s\\in S}$ and $Y_2 = \\SET{s y_{s,2} \\mid s\\in S}$ are two different routings from $S$ to $T$ in $D$.\n\tWithout loss of generality we may assume that $Y_1 \\llless Y_2$. Clearly, $R_1' = \\SET{x_{s,1}s y_{s,2}\\mid s\\in S}$ is \n\ta routing from $C\\BSET{c_1}$ to $T$, too.\\footnote{The fact that this is indeed a routing follows from the fact that the vertices visited by the $y_{s,i}$-paths are different from the vertices visited by the $x_{s,i}$-paths, since otherwise $S$ cannot be a separator; the same situation occurs in the proof of Menger's Theorem~\\ref{thm:MengerGoering}.}\n\tSince the paths in $R_1$ and in $R_1'$ traverse the same set of arcs until they reach the separator $S$, we see that $Y_1 \\llless Y_2$\n\timplies $R_Y \\llless R_1'$, a contradiction to the fact that $R_1$ is the $\\llless$-maximal routing from $C\\BSET{c_1}$ to $T$ in $D$.\n\\end{proof}\n\n\n\n \\begin{definition}\n \tLet $D=(V,A)$ be a digraph, $\\alpha,\\rho \\in\\mathbb{N}$, $w\\colon A\\longrightarrow \\mathbb{R}$ an indeterminate weighting of $D$,\n \t and $\\ll\\,\\,\\subseteq A\\times A$ is a linear order on $A$.\n \tThen $w$ shall be called a \\deftext[ar-compatible indeterminates@$(\\ll,\\alpha,\\rho)$-compatible indeterminates]{$\\bm (\\bm \\ll\\bm,\\bm\\alpha\\bm,\\bm\\rho\\bm)$-compatible indeterminate weighting of $\\bm D$},\n \tif for all $a\\in A$\\remred{TODO: geschicktes absch\u00e4tzen hier; Def. anpassen}\n \t\\[ 2^{2^\\rho \\alpha}\\cdot \\left| \\prod_{a'\\in A,\\,a'\\ll a,\\,a'\\not= a} \\left( w(a') \\right)^\\rho \\right| < w(a) .\\]\n \\end{definition}\n\n\\needspace{4\\baselineskip}\n \\begin{remark}\n \tSince for every $r\\in \\mathbb{R}$ there is some $z\\in \\mathbb{Z}$ such that $r\\cdot z > 1$, and since $X\\subseteq \\mathbb{R}$ is $\\mathbb{Z}$-independent if\n \tand only if for all $x\\in X$ and $z\\in \\mathbb{Z}\\BSET{0}$ the set $(X\\BSET{x})\\cup\\SET{z\\cdot x}$ is $\\mathbb{Z}$-independent\n \tit is clear that we can derive a $(\\ll,\\alpha,\\rho)$-compatible indeterminate weighting $w'$ from any indeterminate weighting\n \t$w$ by setting $$w'(a) = n_a \\cdot w(a)$$ where for all $a\\in A$\n \t$$ n_a = 1 + 2^{2^\\rho \\alpha} \\cdot \\left\\lceil \\left| \\frac{1}{w(a)} \\right| \\cdot \\left| \\prod_{{a'\\in A,\\,a'\\ll a,\\,a'\\not= a}} \\left( w(a')\\cdot \\prod_{a''\\in A,\\,a'' \\ll a'} n_{a''} \\right)^\\rho \\right| \\right\\rceil .$$\n \\end{remark}\n\n \\begin{definition}\n \tLet $A$ be any set, $\\alpha,\\rho\\in \\mathbb{N}$, and $p\\in \\mathbb{Z}[A]$ a polynomial over $\\mathbb{Z}$ with variables in $A$.\n \tThen $p$ shall be called \\deftext[ar-polynomial@$(\\alpha,\\rho)$-polynomial]{$\\bm(\\bm\\alpha\\bm,\\bm\\rho\\bm)$-polynomial},\n \tif \n \t\\( \\sum_{f\\in \\mathbb{N}^{(A)}} \\left| p(f) \\right| \\leq 2^\\alpha\\)\n \tand \\[ \\max\\SET{\\sum_{a\\in A} f(a) ~~\\middle|~~ f\\in \\mathbb{N}^{(A)},\\, p(f) \\not= 0} \\leq \\rho \\]\n \tholds.\n \\end{definition}\n\n \\begin{definition}\n \tLet $A$ be any non-empty set, and $\\ll\\,\\,\\subseteq A\\times A$ is a linear order on $A$. The\n \t\\deftext[monomial order induced by@monomial order induces by $\\ll$]{monomial order induced by $\\bm \\ll$} shall be the order\n \t$\\ll'$ on $\\mathbb{N}^{(A)}$, where for all $f,g\\in \\mathbb{N}^{(A)}$, we have\n \t$f \\ll' g$ if and only if either $f=g$ or there exists $a\\in A$ such that $f(a) < g(a)$ and for all $a'\\in A$ with $f(a') > g(a')$, $a' \\ll a$.\n \\end{definition}\n\n\\begin{remark}\n\tClearly, $\\ll'$ is a linear order on $\\mathbb{N}^{(A)}$, because whenever $f\\not= g$ there is always a $\\ll$-maximal element $a\\in A$ with $f(a) \\not= g(a)$. Then if $f(a) < g(a)$, $f \\ll' g$, otherwise $g \\ll' f$.\n\\end{remark}\n\n\\begin{definition}\n\tLet $A$ be any non-empty set, and $\\ll\\,\\,\\subseteq A\\times A$ is a linear order on $A$, and $p\\in \\mathbb{Z}[A]$ a polynomial over $\\mathbb{Z}$ with variables in $A$ where\n\t$p\\not= 0$.\n\tThe \\deftext[leading monomial of p@$\\ll$-leading monomial of $p$]{$\\bm \\ll$-leading monomial of $\\bm p$} is\n\tdefined to be the $\\ll'$-maximal monomial $f\\in \\mathbb{N}^{(A)}$ with $p(f) \\not = 0$.\n\tBy convention, we say the zero map $z\\colon A\\mapsto \\mathbb{N}, \\,a\\mapsto 0$ is the $\\ll$-leading monomial of the zero polynomial.\n\\end{definition}\n\n \\begin{lemma}\n \tLet $D=(V,A)$ be a digraph, $\\alpha,\\rho\\in \\mathbb{N}$, $\\ll\\,\\,\\subseteq A\\times A$ is a linear order on $A$,\n \t $w\\colon A\\longrightarrow \\mathbb{R}$ an $(\\ll,\\alpha,\\rho)$-compatible indeterminate weighting of $D$, and $p\\in \\mathbb{Z}[A]$\n \t an $(\\alpha,\\rho)$-polynomial over $\\mathbb{Z}$ with variables in $A$ with $\\ll$-leading monomial $f\\in \\mathbb{N}^{(A)}$.\n \t Then\n \t \\[ \\mathrm{sgn} \\left( \\vphantom{x^A_A}{\\mathrm{eval}}_\\mathbb{R}(p)(w) \\right) = \\mathrm{sgn} \\left( p(f) \\cdot \\prod_{a\\in A} w(a)^{f(a)} \\right).\\]\n \\end{lemma}\n\n \\begin{proof}\n \tBy definition \n \t\\[ \t{\\mathrm{eval}}_{{\\mathbb{R}}} (p)(w) = \\sum_{f\\in \\mathbb{N}^{(A)}} p(f)\\cdot \\prod_{a\\in A} w(a)^{f(a)}.\\]\n \tLet $a_f \\in A$ be the $\\ll$-maximal element of $A$ such that $f(a) > 0$, or the minimal element of $A$ if $f$ is the zero map.\n\n \t\\remred{TODO}\n \\end{proof}\n\n\n\n\n\n\\begin{lemma}\n\tLet $(D,T,E,S_A,\\ll)$ be a realizable combinatorial orientation of a gammoid and $I\\subseteq E\\backslash T$,\n\tthen the reorientation \n\t\\[ \\left( {\\mathcal{O}}(D,T,E,S_A,\\ll) \\right)_{-I} \\quad = \\quad {\\mathcal{O}}(D,T,E,S'_A,\\ll) \\]\n\tcorresponds to the combinatorial orientation\n\twhere\n\t\\[ S'_A \\colon A\\longrightarrow \\SET{-1,1},\\quad (u,v)\\mapsto \\begin{cases} -S_A(u,v) & \\quad \\text{if~} \\left| \\SET{u,v}\\cap I \\right| = 1,\\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t \\hphantom{-}S_A(u,v)&\\quad \\text{otherwise.} \\end{cases}\\]\n\\end{lemma}\n\n\\begin{proof}\n\\remred{TODO}\n\\end{proof}\n\n\n\n\\noindent In this section, we investigate in which cases we can take a\n strict gammoid $N$ that is an extension of the gammoid $M$, and use it to find an extension $N'$ of a matroid $M'$ with\n similar $\\alpha$-violations. The concept of similar $\\alpha$-violations is the key for devising a backtrack algorithm for $\\mathrm{Rec}\\Gamma_{\\mathcal{M}}$\n which tries to add elements to a matroid $M$ in order to generate a strict gammoid extension $N$: if the a new matroid encountered in the search process\n is similar to a matroid encountered before, then we may safely ignore the new matroid and continue.\n In this section, we are dealing with multi-elementary extensions of matroids, therefore\n we need the following easy generalization of the concept of a modular cut of $M$.\n\n\\needspace{6\\baselineskip}\n\n\\begin{definition}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, $k\\in\\mathbb{N}$, $E' = \\dSET{e_1,e_2,\\ldots,e_k}$ with $E\\cap E'=\\emptyset$.\n\tA $k$-tuple $K=(C_1,C_2,\\ldots,C_k)$ is called \n\t\\deftext[k-fold modular cut of M@$k$-fold modular cut of $M$]{$\\bm k$-fold modular cut of $\\bm M$ by $\\bm E{\\bm'}$}, if\n\tthere is a sequence of matroids\n\t$M_0,M_1,\\ldots, M_k$ such that $M_0 = M$ and such that for all $i\\in\\SET{1,2,\\ldots,k}$,\n\t$$M_i \\in {\\mathcal{X}}(M_{i-1},e_{i})$$ and $$C_i = \\SET{F\\in{\\mathcal{F}}(M_{i-1})~\\middle|~ e_i\\in\\mathrm{cl}_{M_i}(F)}$$ holds.\n\tThe class of all $k$-fold modular cuts of $M$ by $E'$ shall be denoted by ${\\mathcal{M}}(M,E')$.\\label{n:McalMEp}\n\tIf $K = (C_1,C_2,\\ldots,C_k) \\in {\\mathcal{M}}(M,E')$, then the matroid $M_k$ \n\twith the defining property as above shall be called the\n\t\\deftext[k-fold extension of M@$k$-fold extension of $M$]{$\\bm K$-extension of $\\bm M$ by $\\bm E{\\bm'}$}\n\t --- or shorter a \\deftextX{$\\bm k$-fold extension of $\\bm M$}. Furthermore,\n\t for all finite $E'$ with $E\\cap E'=\\emptyset$ we denote the class of all $\\left| E' \\right|$-fold extensions of $M$ by $E'$ with\n\t \\( {\\mathcal{X}}(M,E') \\).\\label{n:XcalMEp}\n\\end{definition}\n\n\\begin{definition}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid and let $X\\in {\\mathcal{V}}(M)$ be a violation.\n\tThen we denote the \\deftext[non-vanishing flats]{family of all non-vanishing flats of $\\bm M$ below $\\bm X$} \\label{n:nonvanishing}\n\tby \\[ {\\mathcal{F}}_{\\pm}(M,X) = \\SET{F\\in {\\mathcal{F}}(M)~\\middle|~ F\\subsetneq X,\\,\\alpha_M(F) \\not= 0}.\\]\n\t\\remred{Vielleicht auch hier X eventuell zulassen.}\n\\end{definition}\n\n\\begin{definition}\n\tLet $M=(E,{\\mathcal{I}})$ and $N=(E',{\\mathcal{I}}')$ be matroids, and let $X\\in {\\mathcal{V}}(M)$ and $Y\\in{\\mathcal{V}}(N)$ be violations.\n\tWe say that \\deftext[calculates like@$X$ calculates like $Y$]{$\\bm X$ calculates like $\\bm Y$} \n\tif $\\alpha_M(X) = \\alpha_N(Y)$ and if there is a bijection $\\rho \\colon {\\mathcal{F}}_\\pm(M,X) \\longrightarrow {\\mathcal{F}}_\\pm(N,Y)$\n\tthat has the properties\n\t\\[ F \\subseteq G \\Leftrightarrow \\rho(F) \\subseteq \\rho(G)\n\t\\quad{\\mathop{\\text{~and~}}}\\quad \n\t \\alpha_M(F) = \\alpha_N(\\rho(F)) \\]\n\tfor all $F,G\\in {\\mathcal{F}}_\\pm(M,X)$.\n\n\n\n\n\n\n\tIf $X$ calculates like $Y$ and $\\rho$ is a bijection with the properties above, then $\\rho$ shall be called \\deftext[connector!$\\alpha$-violations]{$\\bm X$-$\\bm Y$-connector}.\n\tWe say that \\deftext[violations!similar]{$\\bm X$ is similar to $\\bm Y$} if $X$ calculates like $Y$ such that there is\n\tan $X$-$Y$-connector with the further property\n\t\\[ \\SET{\\vphantom{C'}\\rho[C \\cap {\\mathcal{F}}_\\pm(M,X)] ~\\middle|~ C\\in {\\mathcal{M}}(M)} = \\SET{C'\\cap {\\mathcal{F}}_\\pm(N,Y)~\\middle|~ C'\\in {\\mathcal{M}}(N)} .