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+{"text":"\\section{Introduction and Statement of the Result}\n\nThe curvature of the Weil-Pe\\-ters\\-son\\ metric recently attracted further interest. In this note we will give the precise estimate for the average of the scalar curvature $S_{WP}$ of the Weil-Petersson metric on the moduli space $\\ol{\\mathcal M} _g$ as $g$ tends to infinity. The result is the value\n$$\n \\frac{1}{(g-1)}  \\frac{\\bigintss_{\\ol{\\cM}_g} \\, (-S_{W\\! P})\\,dV_{W \\! P}}{\\bigintss_{\\ol{\\cM}_g} dV_{W\\! P}} =  \\frac{13}{4\\pi} +  \\frac{\\pi}{12} \\frac{1}{g} + \\left( \\frac{1}{4\\pi} +\\frac{\\pi}{12} \\right) \\frac{1}{g^2}+ O\\left(\\frac{1}{g^3}\\right).\n$$\nThe proof of the asymptotics will be based upon methods of Algebraic Geometry. Wolpert showed in \\cite{wo1,wo2} that Mumford's canonical class $\\kappa_1$ (the tautological class obtained from the universal curve) from \\cite{mu1,mu2} (cf.\\ \\cite{ac,ac2}) is the cohomology of the Weil-Pe\\-ters\\-son\\ form extended to the compactification  up to the factor $2\\pi^2$ together with the fact that its restriction to the boundary equals to the Weil-Pe\\-ters\\-son\\ cohomomology of the boundary (interpreted as related to  moduli of punctured surfaces of lower genus).\n\nThe finiteness of the Weil-Pe\\-ters\\-son\\ volume itself is a consequence of Masur's estimates \\cite{ma}, whereas the curvature was computed by Wolpert \\cite{wo3} and Fischer-Tromba \\cite{tro}. These results implied strong negativity properties, in particular the strict negativity of the scalar curvature. It is known that the scalar curvature tends to $-\\infty $ towards the boundary. Precise estimates of the curvature of the Weil-Pe\\-ters\\-son\\ metric towards the boundary are contained in \\cite{s} \\and \\cite{t} with a later developments by Liu-Sun-Yau \\cite{lsy1,lsy2}.\n\nEstimates of the Weil-Pe\\-ters\\-son\\ volume had been given by Mirzakhani \\cite{mi}, Mirzakhani-Zograf \\cite{m-z}, Penner \\cite{pe}, Grushevsky \\cite{gru}, Zograf \\cite{zo2,zo} and previously in \\cite{s-t}. The algebraic aspect is contained in the push-pull formulas by Arbarello and Cornalba \\cite{ac,ac2}.\n\n\nThe Weil-Pe\\-ters\\-son\\ volume of the moduli spaces  $\\ol{\\mathcal M} _{g,n}$ of Riemann surfaces of genus $g$ with $n$ punctures is denoted by\n$$\nV_{g,n}=\\int_{{\\mathcal M}_{g,n}}\\kappa_1^{3g-3+n}.\n$$\nFinally the relationship of intersection numbers and volumes as related to two dimensional gravity ought to pointed out. Pertinent references are \\cite{dij,fp,getz,kon,mz,witt}.\n\n\n\nWe showed the following estimates.\n\\begin{theorem}[\\cite{s-t}]\n\\begin{itemize}\n\\item[(i)]\n Let $g>1$. Then\n\\begin{equation}\\label{eq:upper_est}\nV_{g,0} \\geq \\frac{1}{28} V_{g-1,2} + \\frac{1}{672} V_{g-1,1} +\n\\frac{1}{14} \\sum_{j=2}^{[g\/2]} V_{j,1}V_{g-j,1}\n-\\frac{1}{28} (V_{\\frac{g}{2},1})^2 ,\n\\end{equation}\nwith $V_{\\frac{g}{2},1}=0$, if $g$ is odd.\n\\item[(ii)]\nThere exist constants $0 < c < C$, independent of $n$ such that\n\\begin{equation}\\label{eq:asymp1}\nc^g  (2g)! \\leq \\frac{V_{g,n}}{(3g-3+n)!} \\leq C^g  (2g)!\n\\end{equation}\nfor all fixed $n\\geq 0$ and large $g$.\n\\end{itemize}\n\\end{theorem}\nConcerning \\eqref{eq:asymp1}, a lower estimate for $n=1$ is due to Penner \\cite[Theorem 6.2.2]{pe}, and the upper estimates for $n \\geq 1$ were first shown by Grushevsky in \\cite[Sec.~7]{gru}.\n\nBounds for the curvature of the Weil-Pe\\-ters\\-son\\ metric were proven by Wu and Wolf in \\cite{ww} and Wu in \\cite{wu1,wu2}. A recent result is the following:\n\\begin{theorem}[{Bridgeman-Wu, \\cite{b-w}}]\\label{th:briwu}\nDenote by $S_{W\\!P}$ the scalar curvature of the Weil-Pe\\-ters\\-son\\ form $\\omega_{WP}$, and by $dV_{W\\!P}$ its volume element. There exist constants $0<c<C$ such that\n\\begin{equation}\\label{eq:briwu}\nc\\cdot g \\leq \\frac{\\bigintss_{\\ol{\\cM}_g} \\, (-S_{W\\! P})\\,dV_{W \\! P}}{\\bigintss_{\\ol{\\cM}_g} dV_{W\\! P}} \\leq C\\cdot g\n\\end{equation}\nfor all $g$.\n\\end{theorem}\nWe will show that the Bridgeman-Wu estimate follow in an algebraic way using \\eqref{eq:upper_est}.\n\nThe actual asymptotics of the Weil-Pe\\-ters\\-son\\ volume (containing \\eqref{eq:asymp1}) was computed by Mirzakhani and Zograf. (We use the above normalization for the Weil-Pe\\-ters\\-son\\ volume.)\n\\begin{theorem}[{\\cite[Theorem 1.2]{mz}}]\nThere exists a constant $C\\in(0,\\infty)$ such that for any given $k\\geq 1$, $n\\geq 0$\n\\begin{equation}\\label{eq:mz}\n  V_{g,n}= C_{MZ} \\frac{(3g-3+n)! \\, (2g-3+n)! \\, 2^{g-3+n}}{\\pi^{2g}\\sqrt{g}}\\left( 1+ \\sum_{j=1}^{k} \\frac{c_n^{(j)}}{g^j} + O\\left(\\frac{1}{g^{k+1}}\\right)   \\right)\n\\end{equation}\nas $g\\to\\infty$.\n\\end{theorem}\nThe theorem contains further characterizations of the polynomials $c^{(j)}_n$ -- we will need the case $k\\leq 1$, and $n\\leq 2$. The Mirzakhani-Zograf constant $C_{MZ}$ is conjectured to be $C_{MZ}=1\/\\sqrt\\pi$.\n\nLet again $\\delta$ be the $\\mathbb Q$-divisor related to the boundary components of the moduli space (see also below), and denote by $\\kappa_1$ the Mumford class. Recall that the class $[\\omega_{WP}]$  of the Weil-Pe\\-ters\\-son\\ form was computed in \\cite{wo1,wo2} as $2 \\pi^2 \\kappa_1$. Set $\\eta= [-Ric(\\omega_{WP})\/2\\pi]$.\n\\begin{maintheorem*}\n  The average total scalar curvature of the Weil-Pe\\-ters\\-son\\ metric on the moduli space $\\ol{\\mathcal M} _g$ satisfies the estimate\n  \\begin{eqnarray}\n   \\frac{13}{4\\pi} &\\leq&\n     \\frac{3}{\\pi} \\frac{ \\eta \\cdot \\kappa_1^{3g - 4}}{  \\kappa_1^{3g - 3}} = \\frac{1}{(g-1)}  \\frac{\\bigintss_{\\ol{\\cM}_g} \\, (-S_{W\\! P})\\,dV_{W \\! P}}{\\bigintss_{\\ol{\\cM}_g} dV_{W\\! P}} =  \\frac{13}{4\\pi} +  \\frac{1}{4\\pi}  E_g  , \\label{eq:Eg}  \\quad \\text{ where } \\\\ \\nonumber\n \\frac{1}{4\\pi} E_g &=&  \\frac{\\pi}{12} \\frac{1}{g} + \\left( \\frac{1}{4\\pi} +\\frac{\\pi}{12} \\right) \\frac{1}{g^2}+ O\\left(\\frac{1}{g^3}\\right) \\text{ for } g\\to \\infty.\\\\\\text{ The precise value is } \\nonumber\\\\\n\\label{eq:kappa}\n E_g &=& \\frac{\\kappa_1^{3g-4} \\cdot \\delta }{\\kappa_1^{3g-3}} \\text{ where }\\delta = \\sum_{j=0}^{[g\/2]} \\delta_j.\n\\end{eqnarray}\n\\end{maintheorem*}\nNote that the ampleness of $\\kappa_1$ implies that $E_g$ is positive.\n\n\n\\section{Proof}\n\nLet $D$ denote the compactifying divisor of ${\\mathcal M} _g$ with components $\\Delta_j$ for $j=0,\\ldots,[g\/2]$. These give rise to $\\mathbb Q$-divisors $\\delta_j$ such that in terms of the generally used notation\n$$\n[\\Delta_1]= 2 \\delta_1 \\text{ and } [\\Delta_j]=  \\delta_j \\text{ for } j \\neq 1.\n$$\nThen $\\delta:= \\sum_{j=0}^{[g\/2]} \\delta_j$ so that\n$$\nD= \\delta +\\delta_1.\n$$\nNote that there exist branched two-sheeted coverings $\\ol{\\mathcal M} _{g-1,2} \\to \\Delta_0$, \\quad $\\ol{\\mathcal M} _{g-1,1}\\times \\ol{\\mathcal M} _{1,1} \\to \\Delta_1$, and $\\ol{\\mathcal M} _{g\/2,1}\\times\\ol{\\mathcal M} _{g\/2,1} \\to \\Delta_{g\/2}$ for even $g$ yielding extra factors $1\/2$ in Lemma~\\ref{le:le1}.\n\n.\n\nTwo classical facts are needed. Mumford's direct image bundle $\\lambda$ satisfies \\cite{mu1}\n$$\n\\kappa_1 = 12 \\lambda - \\delta,\n$$\nand by \\cite{hm}  the canonical bundle $K_{\\ol{\\mathcal M} _{g,0}}$ is equal to\n$$\nK_{\\ol{\\mathcal M} _{g,0}} = 13 \\lambda - 2\\delta - \\delta_1.\n$$\n\n\n\nIn \\cite[Corollary 5.5]{t} it was shown by means of a a singular Mumford good hermitian metric that $\\eta$ is the Chern class of the dual of the logarithmic tangent bundle $T_{{\\overline{\\mathcal M} }_{g,0}}({\\log} D)$. Therefore\n\\begin{equation}\\label{eq:ric}\n  [\\eta] = K_{\\ol{\\mathcal M} _{g,0}} + [D].\n\\end{equation}\nNow \\eqref{eq:Eg} can be shown:\n\\begin{gather*}\n\\frac{ \\eta \\cdot \\kappa_1^{3g - 4}}{  \\kappa_1^{3g - 3}} = \\frac{ (K_{\\ol{\\mathcal M} _{g,0}}+\\delta+\\delta_1) \\cdot \\kappa_1^{3g - 4}}{  \\kappa_1^{3g - 3}} = \\frac{(13 \\lambda - \\delta )\\cdot \\kappa_1^{3g - 4}}{  \\kappa_1^{3g - 3}} = \\frac{(\\frac{13}{12} \\kappa_1 + \\frac{1}{12}\\delta)\\cdot \\kappa_1^{3g - 4}}{ \\kappa_1^{3g - 3}}\n\\end{gather*}\nThe main result consists of the estimate for $E_g$. The special value for $V_{1,1}$ etc.\\ are taken from \\cite{s-t}.\n\\begin{lemma}\\label{le:le1}\n\\begin{eqnarray}\n  \\frac{\\kappa_1^{3g-4} \\cdot \\delta_0 }{\\kappa_1^{3g-3}} &=& \\frac{1}{2} \\frac{V_{g-1,2}}{V_{g,0}}\n  \\\\\n  \\frac{\\kappa_1^{3g-4} \\cdot \\delta_1 }{\\kappa_1^{3g-3}} &=&   \\frac{1}{2}  \\cdot \\frac{V_{1,1}\\cdot V_{g-1,1}}{V_{g,0}} = \\frac{1}{48} \\cdot \\frac{ V_{g-1,1}}{V_{g,0}}  \\\\\n  \\frac{\\kappa_1^{3g-4} \\cdot \\delta_j }{\\kappa_1^{3g-3}} &=& \\frac{V_{j,1}\\cdot V_{g-j,1}}{V_{g,0}} \\qquad \\text{ for }\\quad 2\\leq j\\ \\leq [(g-1)\/2]  \\\\\\label{eq:deltaj}\n  \\frac{\\kappa_1^{3g-4} \\cdot \\delta_{g\/2} }{\\kappa_1^{3g-3}} &=&\\frac{1}{2}  \\frac{(V_{g\/2,1})^2}{V_{g,0}}   \\text{ \\ for \\ } g \\text{ \\ even}.\n\\end{eqnarray}\n\\end{lemma}\n\\begin{remark}\n Lemma~\\ref{le:le1}, and \\eqref{eq:Eg} together with \\eqref{eq:upper_est} imply the Bridgeman-Wu Theorem~\\ref{th:briwu}.\n\\end{remark}\n\n\n\nWe will apply the Mirzakhani-Zograf estimate \\eqref{eq:mz}, and the following values contained in \\cite[Remark 1.2]{mz}:\n\\begin{eqnarray}\n  c^{(1)}_0 & = & \\frac{7}{12} -\\frac{17}{6\\pi^2} \\label{eq:c0}\\\\\n  c^{(1)}_1 & = & \\frac{1}{3} - \\frac{5}{6\\pi^2} \\label{e1:c1}\\\\\n  c^{(1)}_2 & = & \\frac{1}{12} + \\frac{1}{6\\pi^2} .\\label{eq:c2}\n\\end{eqnarray}\n\n\n\\begin{lemma}\n  \\begin{eqnarray}\n \n    \\frac{1}{2}\\frac{V_{g-1,2}}{V_{g,0}} &=&\n    \\frac{\\pi^2 }{3g} \\left( 1 + \\left(\\frac{3}{\\pi^2} + 1 \\right)\\frac{1}{g} + O\\left( \\frac{1}{g^2} \\right)     \\right)\n    \\label{eq:g-1_2}\\\\\n    \\frac{1}{48}\\frac{V_{g-1,1}}{V_{g,0}} &=& \\frac{1}{144}\\sqrt{\\frac{g}{g-1}} \\frac{\\pi^2}{(g-1)(3g-4)(2g-3)}\\left(1 + O\\left(\\frac{1}{g}\\right)\\right) = O\\left(\\frac{1}{g^3}\\right)\\\\\n    \\frac{V_{j,1}\\cdot V_{g-j,1}}{V_{g,0}} &=& \\frac{ C_{MZ} }{2}\\sqrt{\\frac{g}{j(g-j)}} \\frac{1}{(3g-3) \\left(3g-4\\atop 3j-2 \\right)(2g-3) \\left(2g-4\\atop 2j-2  \\right) }\\left(1 + O\\left(\\frac{1}{g}\\right)\\right)\\label{eq:Vj1}\\\\ \\nonumber && \\text{ for } 2\\leq j \\leq[(g-1)\/2]\\\\\n    \\frac{1}{2} \\frac{(V_{g\/2,1})^2}{V_{g,0}} &=&  \\frac{ C_{MZ} }{2}{\\frac{1}{\\sqrt{g}}} \\frac{1}{(3g-3) \\left(3g-4\\atop (3g-4)\/2 \\right)(2g-3) \\left(2g-4\\atop g-2  \\right) }\\left(1 + O\\left(\\frac{1}{g}\\right) \\right)\\label{eq:gover2}\\\\ && \\nonumber  \\text{ for g even}.\n  \\end{eqnarray}\n\\end{lemma}\nThe computation of \\eqref{eq:g-1_2} is the following.\n\\begin{gather*}\n  \\frac{1}{2}\\frac{V_{g-1,2}}{V_{g,0}} = \\frac{1}{2} \\frac{(3g-4)!\\,(2g-3)!\\, 2^{g-2} \\pi^{2g}\\, \\sqrt{g}}{(3g-3)!\\,(2g-3)!\\,2^{g-3}\\,\\pi^{2g-2}\\,\\sqrt{g-1}}\\cdot \\widetilde C_g =\\\\ = \\frac{\\pi^2}{3g} \\left(\\frac{g}{g-1} \\right)^{3\/2} \\cdot \\widetilde C_g  = \\frac{\\pi^2}{3 g } \\left( 1 + \\frac{3}{2 g} + O\\left(\\frac{1}{g^2}  \\right)   \\right) \\cdot \\widetilde C_g\n\\end{gather*}\nwhere\n\\begin{gather*}\n  \\widetilde C_g = \\left(1 + c^{(1)}_{2}\\frac{1}{g-1} + O\\left(\\frac{1}{g^2}\\right)\\right) \\Big\/ \\left(1 + c^{(1)}_{0}\\frac{1}{g} + O\\left(\\frac{1}{g^2}\\right)\\right) \\\\\n\\end{gather*}\n\n\n\nThe coefficients $c^{(1)}_n$ do not contribute to the remaining terms up to order three in $1\/g$.\n\nWe need an elementary estimate:\n\nLet $l \\leq k \\leq n$ be positive integers,  then $(n-k)(k-l) \\geq 0,$ i.e. $\\frac{n-k+l}{l} \\geq \\frac{n}{k}.$ So we derive the well known inequality\n \\begin{equation} \\binom{n}{k} = \\prod_{l=1}^{k} \\frac{n-k+l}{l}  \\geq \\left(\\frac{n}{k}\\right)^k.  \\label{binomial} \\end{equation}\n\nThe function\n$$\nf(x) := \\log \\left( \\frac{h}{x} \\right)^x = x ( \\log h - \\log x)\n$$\nattains its maximum value at $x=h\/e$ so that its minimum will be taken at either end of a given interval.\n\nThe binomial coefficients in \\eqref{eq:Vj1} attain asymptotic lower estimates for $2\\leq j\\leq [g\/2]$ at $j=2$ namely\n$$\n \\left(3g-4\\atop 3j-2 \\right)\\geq \\left( \\frac{3g-4}{4}\\right)^4 \\quad \\text{and} \\quad \\left(2g-4\\atop 2j-2  \\right)\\geq (g-2)^2.\n$$\nHence, summing up all terms of the type \\eqref{eq:Vj1} yields a term of growth order at most $O(1\/g^7)$. The last term \\eqref{eq:gover2} has a at most a negative exponential growth with respect to $g$ and can also be disregarded.\n\n\n\n This proves the main theorem. \\qed\n\n{\\it Acknowledgements.} The first named author expresses his thanks to the Department of Mathematics at Tor Vergata for their kind hospitality, and for support by PRIN Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics n.2017JZ2SW5, and by DFG SCHU771\/5-1 (Singular Hermitian Metrics\/Analytic Theory of Moduli Spaces). The second named author is supported, by PRIN Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics n.2017JZ2SW5, and by MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.\u00a0\n\n\n\\frenchspacing\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{Introduction}\nIn this paper we present the Reachable Polyhedral Marching (RPM) algorithm for computing forward and backward reachable sets of deep neural networks with rectified linear unit (ReLU) activation.  This is a critical building block to proving safety properties for autonomous systems with learned perception, dynamics, or control components in the loop.  Specifically, given a set of input values, RPM will compute the set of all output values that can be obtained under the ReLU network (the forward reachable set, or image, of the input set).  Similarly, given a set of output values, RPM will compute the set of all input values that can lead to those output values under the ReLU network (the backward reachable set, or preimage, of the output values).  When the ReLU network is part of a dynamical process, as is common in robots with deep learned perception or control components, RPM can compute such reachable sets for multiple time steps into the future or the past without explicitly iterating over each time step.  Figure \\ref{fig:cell_enum} shows the incremental nature of how RPM enumerates input space polyhedra over which the ReLU network is affine.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width = 0.98\\columnwidth]{incremental.png}\n  \\caption{Snapshots of how RPM incrementally enumerates the 2D input space polyhedra for which a random ReLU network is affine, resulting in the explicit piecewise-affine representation. Polyhedron color is random.}\n  \\label{fig:cell_enum}\n\\end{figure}\n\n\nAll existing algorithms that compute exact reachable sets iterate through the network layer-by-layer \\cite{xiang2017reachable,nnv,yang2020reachability}, which may become intractable if the network is applied iteratively in a feedback loop.  RPM applies a fundamentally different approach to avoid layer-by-layer computation.  Instead, we compute the reachable set one polyhedral cell at a time, where each cell represents a region of the input space over which the network is affine.  Each cell can be computed quickly through a series of linear programs equal to the number of neurons in the network, regardless of width or depth.  Our algorithm computes cells until the explored set of cells fills the desired reachable set. In this way, our method is geometrically similar to fast marching methods in optimal control \\cite{HJBFastMarching}, path planning \\cite{PlanningFastMarching,FMT}, and graphics \\cite{FastMarchingCubes,lei2020analytic}.\n\nWe demonstrate RPM in two examples related to the safety verification of dynamical systems with learned components.  First, we compute forward reachable and backward reachable sets for a learned dynamical model of a pendulum over 50 time steps in the future and past, respectively. Furthermore, we compare with a state of the art reachability method \\cite{yang2020reachability} in a safety verification problem involving ACAS Xu, a neural network policy for aircraft collision avoidance.  Our algorithm finds unsafe inputs for this network many times faster than \\cite{yang2020reachability} and when no unsafe input exists, our algorithm certifies safety approximately 2 times slower.  \n\n\n\n\n\nThe paper is organized as follows. We give related work in Section \\ref{Related Work} and give background and state the problem in Section \\ref{Background}.  Section \\ref{PWA} presents our main algorithm, and explains its derivation. In Section \\ref{reach} we describe how the RPM algorithm can be used to perform forward and backward reachability over multiple time steps. In Section \\ref{Examples} we present the aforementioned examples of RPM.\n\n\n\n\n\\section{Related Work} \\label{Related Work}\nThough the analysis of neural networks is a young field, a broad literature has emerged to address varied questions related to interpretability, verification, and safety. Much work has been dedicated to characterizing the expressive potential of ReLU networks by studying how the number of affine regions scales with network depth and width \\cite{bengio,Hanin_2019,understanding}. Other research includes encoding piecewise-affine (PWA) functions as ReLU networks \\cite{fem,reverseengineering}, learning deep signed distance functions and extracting level sets \\cite{lei2020analytic}, and learning neural network dynamics or controllers that satisfy stability conditions \\cite{zico,jin2020neural}, which may be more broadly grouped with correct-by-construction training approaches \\cite{art,lincolnlab}. \n\n\nSpurred on by pioneering methods such as Reluplex \\cite{reluplex}, the field of neural network verification has emerged to address the problem of analyzing properties of neural networks over continuous input sets. A survey of the neural network verification literature is given by \\cite{survey}. Reachability approaches are subset of this literature and are especially useful for analysis of learned dynamical systems. Reachability methods can be categorized into overapproximate and exact methods. Overapproximate methods often compute neuron-wise bounds either from interval arithmetic or symbolically \\cite{maxsens,fastlip,reluval,neurify,crown}. Optimization based approaches such as (mixed-integer) linear programming are also used to solve for bounds on a reachable output set in \\cite{sherlock,alessio1,alessio2,proveable}. Other approaches include modeling the network as a hybrid system \\cite{verisig}, abstracting the domain \\cite{abstract}, and performing layer-by-layer operations on zonotopes \\cite{AI2}. Further, \\cite{julian2019reachability} demonstrated how \\cite{reluplex, sherlock, reluval, verisig} could be used to perform closed-loop reachability of a dynamical system given a neural network controller.\n\nExact reachability methods have also been proposed, although to a lesser degree \\cite{xiang2017reachable,nnv,yang2020reachability}. These methods generally perform the set operations of affine transformation, intersection\/division, and projection layer-by-layer through a ReLU network. Layer-by-layer approaches have also been proposed to solve for the explicit PWA representation of a ReLU network \\cite{robinson2020dissecting,hanin2019complexity}.\nExact methods, unlike overapproximate methods, can compute backward reachable sets.\n\nOur RPM algorithm inherits all advantages of exact methods, but is unique in that it does not iterate layer-by-layer. This can lead to faster verification decisions when finding unsafe inputs. Layer-by-layer methods must compute the entire reachable set, regardless of whether a network violates a safety property. In contrast, if our algorithm encounters a cell with an unsafe input, it can return that result immediately without computing the entire reachable set. Finally, all intermediate polyhedra and affine map matrices of layer-by-layer methods must be stored in memory until the algorithm terminates (since intermediate polyhedra may be split by later neurons). Conversely, since our method computes the true PWA function incrementally, once a polyhedron-affine map pair is computed it can be sent to external memory and only a binary vector (of the neuron activations) needs to be stored to continue the algorithm. This is especially useful because the number of affine regions is hard to estimate before runtime.\n\n\n\\section{Background and Problem Statement} \\label{Background}\nAn $n$-layer feedforward neural network implements a function $F(\\textbf{x}) : \\mathbb{R}^{k_0} \\rightarrow \\mathbb{R}^{k_n}$ to give a map from inputs $\\mathbf{x}$ to outputs $\\textbf{y} = F(\\textbf{x})$. Each layer in $F$ is a function $F_i(\\textbf{z}_{i-1}) : \\mathbb{R}^{k_{i-1}} \\rightarrow \\mathbb{R}^{k_i}$ where $\\textbf{z}_{i} \\in \\mathbb{R}^{k_i}$ is the hidden layer variable for layer $i$. We assume $\\textbf{z}_0 = \\textbf{x}$ and $\\textbf{z}_n = \\textbf{y}$. The function for layer $i$ is\n\\begin{align}\n    F_i(\\textbf{z}_{i-1}) = \\textbf{z}_{i} = \\sigma_i(\\hat{\\textbf{z}}_{i}) = \\sigma_i(\\textbf{W}_i \\textbf{z}_{i-1} + \\textbf{b}_i),\n\\end{align}\nwhere $\\sigma_i$ is the activation function, $\\hat{\\textbf{z}}_{i}$ is the preactivation value, and $\\textbf{W}_i$ and $\\textbf{b}_i$ are the weights and biases for layer $i$. We assume $\\sigma_{n}$ is an identity map and all hidden layer activations are the rectified linear unit (ReLU)\n\\begin{align}\n    \\sigma_i(\\hat{\\textbf{z}}_{i}) = [max(0,\\hat{z}_{i,1}),...,max(0,\\hat{z}_{i,k_i})]^\\top.\n\\end{align}\n\nFrom the above definition we can augment the weights to construct an equivalent neural network that has no bias terms. This equivalent formulation is such that $\\tilde{F}(\\tilde{\\textbf{x}}) : \\mathbb{R}^{k_0+1} \\rightarrow \\mathbb{R}^{k_n}$ and $\\tilde{\\textbf{x}} = [x_1, x_2, ..., x_n, 1]^\\top$. We then have\n\\begin{subequations}\n    \\begin{align}\n        \\tilde{F}_i(\\tilde{\\textbf{z}}_{i-1}) = \\tilde{\\textbf{z}}_i = \\sigma_i(\\tilde{\\textbf{W}}_i \\tilde{\\textbf{z}}_{i-1}) \\\\\n        \\tilde{\\textbf{W}}_i = \n        \\begin{bmatrix}\n            \\textbf{W}_i & \\textbf{b}_i \\\\\n            \\textbf{0}^\\top & 1\n        \\end{bmatrix} \\\\\n        \\tilde{\\textbf{W}}_n = \n    \\begin{bmatrix}\n        \\textbf{W}_n & \\textbf{b}_n\n    \\end{bmatrix}.\n    \\end{align}\n\\end{subequations}\n\nIt is well-known that the function implemented by a ReLU network is continuous and PWA \\cite{bengio,Hanin_2019,understanding}.  We mathematically characterize PWA functions below.\n\\newtheorem{definition}{Definition}\n\\begin{definition}[Polyhedral Complex]\nA polyhedral complex $\\mathcal{C}$ is a finite set of convex polyhedra where every polyhedron contains its faces and the intersection between two polyhedra is either a face of both $c_1$ and $c_2$ or empty. We henceforth refer to convex polyhedra simply as polyhedra.\n\\end{definition}\n\nA PWA function is a function $M(\\textbf{x}) : \\mathbb{R}^{n} \\rightarrow \\mathbb{R}^{m}$ where\n\\begin{align}\n    M(\\textbf{x}) = \\textbf{C}_i \\textbf{x} + \\textbf{d}_i \\ \\ \\ \\forall \\textbf{x} \\in c_i \\ \\ \\ \\forall c_i \\in \\mathcal{C} \\label{eq:PWA}\n\\end{align}\nfor $\\textbf{C}_i \\in \\mathbb{R}^{m \\times n}$, $\\textbf{d}_i \\in \\mathbb{R}^{m}$, and $c_i$ elements of a polyhedral complex $\\mathcal{C}$. Here we consider only continuous PWA functions. Equation (\\ref{eq:PWA}) is the \\textit{explicit} PWA representation.\n\nThe forward reachability problem is to solve for the image of a specified input set under the neural network map\n\\begin{align}\n    \\mathcal{Y} = \\{F(\\textbf{x}) \\vert \\textbf{x} \\in \\mathcal{X} \\}.\n\\end{align}\nLikewise, the backward reachability problem is to solve for the preimage of a specified output set\n\\begin{align}\n    \\mathcal{X} = \\{x \\vert F(\\textbf{x}) \\in \\mathcal{Y} \\}.\n\\end{align} \n\nOur algorithm transforms a ReLU network into an explicit PWA representation by computing a set of polyhedra ($c_i$) and associated affine maps $(\\textbf{C}_i, \\textbf{d}_i)$.  There exist efficient methods for computing forward and backward reachable sets of polyhedral sets under PWA maps. Accordingly, we restrict our focus to these problems. \n\n\n\\section{Explicit PWA Representation} \\label{PWA}\nFirst, we seek to construct the explicit PWA representation of a ReLU network. Our method constructs a PWA function for a ReLU network by enumerating each polyhedral cell and its associated affine map. In Sec.~\\ref{Cells} we show how cells and affine maps are computed from the \\textit{activation pattern} of a ReLU network. In Sec.~\\ref{Essential} we show how cell representations are reduced to a minimal form, which is used in Sec.~\\ref{Neighbors} to determine neighboring polyhedra given a current polyhedron. This leads to a recursive procedure in which we explore an expanding front of polyhedra, ultimately giving the explicit PWA representation, as explained in Sec.~\\ref{Cell Enumeration}.\n\n\\subsection{Determining Polyhedral Cells from Activation Patterns} \\label{Cells}\nFor a given input to a ReLU network, every hidden neuron has an associated binary neuron activation of zero or one corresponding to whether the preactivation value was nonpositive or positive, respectively\\footnote{This choice is arbitrary and some others use the opposite convention.}. We refer to the vector of all neuron activations as the \\textit{activation pattern} of the network for a given input. For neuron $j$ in layer $i$,\n\\begin{subequations}\n    \\begin{align}\n        AP_{ij}(\\textbf{x}) = \\begin{cases}\n                                 1\\ \\  \\hat{z}_{ij} > 0 \\\\    \n                                 0\\ \\  \\hat{z}_{ij} \\le 0 \n                              \\end{cases} \\label{eq:AP_1} \\\\\n        AP_i(\\textbf{x}) = [ AP_{i,1}(\\textbf{x}), ..., AP_{i,k_i}(\\textbf{x})  ]^\\top \\\\\n        AP(\\textbf{x}) = (AP_1(\\textbf{x}), ..., AP_{n-1}(\\textbf{x})). \\label{eq:AP}\n    \\end{align}\n\\end{subequations}\nFor a network with $N$ hidden neurons there are $2^N$ possible combinations of neuron activations, however not all are realizable by the network. Practically, the number of activation patterns is better approximated by $N^{k_0}$ \\cite{hanin2019deep}.\n\nTo construct an explicit PWA representation we first want to characterize the set of inputs $c$ for which the network reduces to a single affine map. For a fixed activation pattern the ReLU network simplifies to the affine map $\\textbf{C} \\textbf{x} + \\textbf{d}$ where\n\\begin{subequations}\n    \\begin{align}\n        \\begin{bmatrix}\n            \\textbf{C} & \\textbf{d}\n        \\end{bmatrix} = \n        \\tilde{\\textbf{W}}_n \\prod_{i=1}^{n-1} \\tilde{\\textbf{W}}^c_i \\label{eq:linear} \\\\\n        \\tilde{\\textbf{W}}^c_i(\\textbf{x}) = diag(AP_{i}(\\textbf{x}))\\tilde{\\textbf{W}}_i, \\label{eq:diag}\n    \\end{align}\n\\end{subequations}\nand $diag(\\cdot)$ diagonalizes a vector to a matrix. Equation (\\ref{eq:diag}) equivalently sets row $j$ of $\\tilde{\\textbf{W}}_i$ to zero if $AP_{ij} = 0$.\n\nThe activation pattern only changes if some $\\hat{z}_{ij}$ switches from positive to nonpositive (or vice-versa), $c$ is thus given by a set of linear constraints, one for each neuron in the network.  The constraint for neuron $j$ in layer $i$ is\n\\begin{align}\n    \\begin{cases}\n          \\textbf{a}_{ij}^{\\top} \\textbf{x} \\ge b_{ij}\\ \\ \\text{if}\\  AP_{ij} = 1 \\\\    \n          \\textbf{a}_{ij}^{\\top} \\textbf{x} \\le b_{ij}\\ \\ \\text{if}\\  AP_{ij} = 0\n      \\end{cases} \\label{eq:constraints}\n\\end{align}\nwhere\n\\begin{align}\n    \\begin{bmatrix}\n        \\textbf{a}_{ij} & -b_{ij}\n    \\end{bmatrix} =  \n    \\tilde{\\textbf{W}}_i[j,:] \\prod_{l=1}^{i-1} \\tilde{\\textbf{W}}_l, \\label{eq:ab}\n\\end{align}\nand $\\tilde{\\textbf{W}}_i[j,:]$ denotes the $j^{th}$ row of $\\tilde{\\textbf{W}}_i$. Equation (\\ref{eq:ab}) is similar to (\\ref{eq:linear}) but instead simply gives the affine map parameters for a single neuron output. Strict inequalities are not present in (\\ref{eq:constraints}) because the affine map $\\textbf{Cx} + \\textbf{d}$ holds over the interior and boundary of the set where the activation pattern is constant. Since $c$ is an intersection of halfspaces, the set is a polyhedron. A similar formulation is given in \\cite{lei2020analytic} for finding the surface of an object for graphics rendering. Note that it is not uncommon for neurons to have the degenerate linear constraint $\\textbf{0}^{\\top} \\textbf{x} \\le 0$ which is satisfied for all $\\textbf{x} \\in c$, and also for multiple different neurons in the network to have equivalent constraints (perhaps scaled arbitrarily). To address these edge cases we introduce Definitions \\ref{def: hrep}, \\ref{def: duplicate}, \\ref{def: redundant} before describing our fast marching method below.  \n\n\\begin{definition}[H-representation]\nThe halfspace representation (H-representation) of a polyhedron is  \n\\begin{align}\n    P_H = \\{\\textbf{x} \\in \\mathbb{R}^d \\vert \\textbf{A} \\textbf{x} \\le \\textbf{b} \\} \\label{eq:hrep}\n\\end{align}\nwhere $\\textbf{A}$ an $m \\times d$ matrix and $\\textbf{b}$ an $m$-dimensional vector. Equality constraints may also be included if the set $P_H$ is of lower dimension than the ambient dimension. \\label{def: hrep}\n\\end{definition}\n\n\\begin{definition}[Duplicate Constraints]\nA constraint $\\textbf{a}_j^{\\top} \\textbf{x} \\le b_j$ is duplicate if there exists a scalar $\\alpha$ and a prior constraint $\\textbf{a}_i^{\\top} \\textbf{x} \\le b_i$ where $i<j$ such that $\\alpha [\\textbf{a}_j\\ b_j] = [\\textbf{a}_i\\ b_i]$ where $\\alpha > 0$. \\label{def: duplicate}\n\\end{definition}\n\\begin{definition}[Redundant \\& Essential Constraints]\nA constraint $\\textbf{a}_i^{\\top} \\textbf{x} \\le b_i$ is redundant if the feasible set does not change upon its removal or if it is a duplicate constraint. An essential constraint is one which is not redundant. \\label{def: redundant}\n\\end{definition}\n\nWe use (\\ref{eq:constraints}) to construct an H-representation of a cell $c$ by enumerating all constraints associated with hidden layer neurons.  The resulting $\\textbf{a}_{ij}$ vectors and $b_{ij}$ scalars are collected into ($\\textbf{A}$, $\\textbf{b}$) to give the desired H-representation from (\\ref{eq:hrep}).  For an arbitrary ReLU network in an arbitrary activation pattern, we find that the resulting H-representation often has redundant constraints, which need to be removed to give a minimal representation of the polyhedron.  \n\n\n\n\\subsection{Finding Essential Constraints} \\label{Essential}\nWe first normalize all constraints, remove any duplicates, and consider the resulting  H-representation $\\textbf{A}\\textbf{x} \\le \\textbf{b}$. To determine if the $i$th constraint is essential or redundant, define a new set of constraints with the $i$th constraint removed,\n\\begin{subequations}\n    \\begin{align}\n        \\tilde{\\textbf{A}} = [\\textbf{a}_1\\ ...\\ \\textbf{a}_{i-1}\\ \\textbf{a}_{i+1}\\ ...\\  \\textbf{a}_m]^\\top \\\\\n         \\tilde{\\textbf{b}} = [b_1\\ ...\\ b_{i-1}\\ b_{i+1}\\ ...\\  b_m]^\\top,\n    \\end{align}\n\\end{subequations}\n\nand solve the linear program\n\\begin{subequations}\n    \\begin{align}\n        \\max_{\\textbf{x}} \\quad & \\textbf{a}^{\\top}_i \\textbf{x}\\\\\n        \\textrm{subject to} \\quad & \\tilde{\\textbf{A}}\\textbf{x} \\le \\tilde{\\textbf{b}}.\n    \\end{align}\n    If the optimal objective value is less than or equal to $b_i$, constraint $i$ is redundant.\n\\end{subequations}\nIn the worst case, a single LP must be solved for each constraint to determine whether it is essential or redundant. However, heuristics exist to avoid this worst case complexity. We use the bounding box heuristic to quickly find a subset of redundant constraints \\cite{morari}. Empirically, this results in identifying about $90\\%$ of the redundant constraints. We find that other heuristics beyond the bounding box method do not improve performance. Our implementation uses the GLPK open-source LP solver and the JuMP optimization package \\cite{DunningHuchetteLubin2017}.\n\n\n\\subsection{Determining Neighboring Activation Patterns} \\label{Neighbors}\nGiven a cell $c$ in the input space of a ReLU network, we are interested in finding the activation pattern corresponding to a neighboring cell, $c'$. We can then determine the H-representation of $c'$ using the procedure outlined above. The challenge in finding the activation pattern for $c'$ is in determining which individual neuron activations ($AP_{ij}$ for some $ij$) must be flipped when $\\textbf{x}$ transitions from $c$ to $c'$. \n\nThe boundary between $c$ and $c'$ ($c \\cap c'$) is defined by a linear constraint of $c$. We refer to this as the \\textit{neighbor constraint}. Intuitively, to generate $AP^{c'}$ one ought to simply flip the activation of the neuron defining this neighbor constraint (and all neurons defining duplicate constraints). However, this intuitive procedure is incomplete because it does not correctly deal with degenerate constraints $\\textbf{0}^{\\top} \\textbf{x} \\le 0 \\ \\forall \\textbf{x} \\in c$, which are common in practice.  A degenerate constraint holds everywhere over $c$, but it may turn into a non-degenerate constraint for $c'$ due to a neuron activation in an earlier layer being flipped when moving to cell $c'$. Conversely, a non-degenerate constraint in $c$ may turn into a degenerate one when passing to $c'$.  To define a correct procedure for flipping neuron activations we introduce a taxonomy for classifying neurons based on their associated linear constraints.\n\n\\begin{definition}[Type 1, 2, 3 neurons] \\label{types}\nWhen identifying a neighboring cell $c'$ from a current cell $c$, neurons whose linear constraints in $c$ define $c \\cap c'$ are called \\textit{Type 1}. neurons with degenerate linear constraints in $c$, $\\textbf{0}^{\\top} \\textbf{x} \\le 0 \\ \\forall \\textbf{x} \\in c$, are called \\textit{Type 2}. All other neurons are \\textit{Type 3}. \n\\end{definition}\nConstraints labeled by type are illustrated in Figure \\ref{fig:constraints}a. \n\nThough not stated explicitly, the linear constraint parameters from (\\ref{eq:constraints}) are (nonlinear) functions of the input. Given Definition \\ref{types}, for inputs $\\textbf{x} \\in c \\cap c'$,\n\\begin{subequations}\n    \\begin{align}\n        \\textbf{a}_{ij}^{\\top}(\\textbf{x}) \\textbf{x} = b_{ij}(\\textbf{x})\\ \\ \\text{if}\\  ij \\in \\text{Type 1} \\\\\n        \\textbf{a}_{ij}^{\\top}(\\textbf{x}) \\textbf{x} = b_{ij}(\\textbf{x})\\ \\ \\text{if}\\  ij \\in \\text{Type 2} \\\\\n        \\begin{cases}\n            \\textbf{a}_{ij}^{\\top}(\\textbf{x}) \\textbf{x} > b_{ij}(\\textbf{x})\\ \\ \\text{if}\\  ij \\in \\text{Type 3}\\ \\text{and}\\ AP_{ij} = 1 \\\\\n            \\textbf{a}_{ij}^{\\top}(\\textbf{x}) \\textbf{x} < b_{ij}(\\textbf{x})\\ \\ \\text{if}\\  ij \\in \\text{Type 3}\\ \\text{and}\\ AP_{ij} = 0\n        \\end{cases}. \\label{eq:T3}\n    \\end{align}\n\\end{subequations}\nThe function $\\textbf{a}_{ij}^{\\top}(\\textbf{x}) \\textbf{x} - b_{ij}(\\textbf{x})$ is continuous because it is simply the function for computing the preactivation value $\\hat{z}_{ij}$. From this continuity, we know that the strict inequality of (\\ref{eq:T3}) holds for all inputs in $c'$ as well. Therefore, the activation of a Type 3 neuron is not flipped. \n\n\\begin{figure}\n     \\centering\n     \\includegraphics[width=0.95\\columnwidth]{constraints_smaller.png}\n     \\caption{(a) Example $c$ and $c'$ with Type 1 and Type 3 constraints labeled. For (b)-(e), regions in green denote where the nonlinear constraint is strictly satisfied, regions in blue denote where the nonlinear constraint function evaluates to zero, and regions in red denote where the nonlinear constraint is violated. (b) Type 1 constraint where the neuron activation must flip when going from $c$ to $c'$. (c) Type 1 constraint where the neuron activation is set to zero when going from $c$ to $c'$. (d) Type 2 constraint where the neuron activation must flip when going from $c$ to $c'$. (e) Type 2 constraint where the neuron activation remains zero when going from $c$ to $c'$.}\n     \\label{fig:constraints}\n\\end{figure}\n\n \nConversely, inputs in $c'$ may be either feasible or infeasible for Type 1 and Type 2 constraints. Figure \\ref{fig:constraints}b-c illustrates the possible feasible and infeasible regions of a Type 1 constraint. Figure \\ref{fig:constraints}d-e illustrates the possible feasible and infeasible regions of a Type 2 constraint. It follows that the procedure to generate a neighboring activation pattern is to apply (\\ref{eq:constraints}) layer-by-layer updating the activation pattern in place, where Type 1 and Type 2 neuron activations are set to zero if their new linear constraints are $\\textbf{0}^{\\top} \\textbf{x} \\le 0$, otherwise, they are flipped. This procedure is formally defined in Algorithm \\ref{Algo:neighbor_ap}.  We note that applying this neuron flipping algorithm to identify a neighboring cell $c'$ is exceedingly fast, even for networks with 10,000s of neurons, as we only require a single logical check at each neuron. \n\\begin{algorithm}[!h]\n\\SetAlgoLined\n\\DontPrintSemicolon\n \\KwIn{$AP^{c}$, $Type\\ 1$, $Type\\ 2$,  $(W_0, \\ldots, W_N$)}\n \\KwOut{$AP^{c'}$}\n $AP^{c'} \\leftarrow  AP^{c}$ \\algorithmiccomment{Initialize new AP as old AP} \\\\\n \n\\For(\\algorithmiccomment{For each neuron}){$i \\in [1,...,n-1]$, $j \\in [1,...k_i]$}{ \n\\If{$ij \\in Type\\ 1\\ or\\ ij \\in Type\\ 2$}{\n$\\textbf{a}_{ij},\\ b_{ij} \\leftarrow$ Equation \\ref{eq:ab} \\algorithmiccomment{$AP^{c'}_{ij}$ hyperplane} \\\\\n\\uIf{$[\\textbf{a}_{ij}\\ b_{ij}] = \\textbf{0}$}{$AP^{c'}_{ij} \\leftarrow 0$}\n\\Else{$AP^{c'}_{ij} \\leftarrow \\neg AP^{c'}_{ij} $}\n}\n}\n \n\\Return{$AP^{c'}$}\n\\caption{Neighboring Activation Pattern}\n\\label{Algo:neighbor_ap}\n\\end{algorithm}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Reachable Polyhedral Marching} \\label{Cell Enumeration}\nFrom (\\ref{eq:constraints}) and Algorithm \\ref{Algo:neighbor_ap}, we define our main algorithm, Reachable Polyhedral Marching (RPM) in Algorithm \\ref{Algo:cell_enumeration} for explicitly enumerating all polyhedral cells in the input space of a ReLU network, resulting in the explicit PWA representation. First, we start with an initial point in the input space and evaluate the network to find the activation pattern. From this the H-representation is found using (\\ref{eq:constraints}). Essential constraints are then identified as described in Section \\ref{Essential}. For each essential constraint we generate a neighboring activation pattern using Algorithm \\ref{Algo:neighbor_ap}. The process then repeats with each new neighboring activation pattern being added to a working set. A neighbor activation pattern only gets added to the working set if it has not already been visited and it is not already in the working set. For a given starting cell each neighbor cell is enumerated, and since each is connected to another, Algorithm \\ref{Algo:cell_enumeration} is guaranteed to enumerate every cell in the input space. Figure \\ref{fig:cell_enum} shows the result of applying Algorithm \\ref{Algo:cell_enumeration} to a randomly initialized ReLU network.\n\\begin{algorithm}\n\\SetAlgoLined\n\\DontPrintSemicolon\n \\KwIn{$AP^{c_0}$, $(W_0, \\ldots, W_N$)}\n \\KwOut{Explicit PWA representation}\n $input\\ cells$ = $\\emptyset$;  $visited\\ cells$ = $\\emptyset$ \\\\\n $working\\ set$ = \\{$AP^{c_0}$\\} \\\\\n \\While{$working\\ set \\neq \\emptyset$}{\n $AP^{c} \\leftarrow$ pop element off of $working\\ set$ \\\\\n $\\textbf{C}^{c},\\ \\textbf{d}^{c} \\leftarrow$ Equation \\ref{eq:linear} \\algorithmiccomment{Retrieve affine map} \\\\\n $H^{c}_{rep} \\leftarrow$ Equation \\ref{eq:constraints} \\algorithmiccomment{Retrieve H-representation} \\\\\n $H^{c}_{rep} \\leftarrow$ remove redundant $H^{c}_{rep}$ constraints \\\\\n push ($\\textbf{C}^{c},\\ \\textbf{d}^{c},\\ H^{c}_{rep}$) onto $input\\ cells$ \\\\\n \\For{$\\textbf{a}_k, b_k \\in H^{c}_{rep}$}{\n $AP^{c'} \\leftarrow$ Algorithm \\ref{Algo:neighbor_ap} \\\\\n \\If{$AP^{c'} \\notin visited\\ cells \\cup working\\ set$}{\n push $AP^{c'}$ onto $working\\ set$ \\\\\n }\n }\n push $AP^{c}$ onto $visited\\ cells$ \\\\\n }\n \\Return{input cells}\n \\caption{Reachable Polyhedral Marching}\n \\label{Algo:cell_enumeration}\n\\end{algorithm}\n\n\n\\section{Reachability} \\label{reach}\n\\subsection{Forward Reachability} \\label{Forward}\nThe forward reachable set of a PWA function over some input set is simply the union of forward reachable sets of each individual polyhedron under its associated affine map. The image of a polyhedron under an affine map is\n\\begin{align}\n    P_{out} = \\{\\textbf{y} \\vert \\textbf{y}=\\textbf{Cx}+\\textbf{d}, \\textbf{Ax} \\le \\textbf{b}\\}. \\label{eq:forward reach}\n\\end{align}\nFor $\\textbf{C}$ invertible, the H-representation of the image is\n\\begin{align}\n    P_{out} = \\{\\textbf{y} \\vert \\textbf{AC}^{-1}\\textbf{y} \\le \\textbf{b} + \\textbf{AC}^{-1} \\textbf{d}\\}. \\label{eq:image}\n\\end{align}\nIn the case the affine map is not invertible, more general polyhedral projection methods such as block elimination, Fourier-Motzkin elimination, or parametric linear programming can compute the H-representation of the image \\cite{cdd_manual, mpLP}. Our implementation uses the block elimination projection in the case of a non-invertible affine map \\cite{cdd}.\n\nOur RPM algorithm is used to perform forward reachability as follows. We first specify a polyhedral input set whose image through the ReLU network we want to compute.  This is a set over which to perform the RPM algorithm. For each activation pattern $AP^{c}$ we also compute the image of $H^{c}_{rep}$ under the map $\\textbf{C}^{c}\\textbf{x} + \\textbf{d}^{c}$. To do this, an additional line is introduced between lines 9 and 10 of Algorithm \\ref{Algo:cell_enumeration}\n\\begin{align}\n    H^{c}_{rep,\\ forward} \\leftarrow project(H^{c}_{rep}, \\textbf{C}^{c}, \\textbf{d}^{c})\n\\end{align}\nwhere $project(\\cdot, \\cdot, \\cdot)$ applies (\\ref{eq:image}) if $\\textbf{C}^{c}$ invertible and block elimination otherwise.\n\n\n\\subsection{Backward Reachability} \\label{Backward}\nThe preimage of a polyhedron under an affine map is\n\\begin{subequations}\n    \\begin{align}\n        P_{in} = \\{\\textbf{x} \\vert \\textbf{y}=\\textbf{Cx}+\\textbf{d}, \\textbf{Ay} \\le \\textbf{b}\\} \\\\\n        P_{in} = \\{\\textbf{x} \\vert \\textbf{ACx} \\le \\textbf{b}-\\textbf{Ad}\\}. \\label{eq:backward reach}\n    \\end{align}\n\\end{subequations}\nLike forward reachability, performing backward reachability only requires a small modification to Algorithm \\ref{Algo:cell_enumeration}. We first specify a polyhedral output set whose preimage we would like to compute.  For each activation pattern $AP^{c}$, we also compute the intersection of $H^{c}_{rep}$ with the preimage of the given output set under the map $\\textbf{C}^{c}\\textbf{x} + \\textbf{d}^{c}$. Two additional lines are thus introduced between lines 9 and 10 of Algorithm \\ref{Algo:cell_enumeration}\n\\begin{subequations}\n    \\begin{align}\n    P^{c}_{in} \\leftarrow \\text{Equation}\\ \\ref{eq:backward reach} \\label{eq:p_in} \\\\\n    H^{c}_{rep,\\ backward} \\leftarrow H^{c}_{rep} \\cap P^{c}_{in}. \\label{eq:h_back}\n    \\end{align}\n\\end{subequations}\nMultiple backward reachable sets can be solved for simultaneously at the added cost of repeating (\\ref{eq:p_in}) and (\\ref{eq:h_back}) for each output set argument to Algorithm \\ref{Algo:cell_enumeration}.