\\]\n\t\\remred{Das reicht wahrscheinlich noch nicht, es k\u00f6nnte Fernwirkungen geben, die ausserhalb der Violation daf\u00fcr sorgen, dass man sie nicht in beiden F\u00e4llen\n\tfixen kann.} \n\\end{definition}\n\n\\begin{definition}\n\tLet $M=(E,{\\mathcal{I}})$ and $N=(E',{\\mathcal{I}}')$ be matroids. We say that \n\t\\deftext[similar a-violations!similar $\\alpha$-violations]{$\\bm M$ and $\\bm N$ have similar $\\bm \\alpha$-violations},\n\tif there is a bijection $\\rho \\colon {\\mathcal{F}}_\\pm(M,E) \\longrightarrow {\\mathcal{F}}_\\pm(N,E')$ and a bijection $\\phi\\colon {\\mathcal{V}}(M) \\longrightarrow {\\mathcal{V}}(N)$\n\tsuch that $X$ is similar to $\\phi(X)$ for every $X\\in {\\mathcal{V}}(M)$, and if the restriction $\\rho|_{{\\mathcal{F}}_\\pm(M,X)}$\n\t is an $X$-$\\phi(X)$-connector for all $X\\in {\\mathcal{V}}(M)$.\n\\end{definition}\n\n\n\n\\remred{HEAD}\n\n\\remred{TODO}\n\n\\studyremark{\n\n\n\\needspace{6\\baselineskip}\n\n\\noindent First, we shall investigate the kind of $\\alpha_M$-violations that can occur in a gammoid.\nCertainly, not every flat of a given matroid is equally important with respect to its $\\alpha$-function:\n\n\\begin{definition}\n\tLet $M=(E,{\\mathcal{I}})$ be a matroid, and $F\\in{\\mathcal{F}}(M)$. We say that $F$ is a \\deftext[neutral flat in $M$]{neutral flat in $\\bm M$}, if $\\alpha_M(F) = 0$.\n\tFurthermore, we say that $F$ is a \\deftext[hot flat in $M$]{hot flat in $\\bm M$}, if $\\alpha_M(F) > 0$.\n\tFor $X\\subseteq E$, we denote the \\label{n:hotflats}\\deftext[hot flats below $X$]{family of all hot flats below $\\bm X$ in $\\bm M$} by\n\t\\[ {\\mathcal{F}}_+(M,X) = \\SET{F\\in{\\mathcal{F}}(M) ~\\middle|~ F\\subsetneq X,\\,\\alpha_M(F) > 0}. \\]\n\\end{definition}\n\n\\begin{example}\n\tFor $M=(E,{\\mathcal{I}})$, every flat $F\\in {\\mathcal{F}}(M)$ with $F\\in{\\mathcal{I}}$ is neutral.\n\\end{example}\n\\begin{remark}\\label{rem:hotviolation}\n\tFor $M=(E,{\\mathcal{I}})$ and $X\\in{\\mathcal{V}}(M)$, we may cancel out the zero-summands where $\\alpha(F')=0$ for\n\t$F'\\in{\\mathcal{F}}(M)$ with $F'\\subsetneq X$ and obtain\n\t\\[ \\alpha(X) = \\left| X \\right| - \\mathrm{rk}(X) - \\sum_{F\\in{\\mathcal{F}}_+(M,X)} \\alpha(F) .\\]\n\tThe equality also holds whenever $\\alpha|_{2^X} \\geq 0$.\n\\end{remark}\n\n\n\n\\noindent Now, we scrutinize the kind of violation we encountered in $M(K_4)$ (Example~\\ref{ex:violationNonGammoid}).\n\\remred{TODO}\n}\n\n\n\nINGLETON\n\n\n\\noindent Let us dualize the inequality from Theorem~\\ref{thm:IngeltonsCondition} using the rank formula from Lemma~\\ref{lem:rankDual}.\n\n\\begin{align*}\n\t\\mathrm{rk}^\\ast(W) \\,+ \\,&\\mathrm{rk}^\\ast(X) + \\mathrm{rk}^\\ast(W\\cup X\\cup Y) + \\mathrm{rk}^\\ast(W\\cup X\\cup Z) + \\mathrm{rk}^\\ast(Y\\cup Z) \\\\\n\t= & \\hphantom{+\\,} \\left| W \\right| + \\left| X \\right| + \\left| W\\cup X\\cup Y \\right| + \\left| W\\cup X\\cup Z \\right| + \\left| Y\\cup Z \\right|\\\\&\n\t+ \\mathrm{rk}(E\\backslash W) + \\mathrm{rk}(E\\backslash X) + \\mathrm{rk}\\left( \\vphantom{A^A}E\\backslash\\left( W\\cup X\\cup Y \\right) \\right) \\\\ & + \n\t\\mathrm{rk}\\left( \\vphantom{A^A}E\\backslash\\left( W\\cup Y\\cup Z \\right) \\right) + \\mathrm{rk}\\left( \\vphantom{A^A}E\\backslash\\left( Y\\cup Z \\right) \\right) - 5\\cdot\\mathrm{rk}(E),\\\\\n\t\\mathrm{rk}^\\ast(W\\cup X) \\,+\\, &\\mathrm{rk}^\\ast(W\\cup Y) + \\mathrm{rk}^\\ast(W\\cup Z) + \\mathrm{rk}^\\ast(X\\cup Y) + \\mathrm{rk}^\\ast(X\\cup Z) \\\\\n\t= & \\hphantom{+\\,} \\left| W\\cup X \\right| + \\left| W\\cup Y \\right| + \\left| W\\cup Z \\right| + \\left| X\\cup Y \\right| + \\left| X\\cup Z \\right|\\\\&\n\t+ \\mathrm{rk}\\left(\\vphantom{A^A} E\\backslash \\left( W\\cup X \\right) \\right)+ \\mathrm{rk}\\left(\\vphantom{A^A} E\\backslash \\left( W\\cup Y \\right) \\right)+ \n\t\\mathrm{rk}\\left(\\vphantom{A^A} E\\backslash \\left( W\\cup Z \\right) \\right) \\\\ &+ \\mathrm{rk}\\left(\\vphantom{A^A} E\\backslash \\left( X\\cup Y \\right) \\right)\n\t+ \\mathrm{rk}\\left(\\vphantom{A^A} E\\backslash \\left( X\\cup Z \\right) \\right) - 5\\cdot\\mathrm{rk}(E).\n\\end{align*}\nWith the Principle of Inclusion and Exclusion~\\ref{lem:inclusionExclusion}, we obtain the dual inequality\n\\begin{align*}\n\t\\mathrm{rk}(E\\backslash W) \\,+\\, & \\mathrm{rk}(E\\backslash X) + \\mathrm{rk}\\left( \\vphantom{A^A}E\\backslash\\left( W\\cup X\\cup Y \\right) \\right) + \n\t\\mathrm{rk}\\left( \\vphantom{A^A}E\\backslash\\left( W\\cup Y\\cup Z \\right) \\right)\\\\\n\t+\\,& \\mathrm{rk}\\left( \\vphantom{A^A}E\\backslash\\left( Y\\cup Z \\right) \\right)\\\\\n\t \\leq \\,& \\hphantom{+\\,} \\mathrm{rk}\\left( E\\backslash \\left( W\\cup X \\right) \\right)+ \\mathrm{rk}\\left( E\\backslash \\left( W\\cup Y \\right) \\right)+ \n\t\\mathrm{rk}\\left( E\\backslash \\left( W\\cup Z \\right) \\right) \\\\ &+ \\mathrm{rk}\\left( E\\backslash \\left( X\\cup Y \\right) \\right)+ \\mathrm{rk}\\left( E\\backslash \\left( X\\cup Z \\right) \\right) \\\\& + \\left| W\\cap X \\right| + \\left| Y\\cap Z \\right| - \\left| W\\cap X\\cap Y \\right| - \\left| W\\cap X\\cap Z \\right|.\n\\end{align*}\nNow, let $W' = E\\backslash W$, $X' = E\\backslash X$, $Y' = E\\backslash Y$ and $Z' = E\\backslash Z$, then we may write the dual inequality as\n\\begin{align*}\n\t\\mathrm{rk}(W') \\,+\\, & \\mathrm{rk}(X') \n\t+ \\mathrm{rk}\\left( W'\\cap X'\\cap Y' \\right) + \n\t\\mathrm{rk}\\left( W'\\cap Y'\\cap Z' \\right) \n\t+ \\mathrm{rk}\\left( Y'\\cap Z' \\right)\\\\\n\t-\\,& \\left| W'\\right|\n\t - \\left| X' \\right| \n\t- \\left| W'\\cap X'\\cap Y' \\right| \n\t- \\left| W'\\cap X'\\cap Z' \\right| \n\t- \\left| Y'\\cap Z' \\right|\n\t\\\\\n\t \\leq \\,& \\hphantom{+\\,} \\mathrm{rk}\\left( W'\\cap X' \\right)\n\t + \\mathrm{rk}\\left( W'\\cap Y' \\right)+ \n\t\\mathrm{rk}\\left( W'\\cap Z' \\right) +\n\t \\mathrm{rk}\\left( X'\\cap Y' \\right)+ \n\t \\mathrm{rk}\\left( X'\\cap Z'\\right) \\\\& \n\t- \\left| W'\\cap X' \\right|\n\t - \\left| W'\\cap Y' \\right| \n\t - \\left| W'\\cap Z' \\right|\n\t - \\left| X'\\cap Y' \\right|\n\t- \\left| X'\\cap Z' \\right|. \n\t\\end{align*}\nNow let $\\nu(X) = \\left| X \\right| - \\mathrm{rk}(X) = -\\left( \\mathrm{rk}(X) - \\left| X \\right| \\right)$, we obtain\n\\begin{align*}\n\t\\nu(W') \\,+\\, & \\nu(X') + \\nu\\left( W'\\cap X'\\cap Y' \\right) + \n\t\\nu\\left( W'\\cap Y'\\cap Z' \\right) + \\nu\\left( Y'\\cap Z' \\right)\n\t\\\\\n\t \\geq \\,& \\hphantom{+\\,} \\nu\\left( W'\\cap X' \\right)+ \\nu\\left( W'\\cap Y' \\right)+ \n\t\\nu\\left( W'\\cap Z' \\right) + \\nu\\left( X'\\cap Y' \\right)+ \\nu\\left( X'\\cap Z'\\right).\n\\end{align*}\n\n\n\n\n\\begin{lemma}\n\tLet $M=(E,{\\mathcal{I}})$ be a strict gammoid, and let $e\\in E$ such that $C\\not= \\mathrm{cl}(C)$ for every $C\\in{\\mathcal{C}}(M)$ with $e\\in C$.\n\tThen $M| \\left( E\\BSET{e} \\right)$ is a strict gammoid.\n\\end{lemma}\n\\begin{proof}\n \t\\remred{TODO; via $\\alpha$ invariant; bringt auch nicht viel}\n\\end{proof}\n\n\\begin{lemma}\n\tLet $E$ be a finite set, $r\\in \\mathbb{N}$ with $1\\leq r \\leq \\left| E \\right| - 1$, \n\tand let $$U = (E, \\SET{X\\subseteq E ~\\middle|~ \\left| X \\right| \\leq r})$$\n\tbe the uniform matroid of rank $r$ on $E$.\n\tThen \\remred{TODO: schranke} \\[ \\left| E \\right| - 1 \\leq \\mathrm{C}_A(U) .\\]\n\\end{lemma}\n\\begin{proof}\n\tSince $U^\\ast$ is a uniform matroid of rank $\\left| E \\right| - r$ on $E$, \n\tand since every standard representation is duality respecting,\n\twe may assume without loss of generality that $r \\leq \\left| E \\right| - r$.\n\tLet $(D,T,E)$ with $D=(V,A)$ be a standard representation of $U$.\n\n\tIn the proof of Lemma~\\ref{lem:linkage} we saw that every linking $R$ in $D$ onto a fixed set $T$\n\tis uniquely determined by the set of arcs it traverses, i.e. if $R\\colon X\\double{\\rightarrow} T$ and $R'\\colon X'\\double{\\rightarrow} T$ are\n\tlinkings in $D$, then\n\t\\[ R = R' \\,\\, \\Longleftrightarrow \\,\\, \\bigcup_{p\\in R} \\left| p \\right|_A = \\bigcup_{p'\\in R'} \\left| p' \\right|_A \\]\n\tholds. Furthermore, if $\\bigcup_{p\\in R} \\left| p \\right|_A \\subsetneq \\bigcup_{p'\\in R'} \\left| p' \\right|_A$\n\tholds, then there is at least one path $p\\in R$ and a corresponding path $p'=(p'_i)_{i=1}^{n}\\in R'$ \n\tsuch that $p = p_j p_{j+1} \\ldots p_n$\n\tfor some $j\\in \\SET{1,2,\\ldots, n}$. Consequently, at least one element $x\\in \\SET{p_1 ~\\middle|~ p\\in R}$ is not a source in $D$.\n\tFor every $X\\subseteq E$ with $\\left| X \\right| = r$ there is at least one linking $R_X\\colon X\\double{\\rightarrow} T$ in $D$.\n\tLet ${\\mathcal{L}}$ contain exactly one linking from $X$ onto $T$ in $D$ for every $X\\subseteq E\\backslash T$ with $\\left| X \\right| = r$.\n\tSince every vertex $e\\in E\\backslash T$ is a source in $D$, we obtain that for all $L,L'\\in {\\mathcal{L}}$ \n\tthe inclusion $\\bigcup_{p\\in L} \\left| p \\right|_A \\subseteq \\bigcup_{p'\\in L'} \\left| p' \\right|_A$\n\talready implies $L = L'$.\n\t Thus\n\t\\[{\\mathcal{Q}} = \\SET{\\bigcup_{p\\in R} \\left| p \\right|_A ~\\middle|~ R\\in {\\mathcal{L}}}\\]\n\tis an anti-chain in the power set lattice of $A$ with\n\t $\\left| {\\mathcal{Q}} \\right| = \\binom{\\left| E \\right| - r}{r}$ elements,\n\t and where $\\left| Q \\right|\\geq r$ for all $Q\\in {\\mathcal{Q}}$, since no $p\\in Q$ is a trivial path.