\n\nFinally, we address the issue of finding forward and backward reachable sets iterated over multiple time steps.  For this, we note that a ReLU network that is applied iteratively over $T$ timesteps, $\\textbf{x}_{t+1} = F(\\textbf{x}_t)$ for $t = 0, \\ldots, T-1$, is mathematically equivalent to a single ReLU network consisting of $T$ copies of the original network concatenated end to end, $\\textbf{x}_T = F_T(\\textbf{x}_0) := F \\circ \\cdots \\circ F(\\textbf{x}_0)$.  If the original network $F(\\textbf{x})$ has $N$ neurons and $L$ layers, the equivalent multi-time step network $F_T(\\textbf{x})$ has $TN$ neurons and $TL$ layers.  We simply perform RPM for forward or backward reachability on the equivalent multi-time step network $F_T(\\textbf{x})$.\n\n\\section{Examples} \\label{Examples}\nAll examples are run on a 2013 Dell Latitude E6430s laptop with Intel Core i7 3GHz processor and 16GB of RAM. The cell coloring in plots is random.\n\\subsection{Damped Pendulum Example} \\label{Pendulum}\n\nThe forward and backward reachability algorithms can be applied to discrete-time dynamical systems represented as ReLU networks. In this example we analyze a ReLU network that approximates the discrete-time dynamics of a damped pendulum. The function learned by the network is \n\\begin{align}\n    \\bm{x}_{t+1} = f_{nn}(\\bm{x}_t)\n    \\label{pendulum}\n\\end{align}\nwhere $\\bm{x} = [\\theta,  \\dot{\\theta}]^\\top$. The time-step used is $0.1$ seconds. The learned dynamics function $f_{nn}$ is a ReLU network with a single hidden layer of 12 neurons. An example trajectory of the learned system is shown in Figure \\ref{fig:damped}. We can compose multiple copies of $f_{nn}$ together to get a neural network which outputs an arbitrary future state. For instance, composing the network with itself 50 times results in a final network with 600 neurons and output $\\bm{x}_{t+50}$. \n\n\\begin{figure}[!h]\n    \\centering\n    \\includegraphics[width = 0.6\\columnwidth]{damped_sine_f.png}\n    \\caption{50 time-step trajectory from $\\bm{x}_0 = [30, 0]^\\top$}\n    \\label{fig:damped}\n\\end{figure}\n\n\\begin{figure*}\n     \\centering\n     \\begin{subfigure}[b]{0.95\\columnwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{input_50_f.png}\n         \\caption{Input cells.}\n         \\label{fig:pend_in}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.95\\columnwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{output_50_f.png}\n         \\caption{Output cells.}\n         \\label{fig:pend_out}\n     \\end{subfigure}\n        \\caption{(a) Input cells of the 50 time-step pendulum network. (b) image of input cells under the neural network map.}\n        \\label{fig:forward}\n\\end{figure*}\n\nFigure \\ref{fig:damped} suggests that states of the learned system tend toward the origin. We can use the forward reachability algorithm to determine where \\textit{all} states within some set of initial conditions lead to over some time interval. In Figure \\ref{fig:pend_in} we show the result of applying the cell enumeration algorithm and the forward reachability algorithm to the 50 time step system for initial conditions $-90 \\degree \\le \\theta \\le 90 \\degree$ and $-90 \\degree \/ s \\le \\dot{\\theta} \\le 90 \\degree \/ s$. The resulting number of regions is 2781 and the forward reachability version of the RPM algorithm took 87s to complete. It is clear from Figure \\ref{fig:pend_in} that all initial conditions are mapped to a subset of the initial input set, proving that the original input set is forward invariant over 50 time step increments, and indicating that trajectories tend to approach the origin over time.\n\nFurther, we perform backward reachability to find the set of inputs that map to a small neighborhood around the origin after 50 time steps. Figure \\ref{fig:back_reach} shows the inputs that map to the output set $-5 \\degree \\le \\theta \\le 5 \\degree$ and $-2 \\degree \/ s \\le \\dot{\\theta} \\le 2 \\degree \/ s$ after 50 time steps. Interestingly, only initial conditions with positive angles map to the target set. In the true dynamics, the initial angles would be symmetric about zero. This analysis can thus help us identify undesirable modeling artifacts to inform retraining or control of the model.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.9\\columnwidth]{back_reach_f.png}\n    \\caption{Set of initial states that lead to a neighborhood around the origin after 50 time steps for the pendulum network.}\n    \\label{fig:back_reach}\n\\end{figure}\n\n\n\\subsection{Aircraft Collision Avoidance Example} \\label{ACAS}\nThe ACAS Xu networks are 45 distinct policy networks designed to issue advisory warnings to avoid mid-air collisions for unmanned aircarft. Each network has five inputs, five outputs (advisories), and 300 neurons. Inputs are relative distance, angles, and speeds. See \\cite{acas} for more details. \n\nThe appendix of \\cite{reluplex} lists ten safety properties the networks should satisfy. Here we consider Property 3. This property is satisfied if the network never outputs a ``Clear of Conflict\" advisory when the intruder is directly ahead and moving towards the ownship. We use the backward reachability algorithm to verify whether Property 3 is satisfied. A nonempty backward reachable set implies that some allowable inputs will map to unsafe outputs and the safety property is not satisfied. Table \\ref{table:verification} shows the results of verifying Property 3 for four of the networks. A comparison is given against the fastest existing exact reachability method based on the face lattice representation \\cite{yang2020reachability} as well as the Reluplex and Marabou verification algorithms \\cite{reluplex,marabou}. Each algorithm was run on a single core. We note that a parallelized algorithm is also provided for the face lattice approach \\cite{yang2020reachability}, and our RPM algorithm is also amenable to a parallelized implementation to be explored in future work. \n\nThe results in Table \\ref{table:verification} show the advantage of our proposed method when verifying network properties that are found to not hold. Our RPM algorithm is anytime, in the sense that once an unsafe input polyhedron is found, the algorithm can terminate without completing the full reachability computation. This is in contrast to all existing exact reachability methods that proceed layer-by-layer through the network.  They must solve the entire backward reachability problem to conclude any verification result, whether safe or unsafe.\n\n\\begin{table}\n\\centering\n\\caption{ACAS Xu Property 3 Verification Results}\n \\begin{tabular}{|c| c| c| c| c| c|} \n \\hline\n Network & Result & \\begin{tabular}{@{}c@{}}Time (s) \\\\ RPM\\end{tabular} & \\begin{tabular}{@{}c@{}}Time (s) \\\\ Face Lattice\\end{tabular} & \\begin{tabular}{@{}c@{}}Time (s) \\\\ Reluplex\\end{tabular} & \\begin{tabular}{@{}c@{}}Time (s) \\\\ Marabou\\end{tabular} \\\\ [0.5ex] \n \\hline\n $N_{1,7}$ & Unsafe & \\textbf{0.01} & 6.66 & 2.15 & 0.70 \\\\ \n \\hline\n $N_{1,8}$ & Unsafe & \\textbf{0.01} & 5.45 & 4.32 & 1.49 \\\\\n \\hline\n $N_{3,8}$ & Safe & 19.72 & \\textbf{9.35} & 231.28 & 40.13 \\\\\n \\hline\n $N_{5,6}$ & Safe & 31.92 & \\textbf{17.28} & 366.39 & 54.07 \\\\\n \\hline\n\\end{tabular}\n\\label{table:verification}\n\\end{table}\n\n\n\\section{Conclusion}\nWe proposed the Reachable Polyhedral Marching \n(RPM) algorithm to efficiently construct the exact PWA representation of a ReLU network. RPM computes an explicit PWA function representation for a given ReLU network.  This PWA function can then be used to quickly find forward and backward reachable sets over multiple time steps. Solving for multiple backward reachable sets can be done simultaneously at little added cost. RPM is shown to be especially fast when searching for the existence of unsafe inputs during verification.  In the future, we will investigate parallel implementations of RPM to further improve computational speed.\n\n                                 \n                                 \n                                 \n                                 \n                                 \n\n\n\n\n\\bibliographystyle{.\/IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Model  Hamiltonian}\nThere  is a range of values of the strength of the confinement potential\nfor which spontaneous interlayer phase coherence\nis  stable in double quantum dots\\cite{hu}. Outside this stability range\nthe shape of the groundstate is expected to undergo a  reconstruction\\cite{yang0}.\nIt is the\ncompetition between the confinement potential and the\nCoulomb interaction which drives such a reconstruction.  When the confinement\npotential weakens the repulsive Coulomb interaction\npushes electrons from each other, and can create  density depletions inside the\ndroplet. \n\nThe interplay between the reconstruction and the\nformation of merons in the presence of an external potential is\nunclear.  \nWe investigated within a Hartree-Fock (HF) theory how merons\nnucleate in the groundstate when a lateral distortion with a\nreflection symmetry is present  in the confinement potential\\cite{yang0}.  We found that when a sufficiently large distortion\nis present merons can be nucleated.\nBoth density depletions and merons can coexist in the reconstructed states.\nIn the present work we \ndemonstrate by integrating the electron density that these merons have total charge $\\pm1\/2$.\nIn addition we \npresent  numerical evidence for the presence of edge merons, which\nappear to have a half-integer  winding number.\n\n\nAn electron in each dot is in a parabolic potential $\\frac{1}{2}m^*\\Omega^2r^2$.\nIn the strong magnetic field limit the energy of an  electron with the angular momentum $m$ in a  quantum dot is\n$\\epsilon_m=\\frac{\\hbar\\omega_c}{2}+\\hbar\\omega_p(m+1)$, where\n$\\omega_c$ is the cyclotron frequency, $\\hbar\\omega_p=m^*\n\\Omega^2 \\ell^2$, and $\\ell$ is the magnetic length.\nHereafter we will measure $\\hbar\\omega_p$ in units of\n$e^2\/\\varepsilon\\ell$, using\n$\\gamma=\\hbar\\omega_p\/(e^2\/\\varepsilon\\ell)$ which is a dimensionless quantity proportional to\nthe confinement potential strength  (For GaAs $\\gamma=0.131 (\\hbar\\omega_p[meV])^2\/(B[T])^{3\/2}$). The\nreal spins of all electrons are assumed to be frozen aligning with\nthe magnetic field.\nWe consider \na   distortion in the confinement potential of the dot that may be generated by attaching a\nmetallic stripe vertically on the  cylindrical  metallic gate\nsurrounding the vertical quantum dots. The vertical distance between two dots is $d$. \nLet us make a simple model for this  type of distortions.\nWe take a separable form for the distortion potential\n$U_{\\textrm{dis}}(\\vec{r},z)=v g (\\vec{r}) h (z)$, where $v$ is the strength of the distortion.\nThe matrix elements of this potential are\n\\begin{eqnarray}\n\\left<n_1 \\tau_1|U_{\\mathrm{dis}}|n_2 \\tau_2 \\right> = \n\\left(u_{s} \\delta_{\\tau_1 \\tau_2}\\!+\\! u_{d} (1-\\delta_{\\tau_1\n\\tau_2})\\right)<n_1|g (\\vec{r})|n_2>,\n\\end{eqnarray}\nwhere  $\\left|n_j \\tau_j \\right>=c_{n_j \\tau_j}^{\\dagger} \\left|0\\right>$ denotes an electron with the \nangular momentum $n_j$ in the dot labeled by $\\tau_j$.\nThe parameters $u_{s}$ and $u_{d}$ are defined as \n\\begin{eqnarray}\nu_{s} &=& v \\int \\mathrm{d}z h(z) \\left|\n\\varphi\\left(z-\\frac{d}{2}\\right) \\right|^2 \\; ,\\\\\nu_{d} &=& v \\int \\mathrm{d}z h(z) \\varphi^*\\left(z-\\frac{d}{2}\\right) \\varphi \\left(z+\\frac{d}{2}\\right).\n\\end{eqnarray}\nThe parameter $u_{s}$ ($u_{d}$) denotes the strength  of the matrix elements of the disorder\npotential between intra (inter) layer electrons.\nThe wavefunctions\n$\\varphi \\left(z+\\frac{d}{2}\\right)$  and $\\varphi\\left(z-\\frac{d}{2}\\right)$ are, respectively,\nthe left and right dot wavefunctions along the vertical direction.\n\n\nThe total Hamiltonian is $H = H_0 +V$, where\n\\begin{eqnarray}\nH_0 \\!&=&\\! \\sum_{m\\sigma} \\epsilon_m c_{m \\sigma}^\\dagger c_{m\n\\sigma} +\\sum_{m\\sigma}  \\sum_{m'\\sigma'}  \\left< m \\sigma|\nU_{\\mathrm{dis}}|m' \\sigma'\\right> c_{m \\sigma}^\\dagger c_{m'\n\\sigma'},\n\\end{eqnarray}\nand \n\\begin{eqnarray}\nV &=& \\frac{1}{2}\\sum_{m_1 m_2 m_3 m_4} \\sum_{\\sigma=\\uparrow \\downarrow} \n\\left( \\left< m_1 m_2 |V_\\mathrm{s}| m_3 m_4 \\right>\nc_{m_1 \\sigma}^\\dagger c_{m_2 \\sigma}^\\dagger c_{m_3 \\sigma} c_{m_4 \\sigma} \\right. \\nonumber \\\\\n&& \\left. + \\left< m_1 m_2 |V_\\mathrm {d} | m_3 m_4 \\right>\nc_{m_1 \\sigma}^\\dagger c_{m_2,-\\sigma}^\\dagger c_{m_3,-\\sigma}\nc_{m_4 \\sigma} \\right).\n\\end{eqnarray}\nThe first term of $H_0$ represents the confining energy due to the\nparabolic potential. In the second term, $U_{\\mathrm{dis}}$ is the\ndistortion potential. In the interaction\nHamiltonian $V$, the term $V_s$ ($V_d$) represents the Coulomb interaction\npotential between the electrons in the same (different) dots. In\nthe spatial coordinates, those interactions are given by\n$V_s(\\vec{r})=e^2\/\\varepsilon r$ and\n$V_d(\\vec{r})=e^2\/\\varepsilon|\\vec{r}+d\\hat{z}|$, respectively,\nwhere $\\varepsilon$ is the dielectric constant.\nThe spontaneous interlayer phase coherent state (maximum density droplet state\\cite{yang1,yang2}),\nand meron excitations  are  well approximated by a HF theory\\cite{moon}.\nThe pseudospin density is computed with the calculated  density\nmatrix.\n\n\n\n\n\n\n\n\\section{Results}\n\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width= 0.5 \\textwidth]{density3d.g_0.085.vi_0.50b.eps}\n\\includegraphics[width= 0.5 \\textwidth]{density3d.g_0.085.vi_0.60b.eps}\n\\caption{Electron densities in units of $1\/\\ell^2$ for  $(u_{s},u_{d})=(1.5,0.25)(e^2\/\\varepsilon\\ell)$ and\n$(u_{s},u_{d})=(1.8,0.3)(e^2\/\\varepsilon\\ell)$.}\n\\label{fig:density3d.g_0.085.vi_0.50}\n\\end{center}\n\\end{figure}\n\nFig. \\ref{fig:density3d.g_0.085.vi_0.50}\n displays total electron densities\nfor two different strengths of the lateral distortion with the\ncenter at the coordinate $(-5,5)\\ell$ by  a Gaussian lateral potential with the range $a=3\\ell$.\nResults for $\\gamma=0.085$ are shown for two different strengths\nof the distortion\n$(u_{s},u_{d})=(1.5,0.25),(1.8,0.3)(e^2\/\\varepsilon\\ell)$.\nThe maximum\ndensity is bigger than  $1\/2\\pi\\ell^2$, which implies that\nthe total local filling factor is bigger than one where the merons are located.  This is consistent\nwith the presence of merons with negative charges on the background of the total electron\ndensity $1\/2\\pi\\ell^2$.\nAt  $(u_{s},u_{d})=(1.5,0.25)(e^2\/\\varepsilon\\ell)$ a meron is formed\nwith the charge $-e\/2$ and vorticity $-$.\nFor the bigger value $(u_{s},u_{d})=(1.8,0.3)(e^2\/\\varepsilon\\ell)$\ntwo merons appear and the\nvorticities for the lower and upper merons are $-$ and $+$,\nrespectively. \nIntegrating the electron density  over the range that satisfy $n(r)>\\frac{1}{2 \\pi\\ell^2}$,\nwe  find the excess charges\nto be  \n$-0.54 e$ and $-0.98 e$,\nwhich indicates that one meron exists in the upper figure of Fig. \\ref{fig:density3d.g_0.085.vi_0.50}\nand a meron pair exists in the lower figure of Fig. \\ref{fig:density3d.g_0.085.vi_0.50}.\nWe have attempted to evaluate the total excess charge carried by the merons using the expression\\cite{moon} \n\\begin{eqnarray}\n\\Delta N = -\\frac{1}{8 \\pi} \\int {\\mathrm d}^2 \\bf{r} \\\n\\epsilon_{\\mu \\nu} \\bf{m}(\\bf{r}) \\cdot [\\partial_\\mu \\bf{m}(\\bf{r}) \\times \\partial_\\nu \\bf{m}(\\bf{r})],\n\\end{eqnarray}\nwhere the polarization vector $\\vec{m}(r)$ is  $\\vec{\\tau}(r)\/n(r)$ ($\\vec{\\tau}(r)=2\\vec{s}\/\\hbar$ and $\\vec{s}$ is the pseudospin).\nHowever, this method gave inaccurate results.   We believe this is because the expression is valid\nonly for slowly varying pseudospin fields.\n\n\n\\begin{figure}[!htp]\n\\begin{center}\n\\includegraphics[width=1.0  \\textwidth]{meronpair_one_imp.g_0.066.vi_0.08.hmf1.eps}\n\\caption{ Arrows are projections of  $\\vec{\\tau}$ on the $xy$ plane\nand the gray scale indicates the $z$-component of $\\vec{\\tau}$.\nThe dashed circle is the edge of the electron droplet and $+$ and $-$\nindicate the core positions of merons with positive and negative winding numbers.\nThe cross sign is where the impurity is located.\nThe black dot represents the center of an edge meron, shown in the inset, and\nthe small circles show electron density depletions.\n}\n\\label{fig:meronpair_one_imp.g_0.066.vi_0.08}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[!htp]\n\\begin{center}\n\\includegraphics[width=0.6  \\textwidth]{halfmeron.one.imp.g_0.066.vi_0.08.a.eps}\n\\caption{\nMagnified view of the spin texture of an edge meron with nearly a half-integer winding number.}\n\\label{fig:lfmeron.one.imp.g_0.066.vi_0.08}\n\\end{center}\n\\end{figure}\n\n\n\n\n\nFig.~\\ref{fig:meronpair_one_imp.g_0.066.vi_0.08} displays how the\npseudospin texture \nfor $\\gamma=0.066$ and $(u_{s},u_{d})=(0.24, 0.04)(e^2\/\\varepsilon\\ell)$.\nWe  observe merons inside the dots.\nThe values of the polarization $m_z(r)=\\tau_z(r)\/n(r)$ at the meron cores are, within numerical accuracy, $+1$ and $-1$.\nThis implies that the\ncharges should be $-e\/2$ since the vorticities of these two objects are opposite.  \nThe total charge of the merons is thus $-e$.\nNote that there are also five density depletions inside the dots (the open circles in the figure).  \nThese  modulations of electron densities are not simply\nrelated to the shape of\n$U_{\\mathrm{para}}(r)+U_{\\mathrm{dis}}(r)$. Since the densities are\ninhomogeneous the HF single particle potential must be included into\nconsideration. It is non-trivial to anticipate the shape of the HF\nsingle particle potential since it reflects the delicate interplay between the\nHartree self energy, exchange self energy, and confinement potential\\cite{yang3}.\n\nWe also observe  an  edge meron near the edge in Fig.~\\ref{fig:meronpair_one_imp.g_0.066.vi_0.08}.\nA magnified view of the pseudospin texture is shown in Fig.~\\ref{fig:lfmeron.one.imp.g_0.066.vi_0.08}. \nWe   draw a circle around the center of this object and see how the in-plane \npseudospins rotate along the circle.\nThe in-plane\npseudospins nearly flips as   the half circle is followed inside the dot.\nThe magnitude of in-plane pseudospin values decreases to zero as the edge region is passed.\nWe believe thus that the edge meron  has approximately a half integer winding number.\nIt is complicated to calculate accurately the excess charge of the edge meron since\nthe electron density varies strongly near the edge even in the absence of an edge meron.\nIt would be worthwhile to develop an analytical theory for edge merons.\n\n\n\n\\section*{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Geometry}\n     \\subsection{Arrangement}\n        \\subsubsection{Enumeration of all cells in arrangements}\n            \\begin{description}\n                \\item[Input] An arrangement of distinct hyperplanes.\n                \\item[Output] All cells in arrangements.\n                \\item[Complexity] $O(nm\\ell(n,m)N)$ total time and $O(nm)$ space.\n                \\item[Comment] $n$ is the dimension, $m$ is the number of hyperplanes, and $\\ell(n, m)$ is the time for solving an LP with $n$ variables and $m-1$ inequalities.  $N$ is the number of solutions.\n                \\item[Reference] \\cite{Avis1996}\n            \\end{description}\n     \\subsection{Face}\n        \\subsubsection{Enumeration of all arrangements}\n            \\begin{description}\n                \\item[Input] An integer $n$ and $\\mathcal{J} \\subseteq \\{+, -\\}^n$.\n                \\item[Output] The set $\\mathcal{F}$ of faces if $\\mathcal{J}$ is the set of maximal faces of an oriented matroid\n                \\item[Complexity] $O(\\max(n^2 |\\mathcal{J}|^3, n^3 |\\mathcal{J}|^2))$ total time.\n                \\item[Comment] Their paper treats a reconstruction of the location vectors of all faces from the graph.\n                \\item[Reference] \\cite{Fukuda1991}\n            \\end{description}\n     \\subsection{Matching}\n        \\subsubsection{Enumeration of all non-crossing perfect matchings (and convex partitions) in a given points}\n            \\begin{description}\n                \\item[Input] A point sets $P$ in a plane\n                \\item[Output] All non-crossing perfect matchings (and convex partitions) in $P$. \n                \\item[Complexity] After $O(2^nn^4)$ preprocessing time, polynomial delay. \n                \\item[Comment] $n$ is the number of points in $P$\n                \\item[Reference] \\cite{Wettstein2014}\n            \\end{description}\n     \\subsection{Nearest neighbor}\n        \\subsubsection{Enumeration of the $k$ smallest distances between pairs of points}\n            \\begin{description}\n                \\item[Input] $n$ points in a plane.\n                \\item[Output] The $k$ smallest distances between pairs of points in non-decreasing order.\n                \\item[Complexity] $O(n \\log n + k \\log k)$ total time and $O(n + k)$ space.\n                \\item[Reference] \\cite{Dickerson1992}\n            \\end{description}\n     \\subsection{Spanning tree}\n        \\subsubsection{Enumeration of the Euclidean $k$ best spanning trees in a plane with $n$ points}\n            \\begin{description}\n                \\item[Input] $n$ points in a plane\n                \\item[Output] The Euclidean $k$ best spanning trees of the given points.\n                \\item[Complexity] $O(k^2 n + n \\log n)$ total time\n                \\item[Comment] An Euclidean spanning tree in a plane is a spanning tree of the complete graph whose vertices are all given points and the weight of an edge equal to the Euclidean distance between the corresponding vertices.