\n\t \\remred{Kommt man irgendwie auf ein routing f\u00fcr jede injektion von $T$ nach $E\\backslash T$? Dann w\u00e4re man bei \n\t $\\binom{\\left| E \\right|-r}{r}\\cdot r!$\n\n\t Vielleicht auch \u00fcber Spanning Trees und Connectivity? Oder man zeigt erst, dass man den Digraphen beliebig erweitern kann, ohne dass\n\t sich was \u00e4ndert...}\n\\end{proof}\n\n\n\n\n\\begin{lemma}\\remred{Vielleicht auch falsch}\n\tLet $(D,T,V)$ be a representation of a gammoid where $D=(V,A)$ and where all $a\\in A$ are essential arcs of $(D,T,V)$,\n\tand let $p=(p_i)_{i=1}^n \\in {\\mathbf{P}}(D)$ be a path in $D$.\n\tThere is a circuit $C\\in {\\mathcal{C}}(\\Gamma(D,T,V))$\n\twith $p_1\\in C$ and\n\t\\[ C \\subseteq \\left.\\left( \\SET{p_1} \\cup \\bigcup_{i=1}^{n-1} \\SET{v\\in V~\\middle|~ (p_i,v) \\in A} \\right) \\right\\backslash\\SET {\n\t\\vphantom{A^A} p_i~\\middle|~ i\\in \\SET{2,3,\\ldots,n-1}} .\\]\n\\end{lemma}\n\\begin{proof}\n\tBy induction on the length $n$ of $p$.\n\tFor $n=1$ this follows from Theorem~\\ref{thm:IPEssentialStars}.\n\t\\remred{TODO: Two circuits with common elements??!!}\n\\end{proof}\n\n\n\\begin{lemma}\n\tLet $r\\in \\mathbb{N}$, $U=(E,{\\mathcal{I}})$ be a uniform matroid with $r \\leq \\left| E \\right|$,\\linebreak i.e.\n\t${\\mathcal{I}} = \\SET{\\vphantom{A^A}X\\subseteq E~\\middle|~ \\left| X \\right| \\leq r}$.\n\tThen $$\\mathrm{C}_A(U) = r\\cdot\\left( \\left| E \\right| - r \\right).$$\n\\end{lemma}\n\\begin{proof}\n\\remred{WORK IN PROGRESS}\n\tWith Lemmas~ \\ref{lem:uniformArcs} and \\ref{lem:uniformStrictStdRep}\n\t it remains to show that we cannot reduce the number of arcs needed\n\tto represent $U$ by introducing auxiliary vertices.\n\t Assume that $(D,T,E)$ with $D=(V,A)$ is a standard representation of $U$ with fewer than\n\t$r\\cdot \\left( \\left| E \\right| - r \\right)$ arcs.\n\tLet $X = \\SET{e\\in E\\backslash T~\\middle|~ \\SET{e}\\times T \\not\\subseteq A}$,\n\tand let $\\tilde X = E\\backslash \\left( T\\cup X \\right)$. \n\tWe prove that $(D,T,E)$ does not \n\trepresent $U$ by induction on $\\left| X \\right|$ and the auxiliary statement that\n\teach of\n\t$\\left| X \\right| = k$ sources $s'\\in S'$ cannot be (strongly) connected to each of $r$ sinks $t'\\in T'$\n\tin any digraph with\n\tstrictly fewer than $k\\cdot r$ arcs unless there is an $S'$-$T'$-separator\n\twhich consists of fewer than $\\min\\SET{\\left| S' \\right|,\\left| T' \\right|}$ vertices.\n\n\tFirst, we analyze the case where $\\left| X \\right| = 1$.\n\tIn this case, $\\tilde X$ already uses $r\\cdot \\left( \\left| E \\right| - r - 1 \\right)$ arcs,\n\tthus there are $r - 1$ arcs that may be used to connect the unique element $x\\in X$ to $t\\in T$ in $D$.\n\tLet $V_x = \\SET{v\\in V~\\middle|~ {\\mathbf{P}}(D;x,v)\\not= \\emptyset}$ be the set of vertices that may be reached from $x$,\n\tand let $T_x = V_x\\cap T$ be the set of targets that may be reached from $x$.\n\tWe define the undirected graph $G_x = (V_x, E_x)$ where $\\SET{x,y}\\in E_x$ if either $(x,y)\\in A$ or $(y,x)\\in A$.\n\tClearly, $G_x$ consists of a single connected component, and $V_x\\cap \\left( E\\backslash T \\right) = \\SET{x}$, because\n\tall elements from $E\\backslash T$ are sources. \n\tConsequently $\\left| E_x \\right| \\leq r -1$ because the other arcs are incident with vertices \n\tfrom $E\\backslash \\left( T\\cup \\SET{x} \\right)$. \n\tBy connectedness of $G_x$ we obtain that $\\left| V_x \\right| \\leq \\left| E_x \\right| + 1 = r$. \n\tSince $x\\notin T$ and there is no path from $x$ to any $t'\\in T\\backslash T_x$,\n\tthe set $T_x\\cup \\SET{x}$ must contain a circuit. Now $\\left| T_x \\cup \\SET{x} \\right| \\leq \\left| V_x \\right| \\leq r$\n\timplies that $T_x\\cup\\SET{x}\\in {\\mathcal{I}}$ --- a contradiction. Thus $X$ and $T$ cannot be strongly connected in $D$.\n\n\t\\remred{TODO ROUTING ETC}\n\n\n\tNow let $X= \\dSET{x_1,x_2,\\ldots,x_k}$.\n\tIn total we have $k\\cdot r - 1$ arcs which have to provide the desired connectivity between $X$ and $T$.\n\tLet $G_i = (V_i,E_i) = G_{x_k}$ be the undirected graph constructed for $x_k$\n\tas described in the $\\left| X \\right| = 1$ case. Let $\\tilde G_i = \\left(\\tilde V_i,\\tilde E_i \\right)$\n\twhere \n\n\n\t\\remred{TODO}\n\\end{proof}\n\n\\remred{TODO: Infinite class of proper closed subclasses of gammoids via $\\mathrm{W}^k$. as remark}\n\n\n\n\\chapter{Conclusions and Open Problems}\n\n\\noindent In this section, we demonstrate the significance of this work by summing up and contextualizing the main new results and concepts presented\nin this work.\nWe introduced the notion of {\\em duality respecting representations} of gammoids and proved that every gammoid has such a representation.\nFor a long time, it has been a well-known fact that the class of gammoids is closed under duality, and the classical proofs of this\nproperty employ the important insight that strict gammoids are precisely the duals of transversal matroids. By shifting our focus away from\nstrict gammoids, we were able to reveal that a gammoid and its dual are tightly related to each other: they are represented by special pairs of\nopposite digraphs\\footnote{We would like to call such directed graphs dual to each other. This may be justified by observing that for\nlattices and partial orders, which may be considered special binary relations, the concept of duality merely swaps the\nfirst and the second component of that relation. Since directed graphs may be considered binary relations as well, \nit is quite natural to call the opposite digraph dual. Unfortunately, there are other notions of duality with respect \nto directed graphs that have equally good justifications.},\nand these special pairs are easily obtained from any representation. This discovery lead us to the concept of \na {\\em standard representation},\nwhich allowed us to define complexity measures for gammoids as minimal complexity measures of the digraphs that may appear in a standard\nrepresentation of a gammoid. We defined the {\\em arc-complexity} and {\\em vertex-complexity} of gammoids and showed that\nthe derived classes of gammoids with bounded arc- or vertex-complexity are closed under duality and minors, and that these classes are characterized by finitely many excluded minors. But in general, these classes are not closed under direct sums. In order to derive subclasses of gammoids that are closed under direct sums, we defined the {\\em $f$-width} of gammoids. We were able to show \nthat the subclasses of gammoids\nwith bounded $f$-width are closed under direct sums for super-additive functions $f$.\n\n\\medskip\n\\noindent\nRegarding our investigation into the problem of deciding whether a given matroid is a gammoid,\nthe starting point\nwas Mason's $\\alpha$-criterion for strict gammoids. Naturally we were more interested in the situation where the matroid\nunder consideration is not a strict gammoid, and therefore we defined the concept of an {\\em $\\alpha$-violation} that\ncaptures minimal situations in a matroid that are not ``strictly gammoidal''. Unfortunately, we showed that it is not possible\nto classify $\\alpha$-violations into violations that correspond to gammoids and violations that correspond to non-gammoids --- we saw\nthat there are non-gammoids that have two copies of a violation, that may occur in a gammoid as its unique violation.\nWe condensed our gathered experience with the recognition problem of gammoids into the notion of a {\\em matroid tableau} and\nthe corresponding \\ref{lastStep}-step directions for the derivation of a decisive matroid tableau.\n\n\\medskip\n\\noindent\nWe introduced the concept of {\\em lifting} cycles in digraphs of representations of gammoids, in order to find\nacyclic representations of gammoids that have the original gammoid as a contraction minor. \nThis concept may be used to\navoid technicalities with the non-acyclic generalizations of the Lindstr\u00f6m Lemma, but it may generally be applied in\nsituations where the presence of cycles in digraphs complicates matters. One such situation arises when we try to use {\\em heavy arc signatures}\nin order to orient a gammoid. We provided a way to determine orientations of gammoids without having to carry out actual calculations in\n${\\mathbb{Q}}$ or ${\\mathbb{R}}$ as long as the corresponding digraph of the representation of the gammoid has no cycle walk. We also gave an\nexample that this condition may not be dropped. Apart from that, we were able to show that the class of {\\em lattice path matroids} is generalized series-parallel, and therefore $3$-colorable.\n\n\\bigskip\n\\noindent In the following sections, we give some starting points for further research in the field of gammoids.\n\n\\section{Other Complexity Measures}\n\n\\noindent Let $\\mu$ be a measure that assigns every digraph $D=(V,A)$ a value $\\mu(D) \\in \\mathbb{R}$. Analogously to the definitions of\nthe arc-complexity and vertex-complexity of a gammoid, we may define the $\\mu$-complexity of a gammoid $M$ to be\n\\[ \\hat{\\mu}(M) = \\min\\SET{\\vphantom{A^A} {\\mu(D)} ~\\middle|~ \\left( D, T, E \\right)\\text{~is a standard representation of~}M}. \\]\nIf $\\mu(D) = \\mu(D^{\\mathrm{opp}})$ for all digraphs $D$, then the $\\mu$-complexity has the property that $\\hat\\mu(M) = \\hat\\mu(M^\\ast)$.