\n                \\item[Reference] \\cite{Eppstein1992}\n            \\end{description}\n        \\subsubsection{Enumeration of the orthogonal $k$ best spanning trees in a plane with $n$ points}\n            \\begin{description}\n                \\item[Input] $n$ points in a plane\n                \\item[Output] The orthogonal $k$ best spanning trees of the given points.\n                \\item[Complexity] $O(k^2 + k n \\log \\log (n\/k) + n \\log n)$ total time.\n                \\item[Comment] An orthogonal spanning tree in a plane is a spanning tree of the complete graph whose vertices are all given points and the weight of an edge equal to the $L_1$ distance between the corresponding vertices.\n                \\item[Reference] \\cite{Eppstein1992}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning trees of a plane}\n            \\begin{description}\n                \\item[Input] A point set $P$.\n                \\item[Output] All spanning trees of $P$.\n                \\item[Complexity] $O(|P|^3N)$ total time and $O(|P|)$ space.\n                \\item[Comment] $N$ is the number of solutions.\n                \\item[Reference] \\cite{Avis1996}\n            \\end{description}\n        \\subsubsection{Enumeration of the Euclidean $k$ best spanning trees in a plane with $n$ points}\n            \\begin{description}\n                \\item[Input] $n$ points in a plane\n                \\item[Output] The Euclidean $k$ best spanning trees of the given points.\n                \\item[Complexity] $O(n \\log n \\log k + k \\min(k, n)^{1\/2})$ total time\n                \\item[Comment] An Euclidean spanning tree in a plane is a spanning tree of the complete graph whose vertices are all given points and the weight of an edge equal to the Euclidean distance between the corresponding vertices.\n                \\item[Reference] \\cite{Eppstein1997}\n            \\end{description}\n     \\subsection{Triangulation}\n        \\subsubsection{Enumeration of all regular triangulations}\n            \\begin{description}\n                \\item[Input] $n$ points.\n                \\item[Output] All regular triangulations of given points in $(d-1)$-dimensional space.\n                \\item[Complexity] $O(dsLP(r, ds)T)$ time and $O(ds)$ space.\n                \\item[Comment] $s$ is the upper bound of the number of simplcies contained in one regular triangulation, $LP(r, ds)$ denotes the time complexity of solving linear programming problem with $ds$ strict inequality constraints in $r$ variables, and $T$ is the number of regular triangulations.\n                \\item[Reference] \\cite{Masada1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning regular triangulations}\n            \\begin{description}\n                \\item[Input] $n$ points.\n                \\item[Output] All spanning regular triangulations of given points in $(d-1)$-dimensional space.\n                \\item[Complexity] $O(dsLP(r, ds)T')$ time and $O(ds)$ space.\n                \\item[Comment] $s$ is the upper bound of the number of simplcies contained in one regular triangulation, $LP(r, ds)$ denotes the time complexity of solving linear programming problem with $ds$ strict inequality constraints in $r$ variables, and $T'$ is the number of regular triangulations.\n                \\item[Reference] \\cite{Masada1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all triangulations of a point set}\n            \\begin{description}\n                \\item[Input] A point set $P$.\n                \\item[Output] All triangulations of $P$.\n                \\item[Complexity] $O(|P|N)$ total time and $O(|P|)$ space.\n                \\item[Comment] $N$ is the number of solutions.\n                \\item[Reference] \\cite{Avis1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all triangulations in general dimensions}\n            \\begin{description}\n                \\item[Input] Points.\n                \\item[Output] All triangulations.\n                \\item[Complexity] See the paper.\n                \\item[Comment] This algorithm uses the enumeration algorithm for maximal independent sets.\n                \\item[Reference] \\cite{Takeuchi}\n            \\end{description}\n        \\subsubsection{Enumeration of all regular triangulations in general dimensions}\n            \\begin{description}\n                \\item[Input] Points.\n                \\item[Output] All regular triangulations.\n                \\item[Complexity] See the paper.\n                \\item[Comment] This algorithm uses the enumeration algorithm for maximal independent sets.\n                \\item[Reference] \\cite{Takeuchi}\n            \\end{description}\n        \\subsubsection{Enumeration of all triangulation paths of a point set}\n            \\begin{description}\n                \\item[Input] A point set $P$.\n                \\item[Output] All triangulation paths of $P$.\n                \\item[Complexity] $O(N|P|^3\\log |P|)$ time and $O(|P|)$ space.\n                \\item[Comment] $N$ is the number of solutions.\n                \\item[Reference] \\cite{Dumitrescu2001}\n            \\end{description}\n        \\subsubsection{Enumeration of all pseudotriangulations of a finite point set}\n            \\begin{description}\n                \\item[Input] A point set $S$ of size $n$.\n                \\item[Output] All pointed pseudotriangulations of $S$.\n                \\item[Complexity] $O(\\log n)$ time per solution with linear space.\n                \\item[Reference] \\cite{Bereg2005}\n            \\end{description}\n        \\subsubsection{Enumeration of all pseudotriangulations of a point set}\n            \\begin{description}\n                \\item[Input] A point set $P$.\n                \\item[Output] All pseudotriangulations of $P$.\n                \\item[Complexity] $O(n)$ time per pseudotriangulation.\n                \\item[Comment] $n$ is the number of points in $P$.\n                \\item[Reference] \\cite{Bronnimann2006a}\n            \\end{description}\n        \\subsubsection{Enumeration of all triangulations of a convex polygon of $n$ vertices with numbered}\n            \\begin{description}\n                \\item[Input] A convex polygon $P$ with $n$ vertices that are numbered.\n                \\item[Output] All triangulations of $P$.\n                \\item[Complexity] $O(1)$ time per triangulation and $O(n)$ space.\n                \\item[Reference] \\cite{Parvez2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all triangulations of a convex polygon of $n$ vertices}\n            \\begin{description}\n                \\item[Input] A convex polygon $P$ with $n$ vertices that are not numbered.\n                \\item[Output] All triangulations of $P$.\n                \\item[Complexity] $O(n^2)$ time per triangulation and $O(n)$ space.\n                \\item[Reference] \\cite{Parvez2009}\n            \\end{description}\n\\section{Graph}\n     \\subsection{$k$-degenerate graph}\n        \\subsubsection{Enumerate well-ordered (strongly) $k$-generate with $n$ vertices}\n            \\begin{description}\n                \\item[Input] $n$: the number of vertices.\n                \\item[Output] All well-ordered (strongly) $k$-degenerate graphs with $n$ vertices.\n                \\item[Complexity] $O(nm+m^2)$ time per enumerated and printed graph.\n                \\item[Comment] $m$ is the number of edges of printed graphs.\n                \\item[Reference] \\cite{Bauer2010a}\n            \\end{description}\n        \\subsubsection{Enumerate well-ordered (strongly) $k$-generate with $n$ vertices and $m$ edges}\n            \\begin{description}\n                \\item[Input] $n$: the number of vertices, $m$: the number of edges.\n                \\item[Output] All well-ordered (strongly) $k$-degenerate graphs with $n$ vertices and $m$ edges.\n                \\item[Complexity] $O(n^{3\/2}m^2)$ time per enumerated and printed graph.\n                \\item[Reference] \\cite{Bauer2010a}\n            \\end{description}\n     \\subsection{Bounded union}\n        \\subsubsection{Enumeration of all bounded union}\n            \\begin{description}\n                \\item[Input] A graph with $h$ multiedges each with at most $d$ vertices.\n                \\item[Output] All minimal subset $U$ of at most $k$ vertices that entirely includes at least one set from each multiedge.\n                \\item[Complexity] $O(dc^{k+1}h + \\min(kc^{2k}, hkc^k))$ time.\n                \\item[Reference] \\cite{Damaschke2006}\n            \\end{description}\n     \\subsection{Bridge}\n        \\subsubsection{Enumeration of all bridge-avoiding extensions of a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and an edge subset $B \\subseteq E$.\n                \\item[Output] All bridge-avoiding extensions of $G$.\n                \\item[Complexity] $O(K^2 \\log(K) |E|^2 + K^2(|V|+|E|)|E|^2)$ total time.\n                \\item[Comment] $K$ is the number of bridge-avoiding extensions.  $X$ is a bridge-avoiding extensions of $G$ if $X$ is a minimal edge set $X \\subseteq E \\setminus B$ such that no edge $b \\in B$ is a bridge in $(V, B\\cup X)$.\n                \\item[Reference] \\cite{Khachiyan2008a}\n            \\end{description}\n     \\subsection{Bubble}\n        \\subsubsection{Enumeration of all bubbles in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G=(V, E)$ and a source $s$.\n                \\item[Output] All bubbles with a given source $s$.\n                \\item[Complexity] $O(|V| + |E|)$ delay with $O(|V|(|V| + |E|))$ preprocessing time.\n                \\item[Comment] An $(s,t)$-bubble is two disjoint $(s,t)$-paths that only share $s$ and $t$.\n                \\item[Reference] \\cite{Birmele2012}\n            \\end{description}\n        \\subsubsection{Enumeration of all $(s, t, \\alpha_1, \\alpha_2)$-bubble in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$, source vertex $s$, and two upper bounds $\\alpha_1$ and $\\alpha_2$.\n                \\item[Output] All $(s, t, \\alpha_1, \\alpha_2)$-bubble in $G$.\n                \\item[Complexity] $O(|V|(|E|+|V|\\log |V|))$ delay.\n                \\item[Comment] A pair of two vertex-disjoint $(s, t)$-paths $p_1$ and $p_2$ is $(s, t, \\alpha_1, \\alpha_2)$-bubble in $G$, if $|p_1| \\le \\alpha_1$ and $|p_2| \\le \\alpha_2$.\n                \\item[Reference] \\cite{Sacomoto2013b}\n            \\end{description}\n     \\subsection{Chord diagram}\n        \\subsubsection{Enumeration of all non-isomorphic chord diagrams}\n            \\begin{description}\n                \\item[Input] An integer $2n$.\n                \\item[Output] All non-isomorphic chord diagrams with $2n$ points.\n                \\item[Comment] A \\textit{chord diagram} is a set of $2n$ points on an oriented circle (counterclockwise) joined pairwise by $n$ chords.  In an experiment, their algorithm runs in almost CAT, but there is no Mathematical proof.\n                \\item[Reference] \\cite{Sawada2002a}\n            \\end{description}\n     \\subsection{Chordal graph}\n        \\subsubsection{Enumeration of all chordal graphs in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All chordal graphs in $G$.\n                \\item[Complexity] $O(1)$ delay with $O(n^3)$ space.\n                \\item[Comment] Using reverse search.\n                \\item[Reference] \\cite{Kiyomi2006}\n            \\end{description}\n        \\subsubsection{Enumeration of all chordal supergraph that contains a given graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All chordal supergraphs each of which contain $G$.\n                \\item[Complexity] $O(|V|^3)$ time for each and $O(|V|^2)$ space.\n                \\item[Reference] \\cite{Kiyomi2006a}\n            \\end{description}\n        \\subsubsection{Enumeration of all chordal sandwiches in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All chordal sandwiches in $G$.\n                \\item[Complexity] Polynomial delay with polynomial space.\n                \\item[Reference] \\cite{Kijima2010}\n            \\end{description}\n     \\subsection{Clique}\n        \\subsubsection{Enumeration of all cliques in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All cliques in $G$. \n                \\item[Comment] They pointed out Augustson and Minker's algorithm has two errors. \n                \\item[Reference] \\cite{Mulligan1972}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal clique in a graph}\n            \\begin{description}\n                \\item[Input] An undirected graph $G=(V, E)$.\n                \\item[Output] All maximal clique.\n                \\item[Reference] \\cite{Akkoyunlu1973}\n            \\end{description}\n        \\subsubsection{Enumeration of all cliques in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All cliques in $G$.\n                \\item[Reference] \\cite{Bron1973}\n            \\end{description}\n        \\subsubsection{Enumeration of all cliques in an undirected graph}\n            \\begin{description}\n                \\item[Input] An undirected graph $G$. \n                \\item[Output] All cliques in $G$. \n                \\item[Reference] \\cite{Johnston1976}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal cliques in a given graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.  \n                \\item[Output] All maximal cliques in $G$.\n                \\item[Reference] \\cite{Gerhards1979}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximum cliques in a circle graph}\n            \\begin{description}\n                \\item[Input] A circle graph $G = (V, E)$.\n                \\item[Output] All maximum cliques in $G$.\n                \\item[Complexity] $O(1)$ time per maximum clique.\n                \\item[Comment] A graph is a circle graph if it is the intersection graph of chords in a circle. The definition of a circle graph can be found in Information System on Graph Classes and their Inclusions\\footnote{\\url{http:\/\/www.graphclasses.org\/classes\/gc\\_132.html}}.\n                \\item[Reference] \\cite{Rotem1981}\n            \\end{description}\n        \\subsubsection{Enumeration of all cliques in a connected graph}\n            \\begin{description}\n                \\item[Input] A connected graph $G = (V, E)$ and an order $l \\ge 2$ of cliques.\n                \\item[Output] All cliques in $G$.\n                \\item[Complexity] $O(l\\alpha(G)^{l-2}|E|)$ total time and linear space.\n                \\item[Comment] $\\alpha(G)$ is the minimum number of edge-disjoint spanning forests into which $G$ can be decomposed.\n                \\item[Reference] \\cite{Chiba1985}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal cliques in a connected graph}\n            \\begin{description}\n                \\item[Input] A connected graph $G = (V, E)$.\n                \\item[Output] All maximal cliques in $G$.\n                \\item[Complexity] $O(\\alpha(G)|E|)$ total time and linear space.\n                \\item[Comment] $\\alpha(G)$ is the minimum number of edge-disjoint spanning forests into which $G$ can be decomposed.\n                \\item[Reference] \\cite{Chiba1985}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximum cliques of a circular-arc graph}\n            \\begin{description}\n                \\item[Input] A circular-arc graph $G =(V, E)$.\n                \\item[Output] All maximum cliques of $G$.\n                \\item[Complexity] $O(|V|^{3.5} + N)$.\n                \\item[Comment] $N$ is the number of maximum cliques of $G$.  For a family $A$ of arcs on a circle, a graph $G = (V, E)$ is called the \\textit{circular-arc graph} for $A$ if there exists a one-to-one correspondence between $V$ and $A$ such that two vertices in $V$ are adjacent if and only if their corresponding arcs in $A$ intersect.\n                \\item[Reference] \\cite{Kashiwabara1992}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal cliques in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G=(V,E)$.\n                \\item[Output] All maximal cliques included in $G$.\n                \\item[Complexity] $O(M(n))$ time delay and $O(n^2)$ space, or $O(\\delta^4)$ time delay and $O(n+m)$ space, after $O(nm)$ preprocessing.\n                \\item[Comment] $M(n)$: the time needed to multiply two $n \\times n$ matrices, $\\delta$: maximum degree of $G=(V,E)$, $n$: total number of vertices, and $m$: total number of edges.\n                \\item[Reference] \\cite{Makino2004}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal cliques in a bipartite graph}\n            \\begin{description}\n                \\item[Input] A bipartite graph $G=(V,E)$.\n                \\item[Output] All maximal bicliques included in $G$.\n                \\item[Complexity] $O(M(n))$ time delay and $O(n^2)$ space, or $O(\\delta^4)$ time delay and $O(n+m)$ space, after $O(nm)$ preprocessing.\n                \\item[Comment] $M(n)$: the time needed to multiply two $n \\times n$ matrices, $\\delta$: maximum degree of $G=(V,E)$, $n$: total number of vertices, and $m$: total number of edges.\n                \\item[Reference] \\cite{Makino2004}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal cliques in a dynamic graph}\n            \\begin{description}\n                \\item[Input] A graph $G_t = (V, E_t)$.\n                \\item[Output] All maximal cliques in $G_t$.\n                \\item[Reference] \\cite{Stix2004}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal cliques in a dynamic graph}\n            \\begin{description}\n                \\item[Input] A dynamic graph $G$.\n                \\item[Output] All maximal cliques in $G$.\n                \\item[Reference] \\cite{Stix2004}\n            \\end{description}\n        \\subsubsection{Enumeration of all bicliques in a graph in lexicographical order}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All bicliques in $G$.\n                \\item[Complexity] $O(|V|^3)$ delay and $O(2^{|V|})$ space.\n                \\item[Comment] There is no polynomial-delay enumeration algorithm for all bicliques in reverse lexicographical order unless $P = NP$.\n                \\item[Reference] \\cite{Dias2005}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal cliques in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E )$.\n                \\item[Output] All maximal Cliques in $G$.\n                \\item[Complexity] $O(\\frac{\\Delta M_c Tri^2}{P})$ delay and $O(|V| + |E|)$ space.\n                \\item[Comment] $\\Delta$ is the maximum degree in $G$, $M_c$ is the size of the maximum clique in $G$, and $Tri$ is the number of triangles in $G$. This algorithm runs on the computer with $P$ processors.\n                \\item[Reference] \\cite{Du2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all cliques in a chordal graph}\n            \\begin{description}\n                \\item[Input] A chordal graph $G$.\n                \\item[Output] All cliques in $G$.\n                \\item[Complexity] $O(1)$ time per solution on average.\n                \\item[Reference] \\cite{Kiyomi2006}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal cliques in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All maximal cliques in $G$.\n                \\item[Complexity] $O(3^{n\/3})$ total time.\n                \\item[Reference] \\cite{Tomita2006}\n            \\end{description}\n        \\subsubsection{Enumeration of $K$-largest maximal cliques in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] $K$-largest maximal cliques in $G$.\n                \\item[Reference] \\cite{Bulo2007}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal cliques in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All maximal cliques in $G$.\n                \\item[Reference] \\cite{Pan2008}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal bicliques in a bipartite graph}\n            \\begin{description}\n                \\item[Input] A bipartite graph $G = (U, V, E)$.\n                \\item[Output] All maximal bicliques in $G$ in lexicographical order on $U$.\n                \\item[Complexity] $O((|U| + |V|)^2)$ delay with exponential space.\n                \\item[Reference] \\cite{Gely2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal cliques in a comparability graph}\n            \\begin{description}\n                \\item[Input] A comparability graph $G = (V, E)$.\n                \\item[Output] All maximal cliques in $G$ in lexicographical order.\n                \\item[Complexity] $O(|V|)$ delay and $O(|V| + |E|)$ space.\n                \\item[Reference] \\cite{Gely2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal cliques in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All maximal cliques in $G$ in lexicographical order.\n                \\item[Complexity] $O(|V||E|)$ delay with exponential space.\n                \\item[Reference] \\cite{Gely2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal cliques in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All maximal cliques in $G$.\n                \\item[Comment] This algorithm can be paralleled.\n                \\item[Reference] \\cite{Schmidt2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all $c$-isolated maximal clique in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and a positive real number $c$.\n                \\item[Output] All $c$-isolated maximal clique in $G$.\n                \\item[Complexity] $O(c^42^{2c}|E|)$ total time.\n                \\item[Reference] \\cite{Ito2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all $c$-isolated pseudo-clique in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and a positive real number $c$.\n                \\item[Output] All $c$-isolated $PC(k-\\log k, \\frac{k}{\\log k})$ in $G$.\n                \\item[Complexity] $O(c^42^{2c}|E|)$ total time.\n                \\item[Comment] $PC(\\alpha, \\beta)$ is a induced subgraph of $G$ with an average degree at least $\\alpha$ and the minimum degree at least $\\beta$.\n                \\item[Reference] \\cite{Ito2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all $c$-isolated maximal cliques in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and an integer $c$.\n                \\item[Output] All avg-$c$-isolated maximal cliques in $G$.\n                \\item[Complexity] $O(2^cc^5|E|)$ total time.\n                \\item[Comment] A vertex set $S \\subseteq V$ with $k$ vertices is $c$-isolated if it has less than $ck$ outgoing edges, where an outgoing edge is an edge between a vertex in $S$ and a vertex in $V \\setminus S$.\n                \\item[Reference] \\cite{Huffner2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all max-$c$-isolated maximal cliques in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and an integer $c$.\n                \\item[Output] All max-$c$-isolated maximal cliques in $G$.\n                \\item[Complexity] $O(2^cc^5|E|)$ total time.\n                \\item[Comment] A vertex set $S \\subseteq V$ with $k$ vertices is max-$c$-isolated if every vertex in $S$ has less than $c$ neighbors in $V \\setminus S$.\n                \\item[Reference] \\cite{Huffner2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal $c$-isolated cliques in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and an integer $c$.\n                \\item[Output] All maximal $c$-isolated cliques in $G$.\n                \\item[Complexity] $O(2.89^cc^2|E|)$ total time.\n                \\item[Comment] A vertex set $S \\subseteq V$ with $k$ vertices is $c$-isolated if it has less than $ck$ outgoing edges, where an outgoing edge is an edge between a vertex in $S$ and a vertex in $V \\setminus S$.\n                \\item[Reference] \\cite{Komusiewicz2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all cliques in a graph with degeneracy $d$}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ with degeneracy $d$.\n                \\item[Output] All cliques in $G$.\n                \\item[Complexity] $O(d|V|3^{d\/3})$ time.\n                \\item[Comment] They also show the largest possible number of maximal cliques in $G$. The number is $(|V|-d)3^{d\/3}$.\n                \\item[Reference] \\cite{Eppstein2010}\n            \\end{description}\n        \\subsubsection{Enumeration of all pseudo-cliques in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and a threshold $\\theta$.\n                \\item[Output] All pseudo-cliques in $G$, whose densities are no less than $\\theta$.\n                \\item[Complexity] $O(\\Delta |V| + \\mathrm{min}\\{\\Delta^2, |V| + |E|\\})$ delay with $O(|V| + |E|)$ space.\n                \\item[Comment] $\\Delta$ is the maximum degree of $G$. The density of a subgraph $G[S]$ induced by $S \\subseteq V$ is given by the number of edges over the number of all its vertex pairs.\n                \\item[Reference] \\cite{Uno2008a}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal cliques in a graph with limited memory}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All maximal graphs in $G$.\n                \\item[Complexity] See the paper.\n                \\item[Reference] \\cite{Cheng2012b}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal cliques in a graph}\n            \\begin{description}\n                \\item[Input] An undirected graph $G=(V, E)$.\n                \\item[Output] All maximal clique.\n                \\item[Complexity] $O(\\delta\\cdot H^3)$ time delay and $O(n+m)$ space.\n                \\item[Comment] $H$ satisfies $|\\{v\\in V | \\sigma(v)\\ge H\\}| \\le H$.\n                \\item[Reference] \\cite{Chang2013}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal cliques in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All maximal cliques in $G$.\n                \\item[Reference] \\cite{Henry2013}\n            \\end{description}\n     \\subsection{Coloring}\n        \\subsubsection{Enumeration of all the edge colorings in a bipartite graph}\n            \\begin{description}\n                \\item[Input] A bipartite graph $G = (V, E)$.\n                \\item[Output] All edge colorings in $G$.\n                \\item[Complexity] $O(\\Delta|E|)$ time per solution and space.\n                \\item[Comment] $\\Delta$ is the maximum degree in $G$.\n                \\item[Reference] \\cite{Yasuko1994}\n            \\end{description}\n        \\subsubsection{Enumeration of all the edge colorings of a bipartite graph}\n            \\begin{description}\n                \\item[Input] A bipartite graph $B = (V, E)$.\n                \\item[Output] All the edge colorings pf $B$.\n                \\item[Complexity] $O(T(|V|+|E|+\\Delta) + K \\min \\{|V|^2 + |E|, T(|V|, |E|, \\Delta)\\})$ total time and $O(|E|\\Delta)$ space.\n                \\item[Comment] $\\Delta$ is the number of maximum degree and $T(|V|, |E|, \\Delta)$ is the time complexity of an edge coloring algorithm.\n                \\item[Reference] \\cite{Matsui1996b}\n            \\end{description}\n        \\subsubsection{Enumeration of all the edge colorings of a bipartite graph}\n            \\begin{description}\n                \\item[Input] A bipartite graph $B = (V, E)$.\n                \\item[Output] All the edge colorings pf $B$.\n                \\item[Complexity] $O(|V|)$ time per solution and $O(|E|)$ space.\n                \\item[Reference] \\cite{Matsui1996a}\n            \\end{description}\n     \\subsection{Connected}\n        \\subsubsection{Enumeration of all minimal strongly connected subgraphs in a strongly connected subgraph}\n            \\begin{description}\n                \\item[Input] A strongly connected subgraph $G$.\n                \\item[Output] All minimal strongly connected subgraph.\n                \\item[Complexity] Incremental polynomial time.\n                \\item[Reference] \\cite{Boros2004b}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal 2-vertex connected spanning subgraphs in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All minimal 2-vertex connected spanning subgraphs of $G$.\n                \\item[Complexity] Incremental polynomial time.\n                \\item[Reference] \\cite{Khachiyan2006a}\n            \\end{description}\n     \\subsection{Cut}\n        \\subsubsection{Enumeration of all cuts between all pair of vertices in a given graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All cuts bewteen all pair of vertices in $G$\n                \\item[Reference] \\cite{Martelli1976a}\n            \\end{description}\n        \\subsubsection{Enumeration of all cutsets in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All cutsets in $G$.\n                \\item[Complexity] $O((|V| + |E|)(\\mu + 1))$ total time and $O(|V| + |E|)$ or $O(|V|^2)$ space.\n                \\item[Comment] $\\mu$ is the number of solutions.\n                \\item[Reference] \\cite{Tsukiyama1980}\n            \\end{description}\n        \\subsubsection{Enumeration of $k$ best cuts in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$.\n                \\item[Output] $k$ best cuts in $G$.\n                \\item[Complexity] $O(k|V|^4)$ total time.\n                \\item[Reference] \\cite{Hamacher1984a}\n            \\end{description}\n        \\subsubsection{Enumeration of $K$-best cuts in a network}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] $K$-best cuts in $G$.\n                \\item[Complexity] $O(K\\cdot |V|^4)$ time.\n                \\item[Reference] \\cite{Hamacher1984a}\n            \\end{description}\n        \\subsubsection{Enumeration of all articulation pairs in a planar graph}\n            \\begin{description}\n                \\item[Input] An undirected graph $G = (V, E)$.\n                \\item[Output] All articulation pairs in $G$.\n                \\item[Complexity] $O(|V|^2)$ total time.\n                \\item[Reference] \\cite{Laumond1985}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimum-size separating vertex sets in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All minimum-size separating vertex sets.\n                \\item[Complexity] $\\Theta(M|V| + C) = O(2^kn^3)$ total time.\n                \\item[Comment] $M$ is the number of solutions, $k$ is the connectivity of $G$, and $C = k|V| \\min(k(|V|+|E|), A)$, where $A$ is the time complexity of the best maximum network flow algorithm for unit network.\n                \\item[Reference] \\cite{Kanevsky1993}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal separators in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All minimal separators in $G$.\n                \\item[Complexity] $O(|V|^6R)$ total time.\n                \\item[Comment] $R$ is the number of solutions.\n                \\item[Reference] \\cite{Kloks1994}\n            \\end{description}\n        \\subsubsection{Enumeration of all $(s, t)$-cuts in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and two vertices $s, t$ in $G$.\n                \\item[Output] All $(s, t)$-cuts in $G$.\n                \\item[Complexity] $O(|E|)$ time per cut.\n                \\item[Reference] \\cite{Provan1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all $(s, t)$-cuts in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$ and two vertices $s, t$ in $G$.\n                \\item[Output] All $(s, t)$-cuts in $G$.\n                \\item[Complexity] $O(|E|)$ time per cut.\n                \\item[Reference] \\cite{Provan1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all $(s, t)$-uniformly directed cuts in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$ and two vertices $s, t$ in $G$.\n                \\item[Output] All $(s, t)$-uniformly directed cuts in $G$.\n                \\item[Complexity] $O(|V|)$ time per cut.\n                \\item[Comment] An \\textit{undirected directed cut} is also called a UDC. An $(s, t)$-DUC is an $(s, t)$-cut $(X, Y)$ such that $(X, Y) = \\emptyset$, where $(X, Y) = \\{(u, v) \\in E: u\\in X, v\\in Y\\}$.\n                \\item[Reference] \\cite{Provan1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimum weighted $(s, t)$ cuts in an weighted graph}\n            \\begin{description}\n                \\item[Input] An weighted graph $G = (V, E)$ and two vertices $s, t$ in $G$.\n                \\item[Output] All minimum weighted $(s, t)$ cuts in $G$.\n                \\item[Complexity] $O(|V|)$ time per cut.\n                \\item[Reference] \\cite{Provan1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all semidirected $(s, t)$ cuts in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$ and two vertices $s, t$ in $G$.\n                \\item[Output] All semidirected $(s, t)$ cuts in $G$.\n                \\item[Complexity] $O(|V||E|)$ time per cut.\n                \\item[Comment] For a subset $D$ of directed edges, a \\textit{semidirected $(s, t)$ cut} with respect to $D$ is an $(s, t)$ cut $(X, Y)$ such that $(X, Y) \\cup (Y, X)$ defnes an undirected $(s, t)$ cutset and such that $(X, Y) \\cap D = \\emptyset$, where a \\textit{cutset} is an minimal cut set.\n                \\item[Reference] \\cite{Provan1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all strong $(s, K)$ cutsets in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ , $s\\in V$ and $K \\subseteq V \\setminus \\{s\\}$.\n                \\item[Output] All strong $(s, K)$ cutsets in $G$.\n                \\item[Complexity] $O(|E|)$ time per cut.\n                \\item[Comment] An \\textit{$(s, K)$-cut} is defined to be any cut $(X, Y)$ for which $s \\in X$ and $K \\cap Y \\neq \\emptyset$. A \\textit{strong $(s, K)$ cutset} is minimal cuts of the form $(X, Y)$ where $s \\in X$ and $K \\subseteq Y$.\n                \\item[Reference] \\cite{Provan1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all strong $(s, K)$ cutsets in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$ , $s\\in V$ and $K \\subseteq V \\setminus \\{s\\}$.\n                \\item[Output] All strong $(s, K)$ cutsets in $G$.\n                \\item[Complexity] $O(|E|)$ time per cut.\n                \\item[Comment] An \\textit{$(s, K)$-cut} is defined to be any cut $(X, Y)$ for which $s \\in X$ and $K \\cap Y \\neq \\emptyset$. A \\textit{strong $(s, K)$ cutset} is minimal cuts of the form $(X, Y)$ where $s \\in X$ and $K \\subseteq Y$.\n                \\item[Reference] \\cite{Provan1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all quasi $(s, K)$ cuts in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ , $s\\in V$ and $K \\subseteq V \\setminus \\{s\\}$.\n                \\item[Output] All quasi $(s, K)$ cutsets in $G$.\n                \\item[Complexity] $O(|E|)$ time per cut.\n                \\item[Comment] An \\textit{$(s, K)$-cut} is defined to be any cut $(X, Y)$ for which $s \\in X$ and $K \\cap Y \\neq \\emptyset$. A \\textit{Quasi $(s, K)$ cut} is an edge set that is strong $(s, A)$-cutsets for some $A \\subseteq K$ and $A \\neq \\emptyset$.\n                \\item[Reference] \\cite{Provan1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all quasi $(s, K)$ cuts in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$ , $s\\in V$ and $K \\subseteq V \\setminus \\{s\\}$.\n                \\item[Output] All quasi $(s, K)$ cutsets in $G$.\n                \\item[Complexity] $O(|E|)$ time per cut.\n                \\item[Comment] An \\textit{$(s, K)$-cut} is defined to be any cut $(X, Y)$ for which $s \\in X$ and $K \\cap Y \\neq \\emptyset$. A \\textit{Quasi $(s, K)$ cut} is an edge set that is strong $(s, A)$-cutsets for some $A \\subseteq K$ and $A \\neq \\emptyset$.\n                \\item[Reference] \\cite{Provan1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all $(s, K)$ cutsets in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ , $s\\in V$ and $K \\subseteq V \\setminus \\{s\\}$.\n                \\item[Output] All $(s, K)$ cutsets in $G$.\n                \\item[Complexity] $O(|E|)$ time per cut.\n                \\item[Comment] An \\textit{$(s, K)$-cut} is defined to be any cut $(X, Y)$ for which $s \\in X$ and $K \\cap Y \\neq \\emptyset$. A \\textit{$(s, K)$ cutset} is minimal $(s, K)$ cuts.\n                \\item[Reference] \\cite{Provan1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal $a$-$b$ separators in a graph}\n            \\begin{description}\n                \\item[Input] An undirected connected simple graph $G = (V, E)$, non-adjacent vertices $a$ and $b$ in $G$.\n                \\item[Output] All minimal $a$-$b$ separator of $G$.\n                \\item[Complexity] $O(R_{ab}|V|^3)$ total time.\n                \\item[Comment] $R_{ab}$ is the number of minimal $a$-$b$ separators.\n                \\item[Reference] \\cite{Shen1997}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal $a$-$b$ separators in a graph}\n            \\begin{description}\n                \\item[Input] An undirected connected simple graph $G = (V, E)$, non-adjacent vertices $a$ and $b$ in $G$.\n                \\item[Output] All minimal $a$-$b$ separator of $G$.\n                \\item[Complexity] $O(R_{ab}|V|\/\\log|V|)$ total time.\n                \\item[Comment] $R_{ab}$ is the number of minimal $a$-$b$ separators. This algorithm needs $O(|V|^3)$ processors on a CREW PRAM.\n                \\item[Reference] \\cite{Shen1997}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal separators of a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All minimal separators of $G$.\n                \\item[Complexity] $O(|V|^5R)$ total time.\n                \\item[Comment] $R$ is the number of solutions.\n                \\item[Reference] \\cite{Kloks1998}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal separator of a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All minimal separator of $G$.\n                \\item[Complexity] $O(|V|^3)$ time per solution.\n                \\item[Reference] \\cite{Berry2000}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal separator of a chordal graph}\n            \\begin{description}\n                \\item[Input] A chordal graph $G = (V, E)$.\n                \\item[Output] All minimal separator of $G$.\n                \\item[Complexity] $O(|V|+|E|)$ total time.\n                \\item[Reference] \\cite{Chandran2001}\n            \\end{description}\n        \\subsubsection{Enumeration of all cut conjunctions for a given set of vertex pairs in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V,E)$, and a collection $B = \\{(s_1,t_1),...,(s_k,t_k)\\}$ of $k$ vertex pairs $s_i,t_i \\in V$.\n                \\item[Output] All cut conjunctions for $B$ in $G$.\n                \\item[Complexity] Incremental polynomial time.\n                \\item[Comment] A minimal edge sets $X \\subseteq E$ is a \\textit{cut conjunction} if, for all $i = 1, \\dots, k$, vertices $s_i$ and $t_i$ is disconnected in $G' = (V, E\\setminus X)$. A cut conjunction is a generalization of an $s-t$ cut.\n                \\item[Reference] \\cite{Khachiyan2005}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal separators of a 3-connected planar graph}\n            \\begin{description}\n                \\item[Input] A 3-connected planar graph $G = (V, E)$.\n                \\item[Output] All minimal separators of $G$.\n                \\item[Complexity] $O(|V|)$ time per solution.\n                \\item[Reference] \\cite{Mazoit2006}\n            \\end{description}\n        \\subsubsection{Enumeration of all cut conjunctions of a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and a collection $B = \\{(s_1, t_1), \\dots, (s_k, t_k)\\}$.\n                \\item[Output] All cut conjunctions of $G$.\n                \\item[Complexity] $O(K^2 \\log(K) |V||E|^2 + K^2|B|(|V|+|E|)|E|^2)$ total time.\n                \\item[Comment] $K$ is the number of cut conjunctions.  $X$ is a cut conjunctions of $G$ if $X$ is a minimal edge set such that for all $i = 1, \\dots, k$, a pair of vertices $s_i$ and $t_i$ is disconnected in $(V, E\\setminus X)$.\n                \\item[Reference] \\cite{Khachiyan2008a}\n            \\end{description}\n        \\subsubsection{Enumeration of all $(s, t)$-cuts in an weighted directed graph}\n            \\begin{description}\n                \\item[Input] An weighted directed graph $G = (V, E)$.\n                \\item[Output] All cuts in $G$ by non-decreasing weights ordering.\n                \\item[Complexity] $O(|V||E|\\log(|V|^2\/|E|))$ delay.\n                \\item[Reference] \\cite{Yeh2010}\n            \\end{description}\n        \\subsubsection{Enumeration of all $(s, t)$-cuts in an weighted undirected graph}\n            \\begin{description}\n                \\item[Input] An weighted undirected graph $G = (V, E)$.\n                \\item[Output] All cuts in $G$ by non-decreasing weights ordering.\n                \\item[Complexity] $O(|V||E|\\log(|V|^2\/|E|))$ delay.\n                \\item[Reference] \\cite{Yeh2010}\n            \\end{description}\n     \\subsection{Cycle}\n        \\subsubsection{Enumeration of all cycles in an $n$-cube, where $n \\le 4$}\n            \\begin{description}\n                \\item[Input] An integer $n$. \n                \\item[Output] All cycles (closed paths) in an $n$-cube. \n                \\item[Reference] \\cite{Gilbert1958}\n            \\end{description}\n        \\subsubsection{Enumeration of all cycles in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All cycles in $G$. \n                \\item[Reference] \\cite{Welch1965}\n            \\end{description}\n        \\subsubsection{Enumeration of all simple cycles in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.  \n                \\item[Output] All simple cycles in $G$. \n                \\item[Complexity]  \n                \\item[Reference] \\cite{Ponstein1966}\n            \\end{description}\n        \\subsubsection{Enumeration of all circuits in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All circuits in $G$. \n                \\item[Reference] \\cite{Welch1966}\n            \\end{description}\n        \\subsubsection{Enumeration of all Hamiltonian circuits in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All Hamiltonian circuits in $G$. \n                \\item[Reference] \\cite{Yau1967}\n            \\end{description}\n        \\subsubsection{Enumeration of all directed circuits in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G$. \n                \\item[Output] All directed circuits in $G$. \n                \\item[Reference] \\cite{Kamae1967}\n            \\end{description}\n        \\subsubsection{Enumeration of all cycles in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All cycles in $G$. \n                \\item[Reference] \\cite{Danielson1968}\n            \\end{description}\n        \\subsubsection{Enumeration of all cycles in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All cycles in $G$. \n                \\item[Reference] \\cite{Rao1969}\n            \\end{description}\n        \\subsubsection{Enumeration of all elementary circuit in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.  \n                \\item[Output] All elementary circuit in $G$. \n                \\item[Complexity]  \n                \\item[Reference] \\cite{Tiernan1970}\n            \\end{description}\n        \\subsubsection{Enumeration of all cycles in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All cycles in $G$. \n                \\item[Reference] \\cite{Char1970}\n            \\end{description}\n        \\subsubsection{Enumeration of all cycles in an undirected graph}\n            \\begin{description}\n                \\item[Input] An undirected graph $G$. \n                \\item[Output] All cycles in $G$. \n                \\item[Reference] \\cite{Weinblatt1972}\n            \\end{description}\n        \\subsubsection{Enumeration of all cycles in a finite undirected graph}\n            \\begin{description}\n                \\item[Input] A finite undirected graph $G$. \n                \\item[Output] All cycles in $G$. \n                \\item[Comment] He claimed that J. T. Welch, Jr.'s algorithm is wrong. \n                \\item[Reference] \\cite{Weinblatt1972}\n            \\end{description}\n        \\subsubsection{Enumeration of all cycles in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G=(V,E)$.