\nObviously, all complexity measures for directed graphs have this property as soon as they are obtained from measures for undirected graphs by ignoring\nthe orientation of the arcs. This yields a variety of new research questions about the properties of\nsubclasses of gammoids with bounded $\\mu$-complexity: for which measures $\\mu$ are the the classes consisting\nof gammoids $M$ with $\\hat\\mu(M) \\leq k$ closed under minors, under duality, and under direct-sums? If such a class is closed under minors, then what are the excluded minors for that class? Which of these excluded minors are gammoids? Is the class characterized by finitely many excluded minors?\nInteresting choices of $\\mu$ include arboricity, star-arboricity, thickness, degeneracy, girth, tree number, DAG-width, and many more.\nFurthermore, we should consider the same questions with respect to the $f$-$\\mu$-width which may be defined as\n\\[ \\hat\\mu_f(M) = \\max\\SET{\\frac{\\hat\\mu\\left( \\left( M|' Y \\right)| X \\right))}{f\\left( \\left| X \\right|\n\t \\right) }\n\t\t ~\\middle|~ X\\subseteq Y\\subseteq E} \\]\nwhere $M=(E,{\\mathcal{I}})$.\n\n\n\\section{Arc Complexity of Uniform Matroids}\n\n\\noindent The following is the most fundamental open problem that we encountered in the course of this work.\nIt is most promising to be answered positively in the next few years\n-- possibly by utilizing some results from the theory of digraphs -- but the solution of this problem is unfortunately out of\nreach to the author within the schedule of this work. \nWe are not able to show the following conjecture, but we are convinced that it is true.\n\n\\begin{conjecture}\\label{conj:uniformArcs}\n\tLet $r\\in \\mathbb{N}$, $U=(E,{\\mathcal{I}})$ be a uniform matroid of rank $r$ on the ground set $E$, i.e.\n\t${\\mathcal{I}} = \\SET{\\vphantom{A^A}X\\subseteq E~\\middle|~ \\left| X \\right| \\leq r}$.\n\tThen $\\mathrm{C}_A(U) = r\\cdot \\left( \\left| E \\right| -r \\right)$.\n\\end{conjecture}\n\n\\noindent There is the following reformulation with respect to directed graphs.\n\n\\begin{conjecture}\n\tLet $D=(V,A)$ be a digraph, and let $X,Y\\subseteq V$ with $X\\cap Y =\\emptyset$.\n\tIf for every $X'\\subseteq X$ and every $Y'\\subseteq Y$ with $\\left| X' \\right| = \\left| Y' \\right|$\n\tthere is a routing $R\\colon X'\\double{\\rightarrow} Y'$ in $D$ with \n\t\\[ Y \\cap \\bigcup_{p\\in R} \\left| p \\right| = Y',\\] then\n\t $$\\left| A \\right|\\geq \\left| X \\right|\\cdot \\left| Y \\right|.$$\n\\end{conjecture}\n\n\\begin{proof}[Relative proof]\n\tLet $M=\\Gamma(D,Y,X\\cup Y)$, and let $Q\\subseteq X\\cup Y$ with $\\left| Q \\right| \\leq \\left| Y \\right|$.\n\tClearly \\linebreak $\\left| Q \\right| = \\left| Q_X \\right| + \\left| Q_Y \\right|$\n\twhere $Q_X = Q\\backslash Y$ and $Q_Y = Q\\cap Y$. Thus $\\left| Q \\right|\\leq \\left| Y \\right|$ implies\n\t\\linebreak\n\t$\\left| Q_X \\right| \\leq \\left| Y\\backslash Q_Y \\right|$. Therefore there is a subset $Q_Y' \\subseteq Y\\backslash Q$ with $\\left| Q_Y' \\right| = \\left| Q_X \\right|$. By hypothesis we obtain a routing $R\\colon Q_X\\double{\\rightarrow} Q_Y'$ in $D$\n\twhich avoids $Q_Y \\subseteq Y\\backslash Q_Y'$, thus $R\\cup\\SET{q~\\middle|~q\\in Q_Y}$ is a routing from $Q$ to $Y$ in $D$.\n\tConsequently, $Q$ is independent in $M$.\n\tWe showed that every independent subset of $X\\cup Y$ with at most $\\left| Y \\right|$ elements is independent in $M$,\n\tand it is clear that no subset of $X\\cup Y$ with more than $\\left| Y \\right|$ elements is independent in $M$,\n\ttherefore $M$ is a uniform matroid of rank $\\left| Y \\right|$ with\n\t$\\left| X \\cup Y \\right| = \\left| X \\right| + \\left| Y \\right|$ \n\telements. The statement of this conjecture then follows from Conjecture~\\ref{conj:uniformArcs}.\n\\end{proof}\n\n\\noindent Closely related is the following conjecture which would be implied by the previous two conjectures.\n\n\\begin{conjecture}\n\tFor all $k\\in \\mathbb{N}$ there is a gammoid $G=(E,{\\mathcal{I}})$ such that $$\\mathrm{C}_A(G) \\geq k\\cdot \\left| E \\right|.$$\n\\end{conjecture}\n\n\\noindent This conjecture might be easier to proof, and it still would imply that the classes ${\\mathcal{W}}^k$ of gammoids $G$ \nwith $\\mathrm{W}^k(G) \\leq 1$ for $k\\in \\mathbb{N}$ contain an infinite sequence of strictly bigger subclasses of gammoids.\n\n\\section{$\\alpha$-Violations}\n\n\\noindent\nIn Example~\\ref{ex:BiApexMatroid} and Remark~\\ref{rem:BiApex} we saw that $\\alpha$-violations,\nwhich may be resolved into a strict gammoid\nby extension, may overlap in a common matroid. It is possible that this common matroid may not be resolved into a strict gammoid,\nalthough each restriction that encompasses only a single violation may be extended to a strict gammoid. It is an interesting\n open research problem to\ninvestigate in what ways $\\alpha$-violations may overlap, and to determine under which circumstances overlapping obstructs\nthe simultaneous resolution of the respective $\\alpha$-violations into strict gammoid extensions of the matroid exhibiting the overlapping\n$\\alpha$-violations.\n\n\\section{Excluded Minors}\n\n\\noindent Since every matroid of rank $\\leq 2$ is a gammoid, and every gammoid of rank $3$ is a strict gammoid, we may\nuse the formulas from Section~\\ref{sec:alphaNExt} in order to compute all small\\footnote{In this case: up to $10$ elements.} excluded minors of rank $3$ for the class of gammoids,\nas well as the number of isomorphism classes of small gammoids of rank $3$ with $n$-elementary ground sets \\cite{OEIS}. For $n\\leq 10$, this takes less than $4$ hours on modern hardware. The following\nstatements are up to isomorphy:\nThe smallest excluded minor of rank $3$ is $M(K_4)$ with $6$ elements, the second smallest excluded minor is $P_7$ with $7$ elements.\nThere are $3$ excluded minors with $8$ elements, $11$ excluded minors with $9$ elements, and $96$ excluded minors with $10$ elements.\nThe difficult part in obtaining excluded minors with rank and corank greater than $3$ is to prove, that the considered matroid is not a \ngammoid, and apart from $P_8^=$, we do not know any excluded minor with rank and corank greater than $3$, that is representable over ${\\mathbb{R}}$\nand strongly base-orderable. Therefore we ask: Are there other ${\\mathbb{R}}$-representable and strongly base-orderable excluded minors for the class of gammoids with rank and corank greater than $3$? Are there infinitely many such excluded minors?\n\n\\noindent\nFurthermore, we do not know the excluded minors for ${\\mathcal{W}}^k$, the classes of gammoids $G$ with $\\mathrm{W}^k(G) \\leq 1$.\nIs ${\\mathcal{W}}^k$ characterized by finitely many excluded minors? Moreover, let ${\\mathcal{W}}_f$ be the class of gammoids $G$\nwith $\\mathrm{W}_f(G) \\leq 1$ for a super-additive function $f$. What is the smallest growing behavior of $f$, such that ${\\mathcal{W}}_f$\nhas infinitely many excluded minors? And, conversely, what is the biggest growing behavior of $f$, such that ${\\mathcal{W}}_f$ has finitely many excluded minors?\n\n\\section{Complexity Class of Recognition Problems}\n\n\\noindent V.~Chandru, C.R.~Coullard, and D.K.~Wagner showed in \\cite{CHANDRU198575} that the problem of\ndeciding,\nwhether a given matroid $M$ is a bicircular matroid, is NP-hard.\nThe proof involves deciding, whether the frame matroid constructed from a given gain-network\nis a bicircular matroid or not, in order to answer an instance of the {\\em Subset Product Problem}.\nClearly, there are frame matroids which are not gammoids, for instance all Dowling geometries of rank $\\geq 3$ (\\cite{Ox11}, p.663)\nas well as all graphical matroids with an $M(K_4)$ minor.\nOn the other hand, every bicircular matroid is a gammoid,\nand thus it is possible that the additional information, that a given frame matroid is a gammoid, helps to decide\nwhether the given matroid is bicircular within polynomial time.\nIf this is the case, then ruling out that a given matroid is a gammoid must be NP-hard --- which is the most likely scenario\nconsidering D.~Mayhew's result that every gammoid is a minor of an excluded minor for the class of gammoids \\cite{Ma16}.\n\n\\noindent Closely related to the open problem of the complexity class of recognizing gammoids are the open problems\nregarding the complexity classes of finding a representation, finding a representation with\nthe minimal number of arcs when this number is already known, and determining the minimal number of arcs needed to represent a gammoid;\nand all of the above for each of the subclasses ${\\mathcal{W}}^k$ for $k\\in\\mathbb{N}$, too.\n\n\n\\section{Coloring}\n\nEvery gammoid, that is also a binary matroid, is the polygon matroid of a series-parallel network \\cite{In77},\ntherefore graphic gammoids are $3$-colorable.\nThe following conjecture motivated our studies of gammoids in the first place.\n\n\\begin{conjecture}[\\cite{GoHoNe15}, Conjecture~14]\\label{conj:winfried}\n\tEvery simple gammoid of rank $2$ or greater has a quite simple coline.\n\\end{conjecture}\n\n\\noindent\nIf the conjecture holds, then all gammoids are generalized series-parallel matroids and therefore $3$-colorable.\nAlthough we were not able to resolve this conjecture at this point, we are convinced that the newly developed theory in this work\nwill prove helpful for future approaches. A related open problem is the question, whether every gammoid is generalized series-parallel,\nwhich may still be the case even if the Conjecture~\\ref{conj:winfried} is wrong: although no coline of the non-gammoid $P_7$ \nis quite simple, all of its \norientations are still generalized series-parallel.\n\n\\noindent\nWe provided a method of obtaining an orientation of a gammoid from its representation by combinatorial means, but all orientations obtained\nin this way are representable. It has been known for long that there are oriented matroids whose orientations are not representable.\nOne example of a non-representable orientation is the orientation $\\mathtt{RS}(8)$ of the uniform matroid $U_{4,8}$ --- which is a strict gammoid.\nIs there a way to obtain some or all non-representable orientations of gammoids in a purely combinatorial way from their representations, possibly by generalizing the notion of {\\em heavy arc signatures}? And finally, is there a way to deal with cycle walks in the digraph of a given representation of a gammoid other than first lifting all the cycle walks, then orienting using an extended heavy arc signature, and then contracting the oriented extension?\n\n\\vspace*{3cm}\n\\begin{flushright}\n\\textit{Who questions much, shall learn much, and retain much. ~~~~~ \\\\\n\\hfill{~} --- Sir Francis Bacon.}\n\\end{flushright}\n\n\n\n\n\n\n\n\n\n\n\n\\chapter*{Listings}\n\\addcontentsline{toc}{chapter}{Listings}\n\\fancyhead[RE]{Listings}\n\n\\section{Digraph Backtracking Algorithm}\\label{lst:isGammoidSage}\n\n\\PRFR{Mar 7th}\nThe routine ``{\\ttfamily isGammoid}'' performs a backtracking search in the domain of all digraphs with a fixed number of vertices\nin order to determine whether a given input matroid $M$ is a\ngammoid (Algorithm~\\ref{alg:gammoidBackTrack}) --\nor, optionally, whether $M$ is a gammoid representable with certain upper bounds on the number of arcs and vertices occurring. It\nhas been tested with {\\ttfamily SageMath} version 8 running on macOS 10.13.3. We present some runtime measurements of various inputs\nin order to convey a sense of how slow this algorithm actually is. \nWe measured the performance using three matroids, one is a non-gammoid, one is a strict gammoid, and one is a non-strict gammoid, each with\ndifferent upper bounds for the number \nof vertices allowed in a representing digraph candidate. We stopped the each measurement once a time-limit of $48$ hours was reached. \nHere are the results:\n\n\\begin{tabularx}{\\textwidth}{XX}\n(Example~\\ref{ex:MK4}, $M(K_4)$) &\n\\ttfamily sage: time(isGammoid(MK4()))\\\\*\n& \\ttfamily CPU time total: 27.4 ms\\\\\n($M(K_4)$, $\\left| V \\right|=\\left| E \\right|+1$)&\\ttfamily sage: time(isGammoid(MK4(),7))\\\\*\n& \\ttfamily CPU time total: 1.95 s\\\\\n($M(K_4)$, $\\left| V \\right|=\\left| E \\right|+2$)&\\ttfamily sage: time(isGammoid(MK4(),8))\\\\*\n& \\ttfamily CPU time total: 8min 3s\\\\\n($M(K_4)$, $\\left| V \\right|=\\left| E \\right|+3$)&\\ttfamily sage: time(isGammoid(MK4(),9))\\\\*\n& \\ttfamily CPU time total: > 48h\\\\\n(Example~\\ref{ex:nonStrictGammoid}, $\\Gamma(D,T,V)$, $\\left| V \\right|=9$)& \\ttfamily sage: time(isGammoid(strictG,9))\\\\*\n& \\ttfamily CPU time total: 46min 57s\\\\\n(Example~\\ref{ex:nonStrictGammoid}, $\\Gamma(D,T,V)$, $\\left| V \\right|=10$)&\\ttfamily sage: time(isGammoid(strictG,10)) \\\\*\n& \\ttfamily CPU time total: > 48h \\\\\n(Example~\\ref{ex:nonStrictGammoid}, $\\Gamma(D,T,E)$, $\\left| V \\right|=9$)&\\ttfamily sage: time(isGammoid(G),9) \\\\*\n& \\ttfamily CPU time total: 4h 28min 52s\\\\\n(Example~\\ref{ex:nonStrictGammoid}, $\\Gamma(D,T,E)$, $\\left| V \\right|=10$)&\\ttfamily sage: time(isGammoid(G,10)) \\\\*\n& \\ttfamily CPU time total: > 48h \\\\\n\\end{tabularx}\n\n\\PRFR{Mar 7th}\n\\noindent\nThose times suggest that the digraph backtracking method is not suitable for deciding the value of $\\Gamma_{\\mathcal{M}}(M)$ \nfor $M$ defined on ground sets larger than a few elements within a reasonable time frame. The generic bound derived from Remark~\\ref{rem:upperBoundForV}\nis useless in practice, for instance, the matroid defined in Example~\\ref{ex:nonStrictGammoid} has an upper bound of at most $123$ vertices in\na representing digraph.\n We might achieve some slight improvement in performance by utilizing a tree\n structure to store the set of essential paths and the family of maximal essential routings, but this measure would not have any influence on the rapid growth of \n number of digraph \n candidates that have to be traversed by the backtracking method (Remark~\\ref{rem:backTrackSlow}).\n \\vspace*{1\\baselineskip}\n\n\\noindent\n\\lstinputlisting[language=Python,backgroundcolor=\\color{black!5},frame=tlb,basicstyle=\\footnotesize]{backtracklisting.spyx}\n\n\\section{Calculating $\\alpha_N$ for $N\\in {\\mathcal{X}}(M,e)$}\\label{lst:measureDeltaAlpha}\n\n\\begin{figure}[H]\n\\begin{center}\n\\includegraphics[width=.7\\textwidth]{runtimeMeasurement}\n\\end{center}\n\\caption{\\label{fig:Scat1}Scatter plot of runtime measurements for Listing~\\ref{lst:measureDeltaAlpha}. Each $+$-mark corresponds to a principal extension, each $\\times$-mark to a non-principal extension. The red line indicates equal runtime. The logarithms are dyadic.} \n\\end{figure}\n\n\nIn this section, compare the performance of determining the $\\alpha_N$-vector of a single element extension $N\\in {\\mathcal{X}}(M,e)$\nwith the same rank as $M$\nobtained through the formulas \nderived in Section~\\ref{sec:alphaNExt} with the performance of determining the $\\alpha_N$-vector from scratch. The\nperformance has been measured with {\\ttfamily SageMath} version 8 running on macOS 10.13.3. The program code is listed at the end of this section,\nthe code has been compiled with the built-in Cython compiler of {\\ttfamily SageMath} prior to the measurements. \nFor a fixed initial matroid $M=(E,{\\mathcal{I}})$, \nwe measured one representative $N$ of each isomorphism class of the single element extensions of $M$ that have the same rank as $N$.\nThe runtime for each representative has been measured with $3$ repetitions per method. \nIn total we performed runtime measurements on $822$ different single-element extensions,\nthe median of the extension-formula-runtime to from-scratch-runtime ratio is approximately $0.2635$,\nat least $95\\%$ of the measured ratios are smaller than $0.5972$. \nTherefore we expect the method using the formulas from Section~\\ref{sec:alphaNExt} to be almost four times faster on average, \nand to be at least $1\\frac 1 2$ times faster in the usual case, than computation of $\\alpha_N$ from scratch (Figure~\\ref{fig:Scat1} on p.\\pageref{fig:Scat1}).\n\n\n\n\\noindent\nWe give an overview over the measurements with respect to the specific initial matroids in the following table, where $M_0$ is the initial matroid, $k$ is the number of non-isomorphic same-rank single element extensions of $M_0$, $k_1$ is the number of non-isomorphic same-rank single element extensions of $M_0$ that have a principal modular cut, $r_{.5}$ is the median extension-formula-runtime to from-scratch-runtime ratio, and $r_{.95}$ is the $95$th percentile extension-formula-runtime to from-scratch-runtime ratio.\nThe scatter plot depicts the dyadic logarithm of the run-time in seconds using the extension-formula (vertical axis) versus the dyadic logarithm of the run-time in seconds calculating from scratch (horizontal axis).\nThe blue $+$-marks correspond to single element extensions of $M_0$ with principal modular cuts, and the black $\\times$-marks correspond\nto single element extensions of $M_0$ that have no principal modular cuts. The red line indicates the locations that correspond to equal run-time for both methods.\n\\begin{tabularx}{\\textwidth}{p{3cm} |l|l|l|l|l}\n$M_0$ & $k$ & $k_1$ & $ r_{.5}$ & $ r_{.95}$ & \\\\ \n\\hline \n\\hline\nEx.~\\ref{ex:nonStrictGammoid},\n $\\Gamma(D,T,E)$ & $177$ & $22$ & $.24$ & $.65$ & \n \\begin{minipage}[c]{7cm} \\includegraphics[trim={1cm 1cm 1cm 1cm},width=7cm]{Figs\/statsG} \\end{minipage}\n \\\\\n\\hline\nEx.~\\ref{ex:nonStrictGammoid},\n $\\Gamma(D,T,V)$ & $367$ & $26$ & $.26$ & $.55$ & \n \\begin{minipage}[c]{7cm} \\includegraphics[trim={1cm 1cm 1cm 1cm},width=7cm]{Figs\/statsNG} \\end{minipage}\n \\\\\n\\hline\n$\\left(E,2^E\\right)$ with $\\left| E \\right| = 5$ & $6$ & $6$ & $.45$ & $3.49$ & \n \\begin{minipage}[c]{7cm} \\includegraphics[trim={1cm 1cm 1cm 1cm},width=7cm]{Figs\/statsF5} \\end{minipage}\n \\\\\n\\hline\n$M(K_4)$,\nEx.~\\ref{ex:violationNonGammoid} & $7$ & $5$ & $.45$ & $1.81$ & \n \\begin{minipage}[c]{7cm} \\includegraphics[trim={1cm 1cm 1cm 1cm},width=7cm]{Figs\/statsMK4} \\end{minipage}\n \\\\\n\\hline\n$\\!\\!\\!\\!\\!M(K_4)\\oplus M(K_4)$,\nEx.~\\ref{ex:violationNonGammoid} & $28$ & $15$ & $.26$ & $2.87$ & \n \\begin{minipage}[c]{7cm} \\includegraphics[trim={1cm 1cm 1cm 1cm},width=7cm]{Figs\/statsMK4MK4} \\end{minipage}\n \\\\\n\\hline\n$U_{4,2}$,\n\\cite{Ox11},~p.639 & $3$ & $3$ & $.62$ & $1.22$ & \n \\begin{minipage}[c]{7cm} \\includegraphics[trim={1cm 1cm 1cm 1cm},width=7cm]{Figs\/statsU24} \\end{minipage}\n \\\\\n\\hline\n$P_8^=$,\n\\cite{Ox11},~p.651 & $234$ & $9$ & $.27$ & $.52$ & \n \\begin{minipage}[c]{7cm} \\includegraphics[trim={1cm 1cm 1cm 1cm},width=7cm]{Figs\/statsP8pp} \\end{minipage}\n \\\\\n\\end{tabularx}\n\n\n\n\n\n\\vspace*{1\\baselineskip}\n\n\\noindent\n\\lstinputlisting[language=Python,backgroundcolor=\\color{black!5},frame=tlb,basicstyle=\\footnotesize]{calc_alpha_lst.spyx}\n\n\\chapter*{Index of Symbols and Notation}\n\\fancyhead[RE]{Index of Symbols and Notation}\n\\fancyhead[LO]{Index of Symbols and Notation}\n\\addcontentsline{toc}{chapter}{Index of Symbols and Notation}\n\\begin{tabularx}{\\textwidth}{p{4.4cm}X}\n$\\dSET{\\ldots}$ ~\\hrulefill & set consisting of distinct elements given in list, p.\\pageref{n:dset}.\\\\\n$\\subseteq_\\sigma$ ~\\hrulefill & denotes the signed subset relation, p.\\pageref{n:signedsubset}.