\n                \\item[Output] All cycles in $G$.\n                \\item[Complexity] $O((|V|\\cdot |E|)(C+1))$ total time and $O(|V| + |E|)$ space.\n                \\item[Comment] $C$ is the number of cycles included in $G$.\n                \\item[Reference] \\cite{Tarjan1973}\n            \\end{description}\n        \\subsubsection{Enumeration of all cycles in a directed graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$. \n                \\item[Output] All cycles in $G$. \n                \\item[Complexity] $O(|E| + c(|V| \\times |E|))$ total, where $c$ is the number of circuits in $G$. \n                \\item[Reference] \\cite{Ehrenfeucht1973}\n            \\end{description}\n        \\subsubsection{Enumeration of all cycles in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G=(V, E)$.\n                \\item[Output] All cycles in $G$.\n                \\item[Complexity] $O((|V|+ |E|)(C+1))$ total time and $O(|V| + |E|)$ space.\n                \\item[Comment] $C$ is the number of cycles included in $G$.\n                \\item[Reference] \\cite{Johnson1975}\n            \\end{description}\n        \\subsubsection{Enumeration of all cycles in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All cycles in $G$.\n                \\item[Complexity] $O(|E|)$ time per cycle with $O(|E|)$ space.\n                \\item[Reference] \\cite{Read1975}\n            \\end{description}\n        \\subsubsection{Enumeration of all cycles in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$.\n                \\item[Output] All cycles in $G$.\n                \\item[Complexity] $O(|E|)$ time per cycle with $O(|E|)$ space.\n                \\item[Reference] \\cite{Read1975}\n            \\end{description}\n        \\subsubsection{Enumeration of all cycles in a planar graph}\n            \\begin{description}\n                \\item[Input] A planar graph $G=(V,E)$.\n                \\item[Output] All cycles in $G$.\n                \\item[Complexity] $O(|V|(C+1))$ total time and $O(|V|)$ space.\n                \\item[Comment] $C$ is the number of cycles included in $G$.\n                \\item[Reference] \\cite{Syso1981}\n            \\end{description}\n        \\subsubsection{Enumeration of all cycles of a given length in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$ and an integer $k$.\n                \\item[Output] All cycles of length $k$ in $G$.\n                \\item[Complexity] See paper.\n                \\item[Reference] \\cite{Alon1997}\n            \\end{description}\n        \\subsubsection{Enumeration of all cycles in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All cycles in $G$.\n                \\item[Reference] \\cite{Wild2008}\n            \\end{description}\n        \\subsubsection{Enumeration of all small cycles in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All cycles with length at most $5$ in $G$.\n                \\item[Reference] \\cite{Wild2008}\n            \\end{description}\n        \\subsubsection{Enumeration of all chordless cycles in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All chordless cycles in $G$.\n                \\item[Reference] \\cite{Wild2008}\n            \\end{description}\n        \\subsubsection{Enumeration of all Hamiltonian cycles in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All Hamiltonian cycles in $G$.\n                \\item[Reference] \\cite{Wild2008}\n            \\end{description}\n        \\subsubsection{Enumeration of all cycles in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G=(V,E)$.\n                \\item[Output] All cycles in $G$.\n                \\item[Complexity] $O(|E| + \\sum_{c \\in \\mathcal{C}(G)}|c|)$ total time.\n                \\item[Comment] $\\mathcal{C}(G)$ is the set of all cycles in $G$.\n                \\item[Reference] \\cite{Ferreira2012}\n            \\end{description}\n        \\subsubsection{Enumeration of all chordless cycles in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G=(V,E)$.\n                \\item[Output] All chordless cycles in $G$.\n                \\item[Complexity] $\\tilde{O}(|E| +|V|\\cdot C )$ total time.\n                \\item[Comment] $C$ is the number of chordless cycles in $G$.\n                \\item[Reference] \\cite{Ferreira2014}\n            \\end{description}\n        \\subsubsection{Enumeration of all chordless cycles in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G=(V,E)$.\n                \\item[Output] All chordless cycles in $G$.\n                \\item[Complexity] $O(|V|+|E|)$ time per chordless cycle.\n                \\item[Reference] \\cite{Unoc}\n            \\end{description}\n     \\subsection{Dominating set}\n        \\subsubsection{Enumeration of all minimal dominating sets in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All minimal dominating sets in $G$.\n                \\item[Complexity] $O(1.7159^{|V|})$ total time.\n                \\item[Reference] \\cite{Fomin2008}\n            \\end{description}\n        \\subsubsection{Enumeration of $z$ smallest weighted edge dominating sets in a graph}\n            \\begin{description}\n                \\item[Input] An weighted graph $G = (V, E)$, and positive integers $k$ and $z$. Each edge of $G$ has a positive weight.\n                \\item[Output] $z$ smallest weighted edge dominating sets in $G$.\n                \\item[Complexity] $O(5.6^{2k}k^4z^2 +4^{2k}k^3z|V|)$ total time.\n                \\item[Reference] \\cite{Wang2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal dominating sets in a line graph}\n            \\begin{description}\n                \\item[Input] A line graph $G$.\n                \\item[Output] All minimal dominating sets in $G$.\n                \\item[Complexity] $O(||G||^5N^6)$ total time.\n                \\item[Comment] $||G||$ is the size of $G$ and $N$ is the number of minimal dominating sets in $G$.\n                \\item[Reference] \\cite{Kante2012}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal dominating sets in a path graph or $(C_4, C_5, craw)$-free graph}\n            \\begin{description}\n                \\item[Input] A line graph or $(C_4, C_5, craw)$-free graph $G$.\n                \\item[Output] All minimal dominating sets in $G$.\n                \\item[Complexity] $O(||G||^2N^3)$ total time.\n                \\item[Comment] $||G||$ is the size of $G$ and $N$ is the number of minimal dominating sets in $G$.\n                \\item[Reference] \\cite{Kante2012}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal edge-dominating sets in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All minimal edge-dominating sets in $G$.\n                \\item[Complexity] $O(||L(G)||^5N^6)$ total time.\n                \\item[Comment] $L(G)$ is the line graph of $G$, $||L(G)||$ is the size of $L(G)$, and $N$ is the number of minimal edge-dominating sets in $G$.\n                \\item[Reference] \\cite{Kante2012}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal dominating sets in an undirected permutation graph}\n            \\begin{description}\n                \\item[Input] An undirected permutation graph $G=(V, E)$.\n                \\item[Output] All minimal dominating sets.\n                \\item[Complexity] $O(|V|)$ delay with $O(|V|^8)$ pre-processing.\n                \\item[Reference] \\cite{Kante2013}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal dominating sets in an undirected interval graph}\n            \\begin{description}\n                \\item[Input] An undirected interval graph $G=(V, E)$.\n                \\item[Output] All minimal dominating sets.\n                \\item[Complexity] $O(|V|)$ delay with $O(|V|^3)$ pre-processing.\n                \\item[Reference] \\cite{Kante2013}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal edge dominating sets in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All minimal edge dominating sets in $G$.\n                \\item[Complexity] $O(|V|^6|\\mathcal{L}|)$ delay.\n                \\item[Comment] $\\mathcal{L}$ is the set of already generated solutions.\n                \\item[Reference] \\cite{Golovach2013a}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal dominating sets in a line graph}\n            \\begin{description}\n                \\item[Input] A line graph $G = (V, E)$.\n                \\item[Output] All minimal dominating sets in $G$.\n                \\item[Complexity] $O(|V|^2|E|^2|\\mathcal{L}|)$ delay.\n                \\item[Comment] $\\mathcal{L}$ is the set of already generated solutions.\n                \\item[Reference] \\cite{Golovach2013a}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal edge dominating sets in a bipartite graph}\n            \\begin{description}\n                \\item[Input] A bipartite graph $G = (V, E)$.\n                \\item[Output] All minimal edge dominating sets in $G$.\n                \\item[Complexity] $O(|V|^4|\\mathcal{L}|)$ delay.\n                \\item[Comment] $\\mathcal{L}$ is the set of already generated solutions.\n                \\item[Reference] \\cite{Golovach2013a}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal dominating sets in the line graph of a bipartite graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All minimal dominating sets in $G$.\n                \\item[Complexity] $O(|V|^2|E||\\mathcal{L}|)$ delay.\n                \\item[Comment] $\\mathcal{L}$ is the set of already generated solutions.\n                \\item[Reference] \\cite{Golovach2013a}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal dominating sets in a graph of girth at least 7}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ of girth at least 7.\n                \\item[Output] All minimal dominating sets in $G$.\n                \\item[Complexity] $O(|V|^2|E||\\mathcal{L}|^2)$ delay.\n                \\item[Comment] $\\mathcal{L}$ is the set of already generated solutions.\n                \\item[Reference] \\cite{Golovach2013a}\n            \\end{description}\n        \\subsubsection{Enumeration of all 2-dominating sets in a tree}\n            \\begin{description}\n                \\item[Input] A tree $T = (V, E)$.\n                \\item[Output] All 2-dominating sets of $T$.\n                \\item[Complexity] $O(1.3248^n)$ total time.\n                \\item[Comment] If a subset $U\\subseteq V$ is a 2-dominating set if every vertex $v \\in V \\setminus U$ has at least two neighbors in $U$.\n                \\item[Reference] \\cite{Krzywkowski2013}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal dominating sets in a $P_6$-free chordal graph}\n            \\begin{description}\n                \\item[Input] A $P_6$-free chordal graph $G = (V, E)$.\n                \\item[Output] All minimal dominating sets in $G$.\n                \\item[Complexity] Linear delay with $O(|V|^2)$ space.\n                \\item[Reference] \\cite{Kante2014}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal dominating sets in a chordal bipartite graph}\n            \\begin{description}\n                \\item[Input] A chordal bipartite graph $G$. \n                \\item[Output] All minimal dominating sets in $G$. \n                \\item[Complexity] $O(n^3m|\\mathcal{L}|^2)$ delay and the total running time is $O(n^3m|\\mathcal{L}^*|^2)$. \n                \\item[Comment] $n$ is the number vertices in $G$, \n            $m$ is the number of edges in $G$, \n            $\\mathcal{L}$ is the family of already generated minimal dominating sets, and \n            $\\mathcal{L}^*$ is the family of all minimal dominating sets. \n        \n                \\item[Reference] \\cite{Golovach2015}\n            \\end{description}\n     \\subsection{Drawing}\n        \\subsubsection{Enumeration of all rectangle drawings with $n$ faces}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All rectangle drawings with $n$ faces.\n                \\item[Complexity] $O(n)$ time per drawing and space.\n                \\item[Reference] \\cite{Takagi2004}\n            \\end{description}\n     \\subsection{Feedback arc set}\n        \\subsubsection{Enumeration of all minimal feedback arc sets in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All minimal feedback arc sets in $G$. \n                \\item[Reference] \\cite{Yau1967}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal feedback arc sets in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G=(V,E)$.\n                \\item[Output] All minimal feedback arc sets in $G$.\n                \\item[Complexity] $O(|V||E|(|V|+|E|))$ time delay.\n                \\item[Reference] \\cite{Schwikowski2002}\n            \\end{description}\n     \\subsection{Feedback vertex set}\n        \\subsubsection{Enumeration of all minimal feedback vertex sets in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All minimal feedback vertex sets in $G$. \n                \\item[Reference] \\cite{Yau1967}\n            \\end{description}\n        \\subsubsection{Enumeration of all feedback vertex sets in a strongly connected directed graph}\n            \\begin{description}\n                \\item[Input] A strongly connected directed graph $G = (V, E)$ and an integer $k$.\n                \\item[Output] All feedback vertex sets of size $k$ in $G$.\n                \\item[Complexity] $O(|V|^{k-1}|E|)$ total time and $O(|E|)$ space.\n                \\item[Reference] \\cite{Garey1978}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal feedback vertex sets in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G=(V,E)$.\n                \\item[Output] All minimal feedback vertex sets in $G$.\n                \\item[Complexity] $O(|V||E|(|V|+|E|))$ time delay.\n                \\item[Reference] \\cite{Schwikowski2002}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal feedback vertex sets in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G=(V,E)$.\n                \\item[Output] All minimal feedback vertex sets in $G$.\n                \\item[Complexity] $O(|V|^2(|V|+|E|))$ time delay.\n                \\item[Reference] \\cite{Schwikowski2002}\n            \\end{description}\n     \\subsection{General}\n        \\subsubsection{Enumeration of all graphs in an almost sure first order fimiles}\n            \\begin{description}\n                \\item[Input] A first order language $\\theta$.\n                \\item[Output] All graphs in $\\mathcal{G}_\\theta$.\n                \\item[Complexity] Polynomial space and delay.\n                \\item[Reference] \\cite{Goldberg1993b}\n            \\end{description}\n     \\subsection{Independent set}\n        \\subsubsection{Enumeration of all maximal independent sets in a chordal graph}\n            \\begin{description}\n                \\item[Input] A chordal graph $G=(V,E)$.\n                \\item[Output] All maximal independent sets in $G$.\n                \\item[Complexity] $O(|V||E|\\mu)$ total time.\n                \\item[Comment] $\\mu$ is the number of maximal independent sets of $G$. This algorithm can also enumerate all the maximal cliques.\n                \\item[Reference] \\cite{Tsukiyama1977a}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal independent sets in a claw-free graph}\n            \\begin{description}\n                \\item[Input] A claw-free graph $G$.\n                \\item[Output] All maximal independent sets in $G$.\n                \\item[Reference] \\cite{Minty1980}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal independent sets in an undirected graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All maximal independent sets in $G$ in lexicographically.\n                \\item[Reference] \\cite{Loukakis1981}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal independent sets in an interval graph}\n            \\begin{description}\n                \\item[Input] An interval graph $G = (V, E)$.\n                \\item[Output] All maximal independent sets in $G$.\n                \\item[Complexity] $O(|V|^2 + \\beta)$ total time.\n                \\item[Comment] $\\beta$ is the sum of the vertices in all maximal independent sets of $G$.\n                \\item[Reference] \\cite{Leung1984}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal independent sets in a circular-arc graph}\n            \\begin{description}\n                \\item[Input] A circular-arc graph $G = (V, E)$.\n                \\item[Output] All maximal independent sets in $G$.\n                \\item[Complexity] $O(|V|^2 + \\beta)$ total time.\n                \\item[Comment] $\\beta$ is the sum of the vertices in all maximal independent sets of $G$.\n                \\item[Reference] \\cite{Leung1984}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal independent sets in a chordal graph}\n            \\begin{description}\n                \\item[Input] A chordal graph $G = (V, E)$.\n                \\item[Output] All maximal independent sets in $G$.\n                \\item[Complexity] $O((|V| + |E|)N)$ total time.\n                \\item[Comment] $N$ is the number of solutions.\n                \\item[Reference] \\cite{Leung1984}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal independent sets in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G=(V,E)$.\n                \\item[Output] All maximal independent sets included in $G$.\n                \\item[Complexity] $O(n(m+n\\log C))=O(n^3)$ delay and exponential space.\n                \\item[Comment] $C$: total number of maximal independent sets, $n$: total number of vertices, and $m$: total number of edges. Is there no polynomial space and delay algorithm?\n                \\item[Reference] \\cite{Johnson1988}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximum independent sets of a bipartite graph}\n            \\begin{description}\n                \\item[Input] A bipartite graph $B =(V, E)$.\n                \\item[Output] All maximum independent sets of $B$.\n                \\item[Complexity] $O(|V|^{2.5} + N)$.\n                \\item[Comment] $N$ is the number of maximum independent sets of $B$.\n                \\item[Reference] \\cite{Kashiwabara1992}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal independent sets on a tree in lexicographic order}\n            \\begin{description}\n                \\item[Input] Tree $T = (V, E)$.\n                \\item[Output] All maximal independent sets on $T$ in lexicographic order.\n                \\item[Complexity] $O(|V|^2)$ delay with $O(|V|)$ space.\n                \\item[Reference] \\cite{Chang1994a}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximum independent set of a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All maximum independent set of $G$.\n                \\item[Complexity] $O(2^{0.114|E|})$ total time and polynomial space.\n                \\item[Reference] \\cite{Beigel1999}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximum independent set of a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ with maximum degree 3.\n                \\item[Output] All maximum independent set of $G$.\n                \\item[Complexity] $O(2^{0.171|V|})$ total time and polynomial space.\n                \\item[Reference] \\cite{Beigel1999}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximum independent set of a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ with maximum degree 4.\n                \\item[Output] All maximum independent set of $G$.\n                \\item[Complexity] $O(2^{0.228|V|})$ total time and polynomial space.\n                \\item[Reference] \\cite{Beigel1999}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximum independent set of a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All maximum independent set of $G$.\n                \\item[Complexity] $O(2^{0.290|V|})$ total time and polynomial space.\n                \\item[Reference] \\cite{Beigel1999}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal independent sets of a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and a position integer $k$.\n                \\item[Output] Enumeration of all maximal independent sets with at most size $k$ of $G$.\n                \\item[Complexity] $O(3^{4k-|V|} 4^{|V|-3k})$ total time.\n                \\item[Reference] \\cite{Eppstein2003a}\n            \\end{description}\n        \\subsubsection{Enumeration of all independent sets in a chordal graph}\n            \\begin{description}\n                \\item[Input] A chordal graph $G = (V, E)$.\n                \\item[Output] All independent sets in $G$.\n                \\item[Complexity] Constant time per solution on average after $O(|V| + |E|)$ time for preprocessing.\n                \\item[Reference] \\cite{Okamoto2005a}\n            \\end{description}\n        \\subsubsection{Enumeration of all independent sets of size $k$ in a chordal graph}\n            \\begin{description}\n                \\item[Input] A chordal graph $G = (V, E)$ and a positive integer $k$.\n                \\item[Output] All independent sets of size $k$ in $G$.\n                \\item[Complexity] Constant time per solution on average after $O((|V| + |E|)|V|^2)$ time for preprocessing.\n                \\item[Reference] \\cite{Okamoto2005a}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximum independent sets in a chordal graph}\n            \\begin{description}\n                \\item[Input] A chordal graph $G = (V, E)$.\n                \\item[Output] All maximum independent sets in $G$.\n                \\item[Complexity] Constant time per solution on average after $O((|V| + |E|)|V|^2)$ time for preprocessing.\n                \\item[Reference] \\cite{Okamoto2005a}\n            \\end{description}\n        \\subsubsection{Enumeration of all independent sets in an input chordal graph}\n            \\begin{description}\n                \\item[Input] A chordal graph $G=(V,E)$.\n                \\item[Output] All independent sets in $G$.\n                \\item[Complexity] $O(1)$ delay and $O(|V|(|V|+|E|))$ time and space for preprocessing.\n                \\item[Comment] Counting all independent sets in an input chordal graph needs $O(n+m)$ time.\n                \\item[Reference] \\cite{Okamoto2008}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximum independent sets in an input chordal graph}\n            \\begin{description}\n                \\item[Input] A chordal graph $G=(V,E)$.\n                \\item[Output] All maximum independent sets in $G$.\n                \\item[Complexity] $O(1)$ delay and $O(|V|(|V|+|E|))$ time and space for preprocessing.\n                \\item[Comment] Counting all maximum independent sets in an input chordal graph needs $O(n+m)$ time.\n                \\item[Reference] \\cite{Okamoto2008}\n            \\end{description}\n        \\subsubsection{Enumeration of all independent sets with $k$ vertices in an input chordal graph}\n            \\begin{description}\n                \\item[Input] A chordal graph $G=(V,E)$ and an integer $k$.\n                \\item[Output] All maximum independent sets with $k$ vertices in $G$.\n                \\item[Complexity] $O(1)$ delay and $O(k|V|(|V|+|E|))$ time and space for preprocessing.\n                \\item[Comment] The number of independent sets with $k$ vertices in an input chordal graph needs $O(k^2(|V|+|E|))$ time.\n                \\item[Reference] \\cite{Okamoto2008}\n            \\end{description}\n     \\subsection{Interval graph}\n        \\subsubsection{Enumeration of all interval supergraph that contains a given graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All interval supergraphs each of which contain $G$.\n                \\item[Complexity] $O(|V|^3)$ time for each and $O(|V|^2)$ space.\n                \\item[Reference] \\cite{Kiyomi2006a}\n            \\end{description}\n        \\subsubsection{Enumeration of all interval graph of a given graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All interval graphs of $G$.\n                \\item[Complexity] $O((|V|+|E|)^2)$ time for each.\n                \\item[Reference] \\cite{Kiyomi2006a}\n            \\end{description}\n        \\subsubsection{Enumeration of Proper Interval Graphs}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All proper interval graphs with $n$ vertices.\n                \\item[Complexity] $O(1)$ time per proper interval graph and $O(n)$ space, after $O(n^2)$ preprocessing time.\n                \\item[Comment] Preprocessing: generating the complete graph with $n$ vertices.\n                \\item[Reference] \\cite{Saitoh2010a}\n            \\end{description}\n     \\subsection{Matching}\n        \\subsubsection{Enumeration of the $k$ best perfect matchings of a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and an integer $k$.\n                \\item[Output] The $k$ best perfect matchings of $G$ in order.\n                \\item[Complexity] $O(k|V|^3)$ total time.\n                \\item[Reference] \\cite{Chegireddy1987}\n            \\end{description}\n        \\subsubsection{Enumeration of all stable marriage}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All stable marriage of $G$.\n                \\item[Complexity] $O(|V|)$ time per solution and $O(|V|^2)$ space.\n                \\item[Reference] \\cite{Gusfield1989}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimum cost perfect matchings in an weighted bipartite graph}\n            \\begin{description}\n                \\item[Input] An weighted bipartite graph $B = (V, E)$.\n                \\item[Output] All minimum cost perfect matchings in $B$.\n                \\item[Complexity] $O(|V|(|V| + |E|))$ time per solution and $O(|V| + |E|)$ space\n                \\item[Reference] \\cite{Fukuda1992}\n            \\end{description}\n        \\subsubsection{Enumeration of all perfect matchings in a bipartite graph}\n            \\begin{description}\n                \\item[Input] A bipartite graph $B = (U, V, E)$, where $|U| = |V|$.\n                \\item[Output] All perfect matchings in $B$.\n                \\item[Complexity] $O(c(|V| + |E|))$ total time and $O(|V| + |E|)$ space, after $O(n^{2.5})$ preprocessing time.\n                \\item[Comment] $c$ is the number of solutions.\n                \\item[Reference] \\cite{Fukuda1994}\n            \\end{description}\n        \\subsubsection{Enumeration of the $k$ best perfect matchings of a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and an integer $k$.\n                \\item[Output] The $k$ best perfect matchings of $G$ in decreasing order.\n                \\item[Complexity] $O(k|V|^3)$ total time with $O(k|V|^2)$ space.\n                \\item[Reference] \\cite{Matsui1994}\n            \\end{description}\n        \\subsubsection{Enumeration of all perfect, maximum, and maximal matchings in bipartite graphs}\n            \\begin{description}\n                \\item[Input] A bipartite graph $B=(V, E)$.\n                \\item[Output] All perfect, maximum, and maximal matching in $B$.\n                \\item[Complexity] $O(|V|)$ time per matching.\n                \\item[Reference] \\cite{Uno1997}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal matchings in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All maximal matchings in $G$.\n                \\item[Complexity] $O(|V| + |E| + \\Delta N)$ total time and $O(|V| + |E|)$ space.\n                \\item[Comment] $N$ is the number of solutions and $\\Delta$ is the maximum degree in $G$.\n                \\item[Reference] \\cite{Uno2001}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal blocker in a bipartite graph}\n            \\begin{description}\n                \\item[Input] A bipartite graph $G = (U, V, E)$.\n                \\item[Output] All minimal blocker in $G$.\n                \\item[Complexity] Polynomial delay and space.\n                \\item[Comment] A \\textit{blocker} of $G$ is an edge subset $X$ of $E$ such that $G' = (U, V, E \\setminus X)$ has no perfect matching.\n                \\item[Reference] \\cite{Boros2006}\n            \\end{description}\n        \\subsubsection{Enumeration of all basic perfect 2-matchings in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All basic perfect 2-matchings in $G$.\n                \\item[Complexity] Incremental polynomial delay.\n                \\item[Comment] A \\textit{basic 2-matching} of $G$ is a subset of edges that cover the vertices with vertex-disjoint edges and vertex-disjoint odd cycles.\n                \\item[Reference] \\cite{Boros2006}\n            \\end{description}\n        \\subsubsection{Enumeration of all $d$-factor in a bipartite graph}\n            \\begin{description}\n                \\item[Input] A bipartite graph $G = (U, V, E)$ and any non negative function $d: A \\cup B \\to \\{0, 1, \\cdots, |U| + |V|\\}$.\n                \\item[Output] All $d$-factor in $G$.\n                \\item[Complexity] $O(|E|)$ delay.\n                \\item[Comment] A $d$-factor in $G$ is a subgraph $G' = (U, V, X)$ covering all vertices of $G$, whose each vertex $v$ has degree $d(v)$. If for any $v \\in U \\cup V$, $d(v) = 1$, $G'$ is a perfect matching.\n                \\item[Reference] \\cite{Boros2006}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal induced matchings in a triangle-free graph}\n            \\begin{description}\n                \\item[Input] A triangle-free graph $G$. \n                \\item[Output] All maximal induced matchings in $G$. \n                \\item[Complexity] $O(1.4423^n)$ total time with polynomial delay. \n                \\item[Comment] $n$ is the number of vertices in $G$. \n                \\item[Reference] \\cite{Basavaraju2014}\n            \\end{description}\n     \\subsection{Matroid}\n        \\subsubsection{Enumeration of all bases of a graphic matroid in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All bases of a graphic matroid in $G$.\n                \\item[Complexity] $O(|V| + |E| + N)$ total time and $O(|V| + |E|)$ space.\n                \\item[Comment] $N$ is the number of solutions.  If $G$ is connected, any base is a spanning tree.\n                \\item[Reference] \\cite{Uno1998}\n            \\end{description}\n        \\subsubsection{Enumeration of all bases of a linear matroid in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All bases of a linear matroid in $G$.\n                \\item[Complexity] $O(|V|)$ time per solution and $O(|V|^2|E|)$ preprocessing after time.\n                \\item[Reference] \\cite{Uno1998}\n            \\end{description}\n        \\subsubsection{Enumeration of all bases of a matching matroid in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All bases of a matching matroid in $G$.\n                \\item[Complexity] $O(|V| + |E|)$ time per solution.\n                \\item[Reference] \\cite{Uno1998}\n            \\end{description}\n     \\subsection{Ordering}\n        \\subsubsection{Enumeration of all topological sortings of a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G =(V, E)$.\n                \\item[Output] All topological sortings of $G$.\n                \\item[Complexity] $O(|V| + |E|)$ time per sorting and $O(|V| + |E|)$ space.\n                \\item[Reference] \\cite{Knuth1974}\n            \\end{description}\n        \\subsubsection{Enumeration of all topological sortings of a given set in lexicographically}\n            \\begin{description}\n                \\item[Input] An $n$-element set $S$. \n                \\item[Output] All topological sortings of $S$ in lexicographically.\n                \\item[Complexity] $O(m)$ time per solution(?). \n                \\item[Reference] \\cite{Knuth1979}\n            \\end{description}\n        \\subsubsection{Enumeration of all topological sortings of a po set}\n            \\begin{description}\n                \\item[Input] A partial order set $P$.\n                \\item[Output] All topological sortings of $P$.\n                \\item[Complexity] $O(|P|)$ time per solution.\n                \\item[Comment] $|P|$ is the number of objects in $P$.\n                \\item[Reference] \\cite{Varol1981}\n            \\end{description}\n        \\subsubsection{Enumeration of all topological sortings in a directed acyclic graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G$.\n                \\item[Output] All topological sortings in $G$.\n                \\item[Complexity] $O(1)$ amortized time per solution with $O(|G|)$.\n                \\item[Comment] Linear extensions correspond to topological sortings.\n                \\item[Reference] \\cite{Pruesse1991a}\n            \\end{description}\n        \\subsubsection{Enumeration of all topological sortings}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All topological sortings in $G$.\n                \\item[Complexity] $O(1)$ amortized time per solution.\n                \\item[Comment] A topological sorting is also known as a linear extension.\n                \\item[Reference] \\cite{Ruskey1992a}\n            \\end{description}\n        \\subsubsection{Enumeration of all topological sortings}\n            \\begin{description}\n                \\item[Input] A directed acyclic graph $D$.\n                \\item[Output] All topological sortings $D$.\n                \\item[Complexity] $O(1)$ amortized time per topological sorting and $O(|V|)$ space in addition to the space used for $D$.\n                \\item[Comment] Linear sortings correspond to topological sortings.\n                \\item[Reference] \\cite{Pruesse1994}\n            \\end{description}\n        \\subsubsection{Enumeration of all linear extensions of a given poset}\n            \\begin{description}\n                \\item[Input] A poset $P$.\n                \\item[Output] All linear extensions of $P$.\n                \\item[Complexity] $O(1)$ time per solution.\n                \\item[Comment] Their algorithm is a loop-free algorithm.\n                \\item[Reference] \\cite{Canfield1995}\n            \\end{description}\n        \\subsubsection{Enumeration of all topological sortings of an acyclic directed graph}\n            \\begin{description}\n                \\item[Input] An acyclic directed graph $G = (V, E)$.\n                \\item[Output] All topological sortings of $G$.\n                \\item[Complexity] $O(|V|N)$ total time and $O(|V||E|)$ space.\n                \\item[Comment] $N$ is the number of solutions.\n                \\item[Reference] \\cite{Avis1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all perfect elimination orderings}\n            \\begin{description}\n                \\item[Input] A chordal graph $G$.\n                \\item[Output] All perfect elimination orderings of $G$.\n                \\item[Complexity] Constant amortized time per solution.\n                \\item[Reference] \\cite{Chandran2003}\n            \\end{description}\n        \\subsubsection{Enumeration of all forest extensions of a partially ordered set}\n            \\begin{description}\n                \\item[Input] A partially ordered set $P$.\n                \\item[Output] All forest extensions of $P$.\n                \\item[Complexity] $O(|E|^2)$ delay and $O(|E||R|)$ space.\n                \\item[Comment] $E$ is the set of elements. $R$ is the binary relation on $E$.\n                \\item[Reference] \\cite{Szwarcfiter2003b}\n            \\end{description}\n        \\subsubsection{Enumeration of all topological sortings of a directed acyclic graph}\n            \\begin{description}\n                \\item[Input] A directed acyclic graph $D$.\n                \\item[Output] All topological sortings $D$.\n                \\item[Complexity] $O(1)$ delay per topological sorting.\n                \\item[Comment] Linear extensions correspond to topological sortings.\n                \\item[Reference] \\cite{Ono2005}\n            \\end{description}\n        \\subsubsection{Enumeration of all realizer of a triangulated planar graph}\n            \\begin{description}\n                \\item[Input] A triangulated planar graph $G = (V, E)$.\n                \\item[Output] All realizer of $G$.\n                \\item[Complexity] $O(|V|)$ time per realizer.\n                \\item[Reference] \\cite{Yamanaka2006}\n            \\end{description}\n        \\subsubsection{Enumeration of all perfect elimination orderings of a chordal graph}\n            \\begin{description}\n                \\item[Input] A chordal graph $G = (V, E)$.\n                \\item[Output] All perfect elimination orderings of $G$.\n                \\item[Complexity] $O(1)$ time per solution on average with $O(|V|^2)$ space and $O(|V|^3)$ with $O(|V|^2)$ space pre-computation.\n                \\item[Reference] \\cite{Matsui2008}\n            \\end{description}\n        \\subsubsection{Enumeration of all perfect sequences in a chordal graph}\n            \\begin{description}\n                \\item[Input] A chordal graph $G = (V, E)$.\n                \\item[Output] All perfect sequences of $G$.\n                \\item[Complexity] $O(1)$ time per graph with $O(|V|^2)$ space with $O(|V|^3)$ time and $O(|V|^2)$ space pre-computation.\n                \\item[Reference] \\cite{Matsui2010}\n            \\end{description}\n     \\subsection{Orientation}\n        \\subsubsection{Enumeration of all acyclic orientation of a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All acyclic orientation of $G$.\n                \\item[Complexity] $O(N(|V|+|E|))$ total time ($O(|V|(|V|+E|))$ delay) and $O(|V| + |E|)$ space.\n                \\item[Comment] A acyclic orientation of $G$ is an assignment of directions of each edge such that $G$ is acyclic.\n                \\item[Reference] \\cite{Barbosa1999}\n            \\end{description}\n        \\subsubsection{Enumeration of all $(s, t)$-orientations of a biconnected planar graph}\n            \\begin{description}\n                \\item[Input] A biconnected planar graph $G = (V, E)$ and an edge $(s, t)$ in $G$.\n                \\item[Output] All $(s, t)$-orientations of $G$.\n                \\item[Complexity] $O(|V|)$ time per solution.\n                \\item[Reference] \\cite{Setiawan2011}\n            \\end{description}\n     \\subsection{Other}\n        \\subsubsection{Enumeration of all Hamiltonian centers in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All Hamiltonian centers in $G$. \n                \\item[Reference] \\cite{Yau1967}\n            \\end{description}\n        \\subsubsection{Enumeration of all CA-sets of a directed graph}\n            \\begin{description}\n                \\item[Input] Directed graph $G = (V, E)$.\n                \\item[Output] All CA-sets of $G$.\n                \\item[Complexity] $O(|V|^{2.49+} + \\gamma)$.\n                \\item[Comment] $\\gamma$ is the output size. $S \\subset V$ is a \\textit{CA-set} if, for each $v \\in S$, all ancestor of $v$ belongs to $S$.\n                \\item[Reference] \\cite{Kashiwabara1992}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal induced subgraphs for (connected) hereditary graph properties}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All maximal induced subgraphs in $\\mathcal{P}(G)$.\n                \\item[Complexity] See the paper.\n                \\item[Comment] $\\mathcal{P}$ is a set of subgraphs of $G$ with (connected) hereditary graph properties.\n                \\item[Reference] \\cite{Cohen2008}\n            \\end{description}\n     \\subsection{Path}\n        \\subsubsection{Enumeration of all simple paths in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.  \n                \\item[Output] All simple paths in $G$. \n                \\item[Complexity]  \n                \\item[Reference] \\cite{Ponstein1966}\n            \\end{description}\n        \\subsubsection{Enumeration of all Hamiltonian paths in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All Hamiltonian paths in $G$. \n                \\item[Reference] \\cite{Yau1967}\n            \\end{description}\n        \\subsubsection{Enumeration of all directed paths in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G$. \n                \\item[Output] All directed paths in $G$. \n                \\item[Reference] \\cite{Kamae1967}\n            \\end{description}\n        \\subsubsection{Enumeration of all paths in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All paths in $G$. \n                \\item[Reference] \\cite{Kroft1967b}\n            \\end{description}\n        \\subsubsection{Enumeration of all paths in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All paths in $G$. \n                \\item[Reference] \\cite{Danielson1968}\n            \\end{description}\n        \\subsubsection{Enumeration of $k$ shortest paths in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] $K$ shortest paths in $G$.\n                \\item[Complexity] $O(|V|^3)$ total time.\n                \\item[Reference] \\cite{Yen1971}\n            \\end{description}\n        \\subsubsection{Enumeration of $k$ shortest paths in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] $k$ shortest paths in $G$.\n                \\item[Complexity] $O(k|V|c(|V|))$ total time.\n                \\item[Comment] (?) $c(n)$ is the time complexity to find an optimal solution to a problem with $n$ (0, 1) variables.\n                \\item[Reference] \\cite{Lawler1972}\n            \\end{description}\n        \\subsubsection{Enumeration of all paths in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All paths in $G$.\n                \\item[Complexity] $O(|E|)$ time per path with $O(|E|)$ space.\n                \\item[Reference] \\cite{Read1975}\n            \\end{description}\n        \\subsubsection{Enumeration of all paths in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$.\n                \\item[Output] All paths in $G$.\n                \\item[Complexity] $O(|E|)$ time per path with $O(|E|)$ space.\n                \\item[Reference] \\cite{Read1975}\n            \\end{description}\n        \\subsubsection{Enumeration of $k$ shortest paths in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E). $\n                \\item[Output] $K$ shortest paths in $G$.\n                \\item[Complexity] $O(k|V|^3)$ total time.\n                \\item[Reference] \\cite{Shier1979}\n            \\end{description}\n        \\subsubsection{Generation of the $k$-th longest path in a tree}\n            \\begin{description}\n                \\item[Input] A tree $T = (V, E)$ and an integer $k$.\n                \\item[Output] The $k$-th longest path in $T$.\n                \\item[Complexity] $O(n\\log^2 n)$ time.\n                \\item[Reference] \\cite{Megiddo1981}\n            \\end{description}\n        \\subsubsection{Enumeration of all shortest paths in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All shortest paths in $G$.\n                \\item[Reference] \\cite{Florian1981}\n            \\end{description}\n        \\subsubsection{Enumeration of $k$ shortest paths of a directed graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] $k$ shortest paths that may contains cycles in $G$.\n                \\item[Reference] \\cite{Martins1984}\n            \\end{description}\n        \\subsubsection{Enumeration of all quickest paths in a network}\n            \\begin{description}\n                \\item[Input] A network $N = (V, E, c, \\ell)$.\n                \\item[Output] All quickest paths in $N$.\n                \\item[Complexity] $O(rS|V||E| + rS|V|^2\\log|V|)$ total time.\n                \\item[Comment] $c$ is a positive edge weight function and $\\ell$ is a nonnegative edge weight function. $r$ is the number of distinct capacity value of $N$. $S$ is the number of solutions.\n                \\item[Reference] \\cite{Rosen1991}\n            \\end{description}\n        \\subsubsection{Counting all acyclic walks in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G=(V,E)$.