\\\\\n$\\emptyset_{\\sigma E}$ ~\\hrulefill & denotes the empty signed subset of $E$, p.\\pageref{n:emptysigma}.\\\\\n$\\emptyset_{\\mathbb{Z} . E}$ ~\\hrulefill & denotes the empty signed multi-subset of $E$, p.\\pageref{n:emptysetZE}.\\\\\n$\\DclD{\\bullet}{D}$ ~\\hrulefill & outer-extension operator in $D$, p.\\pageref{n:DclD}.\\\\\n$\\partial_D\\bullet$ ~\\hrulefill & outer-margin operator in $D$, p.\\pageref{n:partD}.\\\\\n$\\prod p$ ~\\hrulefill & product of the weights of all traversed arcs of $p$, p.\\pageref{n:prodp}.\\\\ \n$\\llless $ ~\\hrulefill& $(\\sigma,\\ll)$-induced order on the routings of $D$, p.\\pageref{n:llless}.\\\\\n$\\alpha_M\\colon 2^E \\longrightarrow \\mathbb{Z}$ ~\\hrulefill & $\\alpha$-invariant of $M$, p.\\pageref{n:alphaM}.\\\\\n$\\left(\\mathsf{A}_M, \\sqsubseteq_M\\right)$ ~\\hrulefill & $\\alpha_M$-poset, p.\\pageref{n:alphaposet}.\\\\\n$\\left(\\mathsf{B}_M^C, \\sqsubseteq_M^C \\right)$ ~\\hrulefill & extension poset of $C \\in {\\mathcal{M}}(M)$, p.\\pageref{n:BetaMC}.\\\\\n$\\Delta\\alpha_M$ ~\\hrulefill& $\\Delta\\alpha$-invariant of $M$, p.\\pageref{n:Deltaalphainvariant}.\\\\\n$\\tilde{\\Delta}\\alpha_M$ ~\\hrulefill& $\\tilde{\\Delta}\\alpha$-invariant of $M$, p.\\pageref{n:DeltaPalphainvariant}.\\\\\n$\\delta_D(X,T)$ ~\\hrulefill & barrier between $X$ and $T$ in $D$, p.\\pageref{n:barrier}.\\\\\n$\\Gamma(D,T,E)$ ~\\hrulefill & gammoid represented by $(D,T,E)$, p.\\pageref{n:GTDE}.\\\\\n$\\Gamma(D,T,V)$ ~\\hrulefill & strict gammoid, p.\\pageref{n:GDTV}.\\\\\n$\\Gamma_{\\mathcal{M}}$ ~\\hrulefill& class map for recognizing gammoids in ${\\mathcal{M}}$, p.\\pageref{n:GammaM}.\\\\\n$\\mu \\in K^{R\\times C}$ ~\\hrulefill & $R\\times C$-matrix over $K$, p.\\pageref{n:matrix}.\\\\\n$\\mu^\\top \\in K^{C\\times R}$ ~\\hrulefill & transpose of $\\mu$, p.\\pageref{n:transposed}.\\\\\n$\\mu^\\top_c$ ~\\hrulefill & $c$-th column of $\\mu$, p.\\pageref{n:matrix}.\\\\\n$\\mu_{(P,\\leq)}$ ~\\hrulefill & M\u00f6bius-function of the poset $(P,\\leq)$, p.\\pageref{n:moebius}.\\\\\n$\\mu_r$ ~\\hrulefill & $r$-th row of $\\mu$, p.\\pageref{n:matrix}.\\\\\n$\\mu| R_0$ ~\\hrulefill & restriction of $\\mu$ to the rows $R_0$, p.\\pageref{n:matrestrict}.\\\\\n$\\mu| R_0\\times C_0$ ~\\hrulefill & restriction of $\\mu$ to the rows $R_0$ and columns $C_0$, p.\\pageref{n:matrestrict}.\\\\\n$(\\sigma,\\ll)$ ~\\hrulefill & heavy arc signature of $D=(V,A)$, p.\\pageref{n:sigmaLL}.\\\\\n$\\sigma E$ ~\\hrulefill & class of signed subsets of $E$, p.\\pageref{n:signedsubset}.\\\\\n$\\zeta_{(P,\\leq)}$ ~\\hrulefill & zeta-matrix of the poset $(P,\\leq)$, p.\\pageref{n:zeta}.\\\\\n$\\bigcap {\\mathcal{A}} $ ~\\hrulefill & denotes the intersection $\\bigcap_{A\\in{\\mathcal{A}}} A$, p.\\pageref{n:bigcap}.\\\\\n$\\bigcup {\\mathcal{A}} $ ~\\hrulefill & denotes the union $\\bigcup_{A\\in{\\mathcal{A}}} A$, p.\\pageref{n:bigcup}.\\\\\n${\\mathcal{A}}=(A_i)_{i\\in I}\\subseteq E $ ~\\hrulefill & family of subsets of $E$ indexed by $I$, p.\\pageref{n:Afam}.\\\\\n${\\mathcal{A}}_{\\Delta} = (A_i)_{i\\in B} \\subseteq A$ ~\\hrulefill & arc system of $(A\\mathbin{\\dot{\\cup}} B,\\Delta)$, p.\\pageref{n:ArcSystem}.\\\\\n${\\mathcal{A}}_{D,T} = (A^{(D,T)}_i)_{i\\in V\\backslash T}$ ~\\hrulefill& linkage system of $D$ to $T$, p.\\pageref{n:ADT}.\\\\\n$\\mathrm{AC}(D) = (V_{D},A_{D})$ ~\\hrulefill & denotes the arc-cut digraph for $D$, p.\\pageref{n:arcCutDigraph}.\\\\\n$\\mathrm{AC}(D,T,E)$ ~\\hrulefill & denotes the arc-cut matroid for $(D,T,E)$, p.\\pageref{n:ACDTE}.\\\\\n${\\mathcal{A}}_M = (A_i)_{i\\in I} \\subseteq E$ ~\\hrulefill& denotes the $\\alpha$-system of $M=(E,{\\mathcal{I}})$, p.\\pageref{n:alphaSystem}.\\\\\n{\\em (B1)} ~\\hrulefill & (axiom) a base of $M$ exists, p.\\pageref{n:Bx}.\\\\\n{\\em (B2)} ~\\hrulefill & (axiom) equicardinality of bases, p.\\pageref{n:Bx}.\\\\\n{\\em (B3)} ~\\hrulefill & (axiom) base exchange, p.\\pageref{n:Bx}.\\\\\n{\\em (B3')} ~\\hrulefill & (axiom) strong base exchange, p.\\pageref{n:B3p}.\\\\\n$(B,\\rho)$ ~\\hrulefill & denotes an $M$-black box, p.\\pageref{n:Brho}.\\\\\n$\\mathrm{b}(M) = (\\mathrm{b}(M,i))_{i=1}^{N}$ ~\\hrulefill & binary encoding of $M$, p.\\pageref{n:bMenc}.\\\\\n${\\mathcal{B}}(M)$ ~\\hrulefill & family of all bases of $M=(E,{\\mathcal{I}})$, p.\\pageref{n:BcalM}.\\\\\n${\\mathcal{B}}_M(F)$ ~\\hrulefill & family of all bases of $F$ in $M$, p.\\pageref{n:BcalMF}.\\\\\n{\\em (${\\mathcal{C}}$1)} ~\\hrulefill & (o.m. axiom) $\\emptyset_{\\sigma E}$ is not a circuit, p.\\pageref{n:Cx}.\\\\\n{\\em (${\\mathcal{C}}$2)} ~\\hrulefill & (o.m. axiom) circuits closed under negation, p.\\pageref{n:Cx}.\\\\\n{\\em (${\\mathcal{C}}$3)} ~\\hrulefill & (o.m. axiom) incomparability of circuits, p.\\pageref{n:Cx}.\\\\\n{\\em (${\\mathcal{C}}$4)} ~\\hrulefill & (o.m. axiom) strong circuit elimination, p.\\pageref{n:Cx}.\\\\\n{\\em (${\\mathcal{C}}$4')} ~\\hrulefill & (o.m. axiom) weak circuit elimination, p.\\pageref{n:Ccal4p}.\\\\\n{\\em (${\\mathcal{C}}^\\ast$1)} ~\\hrulefill & (o.m. axiom) $\\emptyset_{\\sigma E}$ is not a cocircuit, p.\\pageref{n:Cx}.\\\\\n{\\em (${\\mathcal{C}}^\\ast$2)} ~\\hrulefill & (o.m. axiom) cocircuits closed under negation, p.\\pageref{n:Cx}.\\\\\n{\\em (${\\mathcal{C}}^\\ast$3)} ~\\hrulefill & (o.m. axiom) incomparability of cocircuits, p.\\pageref{n:Cx}.\\\\\n{\\em (${\\mathcal{C}}^\\ast$4)} ~\\hrulefill & (o.m. axiom) strong cocircuit elimination, p.\\pageref{n:Cx}.\\\\\n$\\mathrm{C}_A(M)$ ~\\hrulefill & arc-complexity of the gammoid $M$, p.\\pageref{n:ArcCompl}.\\\\\n$C\\bot D$~\\hrulefill& orthogonality of signed subsets $C,D\\in \\sigma E$, p.\\pageref{n:XorthoY}.\\\\\n$\\mathrm{cl}_M $ ~\\hrulefill & closure operator of $M$, p.\\pageref{n:clM}.\\\\\n${\\mathcal{C}}(M)$ ~\\hrulefill & circuit set of the matroid $M$, p.\\pageref{n:CM}.\\\\\n$\\mathrm{C}_V(M)$ ~\\hrulefill & vertex-complexity of the gammoid $M$, p.\\pageref{n:VertexCompl}.\\\\\n$C_{-X} \\in \\sigma E$ ~\\hrulefill & the $X$-flip of $C\\in\\sigma E$, p.\\pageref{n:Xflip}.\\\\\n$D=(A\\mathbin{\\dot{\\cup}} B, \\Delta)$ ~\\hrulefill & directed bipartite graph for $\\Delta$ from $A$ to $B$, p.\\pageref{n:DABD}.\\\\\n$D=(V,A)$ ~\\hrulefill & directed graph, p.\\pageref{n:DVA}.\\\\\n$D^{{\\mathrm{opp}}}=(V,A^{\\mathrm{opp}})$ ~\\hrulefill & opposite digraph, p.\\pageref{n:Dopp}.\\\\\n$D_{r\\leftarrow s} = (V,A_{r\\leftarrow s})$ ~\\hrulefill & $r$-$s$-pivot of the digraph $D=(V,A)$, p.\\pageref{n:digraphpivot}.\\\\\n${\\mathcal{F}}(\\alpha)$ ~\\hrulefill & family of $\\alpha$-flats for $\\alpha\\colon 2^E\\longrightarrow \\mathbb{Z}$, p.\\pageref{n:FcalAlpha}.\\\\\n${\\mathcal{F}}(M)$ ~\\hrulefill & family of all flats of $M$, p.\\pageref{n:FM}.\\\\\n${\\mathcal{F}}(M,X)$ ~\\hrulefill & flats of $M$ that are proper subsets of $X$, p.\\pageref{n:FcalMX}.\\\\\n$f[X]$ ~\\hrulefill & set of images of $x\\in X$ under $f$, p.\\pageref{n:fsquareX}.\\\\\n$f|_{X'}$ ~\\hrulefill & restriction of the map $f\\colon X\\longrightarrow Y$ to $X'\\subseteq X$, p.\\pageref{n:frestrictX'}.\\\\\n${\\mathcal{I}}_\\alpha$ ~\\hrulefill & zero-family of $\\alpha\\colon 2^E \\longrightarrow \\mathbb{Z}$, p.\\pageref{n:Ialpha}.\\\\\n{\\em (I1)} ~\\hrulefill & (axiom) $\\emptyset$ is independent, p.\\pageref{n:Is}.\\\\\n{\\em (I2)} ~\\hrulefill & (axiom) independence carries over to subsets, p.\\pageref{n:Is}.\\\\\n{\\em (I3)} ~\\hrulefill & (axiom) augmentation of independent sets, p.\\pageref{n:Is}.\\\\\n${\\mathrm{idet}~} \\mu$ ~\\hrulefill & determinant-indicator of $\\mu$, p.\\pageref{n:idet}.\\\\\n$I(D,T,E)$ ~\\hrulefill & matroid on $E$ induced by $D$ from $T=(T_0,{\\mathcal{T}})$, p.\\pageref{n:IDTE}.\\\\\n$K^{m\\times n}$ ~\\hrulefill & class of all $m\\times n$-matrices over $K$, p.\\pageref{n:matrix}.\\\\\n$K^{R\\times C}$ ~\\hrulefill & class of all $R\\times C$-matrices over $K$, p.\\pageref{n:matrix}.\\\\\n$\\mathrm{kth}(n,r)$ ~\\hrulefill & bijection that enumerates all $r$-element subsets of $\\SET{1,2,\\ldots,n}$, p.\\pageref{n:kth}.\\\\\n$M(\\alpha) = (E,{\\mathcal{I}}_\\alpha)$~\\hrulefill & matroid corresponding to the matroid invariant $\\alpha$, p.\\pageref{n:MAlpha}.\\\\\n$M(\\mu)$ ~\\hrulefill & matroid on $E$ represented by $\\mu\\in \\mathbb{K}^{E\\times C}$, p.\\pageref{n:matMmu}\\\\\n$M({\\mathcal{A}}) = (E,{\\mathcal{I}}_{\\mathcal{A}})$~\\hrulefill & transversal matroid presented by ${\\mathcal{A}}$, p.\\pageref{n:MAEIA}.\\\\\n$M|' C$ ~\\hrulefill & contraction of $M$ to $C$, p.\\pageref{n:MC}.\\\\\n$M(\\Delta,M_0)$ ~\\hrulefill & matroid induced by $\\Delta\\subseteq D\\times E$ from $M_0$, p.\\pageref{n:MDM0}.\\\\\n$M=(E,{\\mathcal{I}})$ ~\\hrulefill & (independence) matroid, p.\\pageref{n:EI}.\\\\\n$M^\\ast=(E,{\\mathcal{I}}^\\ast)$ ~\\hrulefill & dual matroid of $M$, p.\\pageref{n:Mdual}.\\\\\n${\\mathcal{M}}(M)$ ~\\hrulefill & class of all modular cuts of $M$, p.\\pageref{n:MM}.\\\\\n$M(K_4)$ ~\\hrulefill & polygon matroid of the complete graph on $4$ vertices, p.\\pageref{ex:MK4}.\\\\\n$M({\\mathcal{O}})$ ~\\hrulefill & underlying matroid of the oriented matroid ${\\mathcal{O}}$, p.\\pageref{n:MOcal}.\\\\\n$M| R$ ~\\hrulefill & restriction of $M$ to $R$, p.\\pageref{n:MR}.\\\\\n${\\mathbf{N}}(M)$ ~\\hrulefill & encoding length of $M$, p.\\pageref{n:encM}.\\\\\n$\\mathbb{N}^X$ ~\\hrulefill & multi-sets over $X$, p.\\pageref{n:multiset}.\\\\\n$\\mathbb{N}^{(X)}$ ~\\hrulefill & finite multi-sets over $X$, p.\\pageref{n:fin-multiset}.\\\\\n{\\em (${\\mathcal{O}}$1)} ~\\hrulefill & (o.m. axiom) orthogonality, p.\\pageref{n:Cx}.\\\\\n{\\em (${\\mathcal{O}}$2)} ~\\hrulefill & (o.m. axiom) underlying matroid, p.\\pageref{n:Cx}.\\\\\n${\\mathcal{O}}=(E,{\\mathcal{C}},{\\mathcal{C}}^\\ast)$ ~\\hrulefill & oriented matroid, p.\\pageref{n:Ocal}.\\\\\n${\\mathcal{O}}^\\ast=(E,{\\mathcal{C}}^\\ast,{\\mathcal{C}})$ ~\\hrulefill & dual oriented matroid of ${\\mathcal{O}}$, p.\\pageref{n:OcalBot}.\\\\\n$[{\\mathcal{O}}]$ ~\\hrulefill & reorientation class of ${\\mathcal{O}}$, p.\\pageref{n:reorientationclass}.\\\\\n${\\mathcal{O}}(\\mu)= (E,{\\mathcal{C}}_\\mu,{\\mathcal{C}}_\\mu^\\ast)$ ~\\hrulefill & oriented matroid represented by $\\mu\\in \\mathbb{R}^{E\\times C}$, p.\\pageref{n:OcalMu}.\\\\\n${\\mathcal{O}}| R$ ~\\hrulefill & restriction of ${\\mathcal{O}}$ to $R$, p.\\pageref{n:OrestrictR}.\\\\\n${\\mathcal{O}}|' Q$ ~\\hrulefill & contraction of ${\\mathcal{O}}$ to $Q$, p.\\pageref{n:OcontractQ}.\\\\\n$(p_{i})_{i=1}^{n}\\in\\{\\mathrm{N},\\mathrm{E}\\}^{n}$ ~\\hrulefill & lattice path on an $\\SET{\\mathrm{N},\\mathrm{E}}$-grid, p.\\pageref{n:latticePath}.\\\\\n $\\left| p \\right|$ ~\\hrulefill & set of vertices visited by $p$, p.\\pageref{n:walk}.\\\\\n $\\left| p \\right|_A$ ~\\hrulefill & set of arcs traversed by $p$, p.\\pageref{n:walk}.\\\\\n ${\\mathbf{P}}(D)$ ~\\hrulefill & set of paths in $D$, p.\\pageref{n:simplePath}.\\\\\n ${\\mathbf{P}}(D; u,v)$ ~\\hrulefill & set of paths from $u$ to $v$ in $D$, p.\\pageref{n:SPathUV}.\\\\\n $p\\preceq q$ ~\\hrulefill & $p$ is never above $q$ with common endpoints, p.\\pageref{n:neverabove}.\\\\\n $\\mathrm{P}\\left[p,q\\right]$ ~\\hrulefill & lattice walks between $p$ and $q$, p.\\pageref{n:LPbetweenPQ}.\\\\\n $\\downarrow_{(P,\\leq)} y$ ~\\hrulefill & the down-set of $y\\in P$ with respect to the poset $(P,\\leq)$, p.\\pageref{n:Pdownset}.\\\\\n{\\em (R1')} ~\\hrulefill & (axiom) $\\mathrm{rk}(\\emptyset) = 0$, p.\\pageref{n:Rxp}.\\\\\n{\\em (R2')} ~\\hrulefill & (axiom) $\\mathrm{rk}$ is unit-increasing, p.\\pageref{n:Rxp}.\\\\\n{\\em (R3')} ~\\hrulefill & (axiom) if two points are dependent, so is their line, p.\\pageref{n:Rxp}.\\\\\n{\\em (R2'')} ~\\hrulefill & unit-increment propagates to subsets, p.\\pageref{n:R2pp}.\\\\\n{\\em (R1)} ~\\hrulefill & (axiom) $\\mathrm{rk}$ is non-negative and subcardinal, p.\\pageref{n:Rx}.\\\\\n{\\em (R2)} ~\\hrulefill & (axiom) $\\mathrm{rk}$ is non-decreasing, p.\\pageref{n:Rx}.\\\\\n{\\em (R3)} ~\\hrulefill & (axiom) $\\mathrm{rk}$ is submodular, p.\\pageref{n:Rx}.\\\\\n{\\em (R4)} ~\\hrulefill & there is a unique maximal superset of same rank, p.\\pageref{n:R4}.\\\\\n{\\em (R5)} ~\\hrulefill & there are rank-cardinality independent subsets, p.\\pageref{n:R5}.\\\\\n$\\mathrm{Rec}\\Gamma_{\\mathcal{M}}$ ~\\hrulefill & gammoid recognition problem for ${\\mathcal{M}}$, p.\\pageref{n:RecGM}.\\\\\n$\\mathrm{rk}_M $ ~\\hrulefill & rank function of $M$, p.\\pageref{n:rkM}.\\\\\n$\\mathbb{R}[X]$ ~\\hrulefill & polynomials over commutative $\\mathbb{R}$ with variables $X$, p.\\pageref{n:polynomring}.\\\\ \n$R\\colon X\\double{\\rightarrow} Y \\subseteq {\\mathbf{W}}(D)$ ~\\hrulefill & routing from $X$ to $Y$ in $D$, p.\\pageref{n:routing}.\\\\\n${\\mathrm{sep}}(C,D)$ ~\\hrulefill & separator of signed subsets $C,D\\in \\sigma E$, p.\\pageref{n:sep}.\\\\\n$\\mathrm{sgn}_\\sigma (R)$ ~\\hrulefill & sign of routing w.r.t. $(\\sigma,\\ll)$, p.\\pageref{n:sgnsigma}.\\\\\n${\\mathbf{T}} = (G,{\\mathcal{G}},{\\mathcal{M}},{\\mathcal{X}},\\simeq)$ ~\\hrulefill & matroid tableau, p.\\pageref{n:mattab}.\\\\\n$\\bigcup_{i=1}^n {\\mathbf{T}}_i$ ~\\hrulefill & joint matroid tableau, p.\\pageref{n:jointTableau}.\\\\\n$[{\\mathbf{T}}]_\\simeq$ ~\\hrulefill & expansion matroid tableau, p.\\pageref{n:expTab}.\\\\\n$[{\\mathbf{T}}]_\\equiv$ ~\\hrulefill & extended matroid tableau, p.\\pageref{n:extTab}.\\\\\n${\\mathbf{T}}!$ ~\\hrulefill & conclusion matroid tableau, p.\\pageref{n:concTab}.\\\\\n${\\mathbf{T}}(M_1\\simeq M_2)$ ~\\hrulefill & identified matroid tableau, p.\\pageref{n:idTab}.\\\\\n$w=(w_i)_{i=1}^n \\in\n V^{n}$ ~\\hrulefill& walk in $D=(V,A)$, p.\\pageref{n:walk}.\\\\\n${\\mathbf{W}}(D)$ ~\\hrulefill & set of walks in $D$, p.\\pageref{n:PbfD}.\\\\\n${\\mathbf{W}}(D; u,v)$ ~\\hrulefill & set of walks from $u$ to $v$ in $D$, p.\\pageref{n:PathUV}.\\\\\n$\\mathrm{W}_f(M)$ ~\\hrulefill & $f$-width of the gammoid $M$, p.\\pageref{n:arcWfM}.\\\\\n $(w_1w_2\\ldots w_n)^i$ ~\\hrulefill & shorthand for $i$-iterations of $w_1w_2\\ldots w_n\\in V^n$, p.\\pageref{n:walk}.\\\\ \n $w.q$ ~\\hrulefill & concatenated walk $w_{1}w_{2}\\ldots w_{n}q_{2}q_{3}\\ldots q_{m}$, p.\\pageref{n:pdotq}.\\\\\n$X\\colon E\\longrightarrow \\SET{-1,0,1}$~\\hrulefill & signed subset of $E$, p.\\pageref{n:signedsubset}.\\\\\n$X_+$~\\hrulefill & positive elements of $X$, p.\\pageref{n:xplus}.\\\\\n$X_-$~\\hrulefill & negative elements of $X$, p.\\pageref{n:xminus}.\\\\\n$X_\\pm$~\\hrulefill & support of $X$, p.\\pageref{n:xpm}.\\\\\n$X_0$~\\hrulefill & zero-set of $X$, p.\\pageref{n:xzero}.\\\\\n$-X$~\\hrulefill & negation of $X$, p.\\pageref{n:minusx}.\\\\\n${\\mathcal{X}}(M,e)$ ~\\hrulefill & class of single-element extensions of $M$ by $e$, p.\\pageref{n:XMe}.\\\\\n${\\mathcal{V}}(M)$~\\hrulefill & family of all $\\alpha_M$-violations, p.\\pageref{n:VM}.\\\\\n$\\mathbb{Z} .E$ ~\\hrulefill & family of signed multisets $S\\colon E\\longrightarrow \\mathbb{Z}$, p.\\pageref{n:ZE}.\\\\\n\\end{tabularx}\n\n\\chapter*{\\bibname}%\n \\addcontentsline{toc}{chapter}{\\bibname}\n \\@mkboth{\\MakeUppercase\\bibname}{\\MakeUppercase\\bibname}%\n \\settowidth{\\dimen0}{\\@biblabel{#1}}%\n \\begin{enumerate}[\n leftmargin=!,\n labelwidth=2.5cm,\n align=right,\n before={\\@openbib@code\n \\sloppy\n \\clubpenalty 4000 \\@clubpenalty\\clubpenalty\n \\widowpenalty 4000\n \\sfcode `\\.\\@m}\n ]%\n }\n{%\n \\def\\@noitemerr{\\@latex@warning{Empty `thebibliography' environment}}%\n \\end{enumerate}%\n}\n\\renewenvironment{cases}[1][l]{\\matrix@check\\cases\\env@cases{#1}}{\\endarray\\right.}\n\\def\\env@cases#1{%\n \\let\\@ifnextchar\\new@ifnextchar\n \\left\\lbrace\\def\\arraystretch{1.2}%\n \\array{@{}#1@{\\quad}l@{}}}\n\n \\def\\@mparswitchfalse{\\@mparswitchfalse}\n \\def\\@mparswitchtrue{\\@mparswitchtrue}\n\\makeatother\n\n\\begin{document}\n\\mainmatter\n\\@mparswitchfalse\n\n\n\\renewcommand{\\contract}{.}\\renewcommand{\\PRFR}[1]{\\ignorespaces}\\renewcommand{\\PRFRC}{\\ignorespaces}\\renewcommand{\\PRFRA}{\\ignorespaces}\\renewcommand{\\PRFRB}{\\ignorespaces}\\renewcommand{\\goldstar}[1]{\\ignorespaces}\\renewcommand{\\remblue}[1]{\\ignorespaces}\\renewcommand{\\remred}[1]{\\ignorespaces}\\renewcommand{\\studyremark}[1]{\\ignorespaces}\n\n\n\\begin{centering}\n \\thispagestyle{empty}\n \n \\vspace*{0.6in}\n \\bgroup\n \\Huge\\bfseries Contributions to the Problems of Recognizing and Coloring Gammoids \\par\n \\egroup\n \\vspace{0.5in}\n \\bgroup\n \\normalsize Dissertation \\\\[0.1in]\n zur Erlangung des Grades\\\\[0.1in]\n \\textsc{Doktor der Naturwissenschaften}\\\\[0.1in]\n \\textsc{(Dr. rer. nat.)}\\\\[0.1in]\n \\egroup\n \\vspace{0.5in}\n \\bgroup\n \\normalsize an der Fakult\u00e4t f\u00fcr \\\\[0.1in]\n Mathematik und Informatik der \\\\[0.1in] FernUniversit\u00e4t in Hagen\\\\[0.1in]\n \\egroup\n \\vspace{0.5in}\n \\bgroup\n \\normalsize verfasst von Herrn\\\\[0.1in]\n \\textsc{Diplommathematiker}\\\\[0.1in]\n \\Large Immanuel Albrecht\\\\[0.1in]\n \\normalsize aus Dresden\\\\[0.1in]\n \\egroup\n \\vspace{0.7in}\n Hagen, 2018\\\\[0.1in]\n\\end{centering}\n\n\\cleardoublepage\n\\begin{centering}\n \\thispagestyle{empty}\n \n \\vspace*{1.2in}\n \n \\begin{flushleft}\n \\textit{``That may be impossible, sir.'' \\\\ ~~~~ --- Data. \\hfill{~}}\n \\textit{``Things are only impossible until they're not!'' ~~~~~ \\\\ \\hfill{~} --- Jean-Luc Picard.}\n \\end{flushleft}\n \n\\end{centering}\n\n\\input{Text\/00_Abstract}\n\\input{Text\/01_Acknowledgements}\n\n\n\n\n\n\n\\tableofcontents\n\n\n\n\n\n\n\\cleardoublepage\n\\input{Text\/10_Preliminaries}\n\\input{Text\/110_General}\n\\cleardoublepage\n\\input{Text\/120_Matroid_Basics}\n\\input{Text\/121_Independence_Axioms}\n\\input{Text\/122_Rank_Axioms}\n\\input{Text\/123_From_Submodular_Functions}\n\\input{Text\/124_Duality}\n\\input{Text\/125_Minors}\n\\input{Text\/126_Representable}\n\\clearpage\n\\input{Text\/13_Single_Element_Extensions}\n\\clearpage\n\\input{Text\/140_Rado_Hall}\n\\input{Text\/141_Bipartite_Matroid_Induction}\n\\input{Text\/142_Transversal_Matroids}\n\\clearpage\n\\input{Text\/150_Directed_Graphs}\n\\input{Text\/151_Routings_And_Transversals}\n\\input{Text\/160_Mengers_Theorem}\n\\input{Text\/161_Routing_Crossover_Lemma}\n\\input{Text\/162_Separator_Lemma}\n\\cleardoublepage\n\\input{Text\/300_Gammoids}\n\\input{Text\/301_Different_Representations}\n\\input{Text\/301_q_Size_of_Representation}\n\\input{Text\/301b_Duality_Respecting_Representations}\n\\input{Text\/301bx_Subclasses_With_Bounded_Complexity}\n\\input{Text\/301by_Essential_Arcs}\n\\input{Text\/301c_Blackbox_Representations}\n\\clearpage\n\\input{Text\/31_Strict_Gammoids}\n\\input{Text\/31b_Masons_Alpha}\n\\clearpage\n\\input{Text\/32_Transversal_Matroids}\n\\clearpage\n\\input{Text\/32a_Constructions}\n\\clearpage\n\\input{Text\/32bb_Recognition}\n\\input{Text\/32c_Special_Cases}\n\\clearpage\n\\input{Text\/330_General_Gammoids}\n\\input{Text\/3310_Violations_And_Extensions.tex}\n\\clearpage\n\\input{Text\/3311_Solutions.tex}\n\\input{Text\/332_Slack_and_Single_Element_Extensions}\n\\input{Text\/333_Stuck_Families_and_Persistent_Violations}\n\\input{Text\/334_M_Stuck_Families_and_M_Persistent_Violations}\n\\clearpage\n\\input{Text\/36_Representations_Over_Fields}\n\\cleardoublepage\n\\input{Text\/4000_Oriented_Matroids}\n\\input{Text\/4005_Minors}\n\\input{Text\/4010_Duality}\n\\clearpage\n\\input{Text\/41_Colorings}\n\\clearpage\n\\input{Text\/42_Lattice_Path_Matroids_are_3_Colorable}\n\\clearpage\n\\input{Text\/43_Orientations_of_Gammoids}\n\\input{Text\/43a1_Heavy_Arc_Orientations}\n\\cleardoublepage\n\\input{Text\/50_Where_To_Go_To_Next}\n\n\n\\cleardoublepage\n\\input{Text\/W0_Listings.tex}\n\n\\begin{spacing}{0.9}\n\n\n\\bibliographystyle{alpha}\n\n\\cleardoublepage\n\\fancyhead[RE]{References}\n\\fancyhead[LO]{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nMany recent studies have focused on the coisotropic intersection problem. In particular, there has been growing interest in the leafwise intersection property described below.\\\\[-1ex]\n\n\\noindent\\textbf{Leafwise intersection.}\nLet $(M,\\om)$ be a $2n$ dimensional symplectic manifold and $\\Sigma$ be a coisotropic submanifold of codimension $k$. Then the symplectic structure $\\om$ determines a symplectic orthogonal bundle $T\\Sigma^\\om$, which is a subbundle of $T\\Sigma$ by the definition of coisotropic:\n\\beqn\nT\\Sigma^\\om:=\\{(x,\\xi)\\in T\\Sigma\\,|\\,\\om_x(\\xi,\\zeta)=0 \\textrm{ for all } \\zeta\\in T_x\\Sigma\\}\n\\eeq\nSince $\\om$ is closed, $T\\Sigma^\\om$ is integrable, thus $\\Sigma$ is foliated by the leaves; we denote by $L_x$ the leaf through $x\\in\\Sigma$. We call $x\\in\\Sigma$ a {\\em leafwise intersection} of $\\phi\\in\\Ham_c(M,\\om)$ if $x\\in L_x\\cap\\phi(L_x)$. $\\Ham_c(M,\\om)$ is defined below in Conventions and Notations.\n\nIn this article, we focus on the case that $\\Sigma$ is a restricted contact type hypersurface with contact form $\\lambda$, i.e. $d\\lambda=\\om$ and $\\lambda\\wedge\\om^{n-1}|_\\Sigma\\neq0$, which bounds a compact region in $M$. The argument developed in this article would continue to hold in general contact case with minor modification (see \\cite{Ka1} for the generalized Rabinowitz Floer theory). In that case, $T\\Sigma^\\om$ is nothing but the characteristic line bundle spanned by the Reeb vector field $R$. A leafwise intersection $x\\in\\Sigma$ is called {\\em a periodic leafwise intersection} if the leaf $L_x$ through $x$ is a closed Reeb orbit.\\\\[-1ex]\n\nThe problem of finding leafwise intersection points was initiated by Moser \\cite{Mo} and pursued further in \\cite{Ba,Dr,EH}. Recently there are three different techniques to this problem; \\cite{Zi} approached the leafwise intersection problem with Lagrangian Floer homology and \\cite{Gi,Gu} took advantage of Symplectic Floer homology. Another effective way to investigate leafwise intersections is Rabinowitz Floer homology developed by Cieliebak-Frauenfelder \\cite{CF}. Albers-Frauenfelder \\cite{AF1} observed that critical points of a perturbed Rabinowitz action functional give rise to leafwise intersections; \\cite{AF1,AF2,AF3,AF4,AMo,AMc,Ka,Ka1,Ka2,Me} obtained many results on the leafwise intersection problem using Rabinowitz Floer homology. Rabinowitz Floer homology is the Floer homology of the Rabinowitz action functional $\\AA^H_F$. Thus, the dimension of Rabinowitz Floer homology gives a lower bound on the number of leafwise intersections as in the Morse inequalities in case the Rabinowitz action functional is Morse. Interestingly enough, Rabinowitz Floer homology has infinite dimension in some cases \\cite{CFO,Me,AMc,Ka2}. In such cases, we know that for $F\\in\\Ham_c(M,\\om)$ which makes $\\AA^H_F$ Morse, $\\AA^H_F$ has infinitely many critical points. It turned out that the Rabinowitz action functional is Morse for a generic $F\\in\\Ham_c(M,\\om)$ \\cite{AF1}; therefore a generic perturbation has either infinitely many leafwise intersections or a periodic leafwise intersection (see Proposition \\ref{prop:critical point answers question}); furthermore, \\cite{AF2} showed that generically there is no periodic leafwise intersection points.\\\\[-1ex]\n\n\n\\noindent\\textbf{Main Theorem.} {\\em If Rabinowitz Floer homology has infinite dimension, then for $\\phi_F\\in\\Ham_c(M,\\om)$ there exists infinitely many leafwise intersections or a periodic leafwise intersection.}\\\\[-1ex]\n\nIt is noteworthy that Main Theorem does not assume any kind of non-degeneracy for $\\phi_F$; that is, the perturbed Rabinowitz action functional $\\AA^H_F$ is not necessarily Morse. When it is Morse, the theorem follows immediately from the Morse inequalities.\n\\begin{Rmk}\nAn analogous result continues to hold without doubt in any other Morse or Floer homology whenever the action functional is a Morse (resp. Morse-Bott) and it has only finitely many critical points (resp. finitely many compact critical components) in a compact action interval.\n\\end{Rmk}\n\\begin{Rmk}\nThe Main theorem subsumes a result in \\cite{AF3} which showed the above result for a (unit) cotangent bundle by means of the spectral invariants in Rabinowitz Floer homology.\n\\end{Rmk}\n\n\n\\noindent\\textbf{Convention and Notations.}\n\\begin{itemize}\n\\item The {\\em Reeb vector field} $R$ is characterized by $\\lambda(R)=1$ and $i_R d\\lambda=0$.\n\\item The {\\em Hamiltonian vector field} $X_F$ associated to a Hamiltonian function $F\\in C^\\infty(S^1\\x M)$ is defined implicitly by $i_{X_F}\\om=dF$.\n\\item $\\phi_F$ is the time one map of the flow of $X_F$ and called a {\\em Hamiltonian diffeomorphism}.\n\\item We denote by $\\Ham_c(M,\\om)$ the group of Hamiltonian diffeomorphisms generated by compactly supported Hamiltonian functions.\n\\end{itemize}\n\n\\subsection{Filtered Rabinowitz Floer homology}\nSince $\\Sigma$ is a contact hypersurface, there exists a Liouville vector field $Y$ such that $L_Y\\om=\\om$ and $Y\\pitchfork\\Sigma$; we denote by $\\phi_Y^t$ the flow of $Y$ and fix $\\delta>0$ such that $\\phi_Y^t|_\\Sigma$ is defined for $|t|<\\delta$. Since $\\Sigma$ bounds a compact region in $M$, we are able to define a Hamiltonian function $G\\in C^\\infty(M)$ so that\n\\begin{enumerate}\n\\item $G^{-1}(0)=\\Sigma$ is a regular level set;\n\\item $G(\\phi_Y^t(x))=t$ for all $x\\in\\Sigma$ and $|t|<\\delta$;\n\\item $dG$ has compact support.\n\\end{enumerate}\n\n\\begin{Def}\nGiven time-dependent Hamiltonian functions $H\\in C^\\infty(S^1\\x M)$ and $F\\in C_c^\\infty(S^1\\x M)$, a pair $(H,F)$ is called a {\\em Moser pair} if it satisfies\n\\begin{enumerate}\n\\item $H$ is a weakly time-dependent Hamiltonian function. That is, $H$ is of the form $H(t,x)=\\chi(t)G(x)$ for $G\\in C^\\infty(M)$ defined above and $\\chi:S^1\\to S^1$ with $\\int_0^1\\chi dt=1$ and $\\Supp\\chi\\subset(\\frac{1}{2},1)$;\n\\item their time supports are disjoint, i.e.\n\\beqn\nH(t,\\cdot)=0 \\quad\\textrm{for}\\,\\,\\, \\forall t\\in\\big[0,\\frac{1}{2}\\big] \\quad\\textrm{and}\\quad F(t,\\cdot)=0 \\quad\\textrm{for}\\,\\,\\, \\forall t\\in\\big[\\frac{1}{2},1\\big].\n\\eeq\n\\end{enumerate}\n\\end{Def}\n\\begin{Rmk}\nWe can easily check that every element in $\\Ham_c(M,\\om)$ is generated by a compactly supported Hamiltonian function with time support on $\\big[0,\\frac{1}{2}\\big]$.\n\\end{Rmk}\n\nWe define a perturbed Rabinowitz Floer functional $\\AA^H_F:C^\\infty(S^1,M)\\x\\R\\pf\\R$ for a Moser pair $(H,F)$ as follows:\n\\beqn\n\\AA^H_F(v,\\eta)=-\\int_0^1v^*\\lambda-\\eta\\int_0^1H(t,v)dt-\\int_0^1F(t,v)dt.\n\\eeq\nAlbers-Frauenfelder observed that a critical point of $\\AA^H_F$ gives rise to a leafwise intersection.\n\\begin{Prop}\\label{prop:critical point answers question}\n\\cite{AF1} Let $(v,\\eta)\\in\\Crit\\AA_{F}^{H}$. Then $x:=v(0)\\in\\Sigma$ and satisfies $\\phi_F(x)\\in L_x$. Thus, $x$ is a leafwise intersection point. Moreover, the map\n\\beqn\n\\Crit\\AA^{H}_F\\pf\\big\\{\\textrm{leafwise intersections}\\big\\}\n\\eeq\nis injective unless there exists a periodic leafwise intersection.\n\\end{Prop}\n\n\n\n\\begin{Def}\nWe define the {\\em action spectrum} for a Moser pair $(H,F)$ by\n\\beqn\n\\mathrm{Spec} (H,F):=\\big\\{\\AA^H_F(v,\\eta)\\in\\R\\,\\big|\\,(v,\\eta)\\in\\Crit(\\AA^H_F)\\big\\}.\n\\eeq\n\\end{Def}\n\nNow, we roughly describe a filtered Rabinowitz Fleor homology for a Morse action functional $\\AA^H_F$ and $aB-||F-f||_-$. Then $\\CF^{(b+||F-f||_-,\\infty)}(\\AA^H_F)$ is empty and thus its homology $\\HF^{(b+||F-f||_-,\\infty)}(\\AA^H_F)$ is well-defined although $\\AA^H_F$ could have degenerate critical points with action outside $(b+||F-f||_-,\\infty)$. Due to Proposition \\ref{prop:continuation} together with the ``homotopy of homotopies'' argument, we have the following commutative diagram \\eqref{eq:diagram}.\n\\bea\\label{eq:diagram}\n\\xymatrix{ \\HF^{(b,\\infty)}(\\AA^H_f) \\ar[dd]^{\\big(\\pi^\\infty_{b,b+||F-f||}\\big)*} \\ar[dr]^\\Phi & \\hspace{8cm} \\\\\n & \\HF^{(b+||F-f||_-,\\infty)}(\\AA^H_F) \\ar[dl]^\\Psi \\\\\n\\HF^{(b+||F-f||,\\infty)}(\\AA^H_f) & \\hspace{8cm} }\n\\eea\nSince $\\HF^{(b+||F||_-,\\infty)}(\\AA^H_F)$ vanishes, accordingly $\\big(\\pi^\\infty_{b,b+||F-f||}\\big)*$ also vanishes and we have the following short exact sequence:\n\\beqn\n0\\stackrel{\\pi_*}{\\pf}\\HF^{(b+||F-f||,\\infty)}(\\AA^H_f)\\stackrel{\\delta_*}{\\pf}\\HF^{(b,b+||F-f||)}(\\AA^H_f)\\stackrel{i_*}\n{\\pf}\\HF^{(b,\\infty)}(\\AA^H_f)\\stackrel{\\pi_*}{\\pf}0\n\\eeq\nwhere $\\pi_*=\\big(\\pi^\\infty_{b,b+||F-f||}\\big)_*$, $i_*=\\big(i_{b}^{b+||F-f||,\\infty}\\big)_*$, and $\\delta_*$ is a connecting homomorphism. Since $||F||<\\infty$, the dimension of $\\HF^{(b,b+||F-f||)}(\\AA^H_f)$ is finite. Since $i_*$ is surjective, $\\HF^{(b,\\infty)}(\\AA^H_f)$ is finite dimensional.\n\nIn the same way, we choose $a\\in\\R$ so that $a