\n                \\item[Output] The number of acyclic walks in $G$.\n                \\item[Reference] \\cite{Babic1993}\n            \\end{description}\n        \\subsubsection{Enumeration of the $k$ shortest paths in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and an integer $k$.\n                \\item[Output] The $k$ smallest shortest paths in $G$.\n                \\item[Complexity] $O(k|E|)$ total time.\n                \\item[Reference] \\cite{Azevedo1993}\n            \\end{description}\n        \\subsubsection{Enumeration of all constrained quickest paths in a network}\n            \\begin{description}\n                \\item[Input] Network $N = (V, E)$ and constraints $L$ and $C$.\n                \\item[Output] All quickest paths in $N$.\n                \\item[Complexity] $O(k|V|^2|E|)$ total time.\n                \\item[Comment] $k$ is the number of solutions. A quickest path is a variant of a shortest path.\n                \\item[Reference] \\cite{Gen-Huey1994}\n            \\end{description}\n        \\subsubsection{Enumeration of the $k$ shortest paths in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$ and an integer $k$.\n                \\item[Output] The $k$ smallest shortest paths in $G$.\n                \\item[Complexity] $O(k|E|)$ total time\n                \\item[Reference] \\cite{Eppstein1998}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal path conjunctions in a graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$, $s_1, s_2, t_1 \\in V$, $T_2 \\subseteq V$, and $\\mathcal{P} = \\{ (s_1, t_1)\\} \\cup \\{(s_2, t): t\\in T_2\\}$.\n                \\item[Output] All minimal path conjunctions in $G$.\n                \\item[Complexity] Polynomial delay.\n                \\item[Comment] A path conjunction is a edge subset $E' \\subseteq E$ such that for all $(s, t) \\in \\mathcal{P}$, $s$ is connected to $t$ in the graph $G' = (V, E')$.\n                \\item[Reference] \\cite{Boros2004a}\n            \\end{description}\n        \\subsubsection{Enumeration of all st-paths in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G=(V,E)$ and $s, v \\in V$.\n                \\item[Output] All st-paths in $G$.\n                \\item[Complexity] $O(|E| + \\sum_{\\pi \\in \\mathcal{P}_{st}(G)}|\\pi|)$ total time.\n                \\item[Comment] $\\mathcal{P}_{st}(G)$ is the set of all st-paths in $G$.\n                \\item[Reference] \\cite{Ferreira2012}\n            \\end{description}\n        \\subsubsection{Enumeration of all $P_3$'s in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All of all $P_3$'s in $G$.\n                \\item[Complexity] $O(|E|^{1.5} + p_3(G))$ total time.\n                \\item[Comment] $P_3$ is a induced path of $G$ with three vertices and $p_3(G)$ is the number of $P_3$ in $G$.\n                \\item[Reference] \\cite{Hoang2013}\n            \\end{description}\n        \\subsubsection{Enumeration of all $P_k$'s in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$ and an integer $k \\ge 4$.\n                \\item[Output] All of all $P_k$'s in $G$.\n                \\item[Complexity] $O(|V|^{k-1} + p_k(G) + k\\cdot c_k(G))$ total time.\n                \\item[Comment] $P_k$ and $C_k$ are a induced path and cycle of $G$ with $k$ vertices, respectively. $p_k(G)$ and $c_k(G)$ are the number of $P_k$ and $C_k$ in $G$, respectively.\n                \\item[Reference] \\cite{Hoang2013}\n            \\end{description}\n        \\subsubsection{Enumeration of all $C_k$'s in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$ and an integer $k \\ge 4$.\n                \\item[Output] All of all $C_k$'s in $G$.\n                \\item[Complexity] $O(|V|^{k-1} + p_k(G) + c_k(G))$ total time.\n                \\item[Comment] $P_k$ and $C_k$ are a induced path and cycle of $G$ with $k$ vertices, respectively. $p_k(G)$ and $c_k(G)$ are the number of $P_k$ and $C_k$ in $G$, respectively.\n                \\item[Reference] \\cite{Hoang2013}\n            \\end{description}\n        \\subsubsection{Enumeration of all chordless $st$-paths in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G=(V,E)$ and two vertices $s, t \\in V$.\n                \\item[Output] All chordless $st$-paths in $G$.\n                \\item[Complexity] $\\tilde{O}(|E| +|V|\\cdot P )$ total time.\n                \\item[Comment] $P$ is the number of chordless $st$-paths in $G$.\n                \\item[Reference] \\cite{Ferreira2014}\n            \\end{description}\n        \\subsubsection{Enumeration of all chordless st-paths in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G=(V,E)$ and $s, t \\in V$.\n                \\item[Output] All chordless st-paths (from $s$ to $t$) in $G$.\n                \\item[Complexity] $O(|V|+|E|)$ time per chordless st-path.\n                \\item[Reference] \\cite{Unoc}\n            \\end{description}\n     \\subsection{Permutation graph}\n        \\subsubsection{Enumeration of all connected bipartite permutation graphs with $n$ vertices}\n            \\begin{description}\n                \\item[Input] A graph size $n$.\n                \\item[Output] All connected bipartite permutation graphs.\n                \\item[Complexity] $O(1)$ time per graph with $O(n)$ space.\n                \\item[Reference] \\cite{Saitoh2010b}\n            \\end{description}\n     \\subsection{Pitch}\n        \\subsubsection{Enumeration of all stories in a graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E, S, T)$ that has the set of source vertices $S$ and the set of target vertices of $T$.\n                \\item[Output] All stories in $G$.\n                \\item[Comment] A \\textit{pitch} $P$ of $G$ is a set of arcs $E' \\subseteq E$, such that the subgraph $G' =(V', E')$ of $G$, where $V' \\subseteq V$ is the set of vertices of G having at least one out-going or in-coming arc in $E'$, is acyclic and for each vertex $w \\in V' \\setminus S$, $w$ is not a source in $G'$, and for each vertex $w \\in V' \\setminus T$ ,$w$ is not a target in $G'$. $P$ is a \\textit{story} if $P$ is maximal.\n                \\item[Reference] \\cite{Acuna2012}\n            \\end{description}\n        \\subsubsection{Enumeration of all pitches}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E, S, T)$ that has the set of source vertices $S$ and the set of target vertices of $T$.\n                \\item[Output] All pitches in $G$.\n                \\item[Complexity] $O(|V| + |E|)$ delay with $O(|V| + |E|)$ space.\n                \\item[Comment] A pitch $P$ of $G$ is a set of arcs $E' \\subseteq E$, such that the subgraph $G' =(V', E')$ of $G$, where $V' \\subseteq V$ is the set of vertices of G having at least one out-going or in-coming arc in $E'$, is acyclic and for each vertex $w \\in V' \\setminus S$, $w$ is not a source in $G'$, and for each vertex $w \\in V' \\setminus T$ ,$w$ is not a target in $G'$. Enumeration of all stories (maximal pitches) is still open.\n                \\item[Reference] \\cite{Borassi2013}\n            \\end{description}\n     \\subsection{Planar graph}\n        \\subsubsection{Enumeration of all maximal planar graphs with $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All maximal planar graph with $n$ vertices.\n                \\item[Complexity] $O(n^3)$ time per graph with $O(n)$ space.\n                \\item[Comment] A planar graph with $n$ vertices is \\textit{maximal} if it has exactly $3n -6$ edges.\n                \\item[Reference] \\cite{Nakano2004a}\n            \\end{description}\n        \\subsubsection{Enumeration of all based floorplans with at most $n$ faces}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All based floorplans with at most $n$ faces.\n                \\item[Complexity] $O(1)$ time per solution with $O(n)$ space.\n                \\item[Comment] A planar graph is called a \\textit{floorplan} if every face is a rectangle. A \\textit{based floorplan} is a floorplan with one designated base line segment on the outer face.\n                \\item[Reference] \\cite{Nakano2001a}\n            \\end{description}\n        \\subsubsection{Enumeration of all based floorplans with exactly $n$ faces}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All based floorplans with exactly $n$ faces.\n                \\item[Complexity] $O(1)$ time per solution with $O(n)$ space.\n                \\item[Comment] A planar graph is called a \\textit{floorplan} if every face is a rectangle. A \\textit{based floorplan} is a floorplan with one designated base line segment on the outer face.\n                \\item[Reference] \\cite{Nakano2001a}\n            \\end{description}\n        \\subsubsection{Enumeration of all floorplans with exactly $n$ faces}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All floorplans with exactly $n$ faces.\n                \\item[Complexity] $O(1)$ time per solution with $O(n)$ space.\n                \\item[Comment] A planar graph is called a \\textit{floorplan} if every face is a rectangle.\n                \\item[Reference] \\cite{Nakano2001a}\n            \\end{description}\n        \\subsubsection{Enumeration of all internally triconnected planar graphs}\n            \\begin{description}\n                \\item[Input] Integers $n$ and $g$.\n                \\item[Output] All internally triconnected planar graphs with exactly $n$ vertices such that $\\kappa(G) = 2$ and the size of each inner face is at most $g$.\n                \\item[Complexity] $O(n^3)$ time per solution on average with $O(n)$ space.\n                \\item[Reference] \\cite{Zhuang2010a}\n            \\end{description}\n     \\subsection{Plane graph}\n        \\subsubsection{Enumeration of all plane straight-line graphs on a given point set in the plane}\n            \\begin{description}\n                \\item[Input] A point set $P$ in the plane.\n                \\item[Output] All plane straight-line graphs on $P$.\n                \\item[Complexity] $O(|P|\\log|P|)$ time per solution.\n                \\item[Comment] Use gray code.\n                \\item[Reference] \\cite{Aichholzer2007}\n            \\end{description}\n        \\subsubsection{Enumeration of all plane and connected straight-line graphs on a given point set in the plane}\n            \\begin{description}\n                \\item[Input] A point set $P$ in the plane.\n                \\item[Output] All plane and connected straight-line graphs on $P$.\n                \\item[Complexity] $O(|P|\\log|P|)$ time per solution.\n                \\item[Comment] Use gray code.\n                \\item[Reference] \\cite{Aichholzer2007}\n            \\end{description}\n        \\subsubsection{Enumeration of all plane spanning trees on a given point set in the plane}\n            \\begin{description}\n                \\item[Input] A point set $P$ in the plane.\n                \\item[Output] All plane spanning trees on $P$.\n                \\item[Complexity] $O(|P|\\log|P|)$ time per solution.\n                \\item[Comment] Use gray code.\n                \\item[Reference] \\cite{Aichholzer2007}\n            \\end{description}\n        \\subsubsection{Enumeration of all plane graphs}\n            \\begin{description}\n                \\item[Input] $m$: the maximum number of edges.\n                \\item[Output] All connected rooted plane graphs with at most $m$ edges.\n                \\item[Complexity] amortized $O(1)$ time per graph with $O(m)$ space.\n                \\item[Comment] This algorithm does not outputs the entire graph but the difference from previous one.\n                \\item[Reference] \\cite{Yamanaka2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all plane graphs}\n            \\begin{description}\n                \\item[Input] $m$: the maximum number of edges.\n                \\item[Output] All connected non-rooted plane graphs with at most $m$ edges.\n                \\item[Complexity] $O(m^3)$ time per graph with $O(m)$ space.\n                \\item[Comment] This algorithm does not outputs the entire graph but the difference from previous one.\n                \\item[Reference] \\cite{Yamanaka2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all plane graphs on a given point set in the plane}\n            \\begin{description}\n                \\item[Input] A fixed point set $P$.\n                \\item[Output] All plane graphs on $P$.\n                \\item[Complexity] $O(N)$ total time.\n                \\item[Comment] $N$ is the number of solutions.\n                \\item[Reference] \\cite{Katoh2009a}\n            \\end{description}\n        \\subsubsection{Enumeration of all non-crossing spanning connected graphs on a given point set in the plane}\n            \\begin{description}\n                \\item[Input] A fixed point set $P$.\n                \\item[Output] All non-crossing spanning connected graphs on $P$.\n                \\item[Complexity] $O(N)$ total time.\n                \\item[Comment] $N$ is the number of solutions.\n                \\item[Reference] \\cite{Katoh2009a}\n            \\end{description}\n        \\subsubsection{Enumeration of all non-crossing spanning trees on a given point set in the plane}\n            \\begin{description}\n                \\item[Input] A fixed point set $P$.\n                \\item[Output] All non-crossing spanning trees on $P$.\n                \\item[Complexity] $O(N + |P|tri(P))$ total time.\n                \\item[Comment] $N$ is the number of solutions and $tri(P)$ is the number of triangulations of $P$.\n                \\item[Reference] \\cite{Katoh2009a}\n            \\end{description}\n        \\subsubsection{Enumeration of all non-crossing minimally rigid frameworks on a given point set in the plane}\n            \\begin{description}\n                \\item[Input] A fixed point set $P$.\n                \\item[Output] All non-crossing minimally rigid frameworks on $P$.\n                \\item[Complexity] $O(|P|^2N)$ total time.\n                \\item[Comment] $N$ is the number of solutions.\n                \\item[Reference] \\cite{Katoh2009a}\n            \\end{description}\n        \\subsubsection{Enumeration of all non-crossing perfect matchings on a given point set in the plane}\n            \\begin{description}\n                \\item[Input] A fixed point set $P$.\n                \\item[Output] All non-crossing perfect matchings on $P$.\n                \\item[Complexity] $O(|P|^{3\/2}tri(P) + |P|^{5\/2}N)$ total time.\n                \\item[Comment] $N$ is the number of solutions and $tri(P)$ is the number of triangulations of $P$.\n                \\item[Reference] \\cite{Katoh2009a}\n            \\end{description}\n        \\subsubsection{Enumeration of all triconnected rooted plane graphs}\n            \\begin{description}\n                \\item[Input] Integers $n$ and $g$.\n                \\item[Output] All triconnected rooted plane graphs with $n$ vertices, whose each inner face has the length at most $g$.\n                \\item[Complexity] $O(1)$ delay with $O(n)$ space after $O(n)$ time preprocessing.\n                \\item[Reference] \\cite{Zhuang2010}\n            \\end{description}\n        \\subsubsection{Enumeration of all triconnected rooted plane graphs}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All triconnected rooted plane graphs with $n$ vertices.\n                \\item[Complexity] $O(n^3)$ delay with $O(n)$ space after $O(n)$ time preprocessing.\n                \\item[Reference] \\cite{Zhuang2010}\n            \\end{description}\n        \\subsubsection{Enumeration of all biconnected rooted plane graphs}\n            \\begin{description}\n                \\item[Input] Integers $n$ and $g$.\n                \\item[Output] All biconnected rooted plane graphs with exactly $n$ vertices such that each inner face is of length at most $g$.\n                \\item[Complexity] $O(1)$ delay with $O(n)$ space, after an $O(n)$ time preprocessing.\n                \\item[Reference] \\cite{Zhuang2010c}\n            \\end{description}\n        \\subsubsection{Enumeration of all biconnected plane graphs}\n            \\begin{description}\n                \\item[Input] Integers $n$ and $g$.\n                \\item[Output] All biconnected plane graphs with at most $n$ vertices such that each inner face is of length at most $g$.\n                \\item[Complexity] $O(n^3)$ time per solution on average with $O(n)$ space.\n                \\item[Reference] \\cite{Zhuang2010c}\n            \\end{description}\n        \\subsubsection{Enumeration of all biconnected rooted plane graphs}\n            \\begin{description}\n                \\item[Input] Integers $n$ and $g$.\n                \\item[Output] All biconnected rooted plane graphs with at most $n$ vertices such that each inner face is of length at most $g$.\n                \\item[Complexity] $O(1)$ delay with $O(n)$ space.\n                \\item[Reference] \\cite{Zhuang2010c}\n            \\end{description}\n        \\subsubsection{Enumeration of all rooted plane graphs in $\\mathcal{G}_{\\text{int}}(n, g) - \\mathcal{G}_3(n, g)$}\n            \\begin{description}\n                \\item[Input] Integers $n$ and $g$.\n                \\item[Output] All rooted plane graphs in $\\mathcal{G}_{\\text{int}}(n, g) - \\mathcal{G}_3(n, g)$\n                \\item[Complexity] $O(1)$ delay with $O(n)$ space and time preprocessing.\n                \\item[Reference] \\cite{Zhuang2010a}\n            \\end{description}\n     \\subsection{Polytope}\n        \\subsubsection{Enumeration of all 3-polytopes of a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All 3-polytopes of $G$.\n                \\item[Reference] \\cite{Deza1993}\n            \\end{description}\n     \\subsection{Quadrangle}\n        \\subsubsection{Enumeration of all quadrangles in a graph}\n            \\begin{description}\n                \\item[Input] A connected graph $G = (V, E)$.\n                \\item[Output] All quadrangles in $G$.\n                \\item[Complexity] $O(\\alpha(G)|E|)$ total time and $O(|E|)$ space.\n                \\item[Comment] $\\alpha(G)$ is the minimum number of edge-disjoint spanning forests into which $G$ can be decomposed.\n                \\item[Reference] \\cite{Chiba1985}\n            \\end{description}\n     \\subsection{Quadrangulation}\n        \\subsubsection{Enumeration of all based biconnected plane quadrangulations with at most $f$ faces}\n            \\begin{description}\n                \\item[Input] An integer $f$.\n                \\item[Output] All based biconnected plane quadrangulations with at most $f$ faces.\n                \\item[Complexity] $O(1)$ time per quadrangulation and $O(f)$ space.\n                \\item[Comment] A \\textit{plane quadrangulation} is a plane graph such that each inner face has exactly four edges on its contour. A \\textit{based plane quadrangulation} is a plane quadrangulation with one designated edge on the outer face.\n                \\item[Reference] \\cite{Li2003}\n            \\end{description}\n     \\subsection{Regular graph}\n        \\subsubsection{Enumeration of all cubic graphs with less than or equal to $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$. \n                \\item[Output] All cubic graphs with less than or equal to $n$ vertices. \n                \\item[Reference] \\cite{Brinkmann2011}\n            \\end{description}\n     \\subsection{Series-parallel}\n        \\subsubsection{Enumeration of all series-parallel graphs with at most $m$ edges}\n            \\begin{description}\n                \\item[Input] An integer $m$.\n                \\item[Output] All series-parallel graphs with at most $m$ edges.\n                \\item[Complexity] $O(m)$ time per graph.\n                \\item[Reference] \\cite{Kawano2005a}\n            \\end{description}\n     \\subsection{Spanning subgraph}\n        \\subsubsection{Enumeration of all minimal $k$-vertex connected spanning subgraphs in a $k$-connected graph.}\n            \\begin{description}\n                \\item[Input] A $k$-connected graph $G$.\n                \\item[Output] All minimal $k$-vertex connected spanning subgraphs in $G$.\n                \\item[Complexity] $O(K^3|E|^3|n| + K^2|E|^5|V|^4 + K|V|^k|E|^2)$ total time.\n                \\item[Comment] $K$ is the number of solutions.  A graph $G$ is $k$-connected if a subgraph of $G$ obtained by removing at most $k-1$ vertices is still connected.\n                \\item[Reference] \\cite{Boros2007}\n            \\end{description}\n     \\subsection{Spanning tree}\n        \\subsubsection{Enumeration of all spanning trees in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All spanning trees in  $G$. \n                \\item[Complexity]  \n                \\item[Reference] \\cite{Hakimi1961}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning trees in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G=(V,E)$.\n                \\item[Output] All spanning trees of $G$.\n                \\item[Comment] In this paper, 'trees' indicate 'spanning trees'. \n                \\item[Reference] \\cite{Minty1965}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning trees of a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All spanning trees of $G$. \n                \\item[Reference] \\cite{Mayeda1965a}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning trees in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All spanning trees in $G$.\n                \\item[Complexity] $O(|V||E|^2)$ time per spanning tree with $O(|V||E|)$ space.\n                \\item[Reference] \\cite{Read1975}\n            \\end{description}\n        \\subsubsection{Enumeration of the $k$ smallest weight spanning trees in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and an integer $k$.\n                \\item[Output] The $k$ smallest weight spanning trees in $G$.\n                \\item[Complexity] $O(k|E|\\alpha (|E|,|V|) + |E|\\log |E|)$ total time and $O(k + |E|)$ space, $\\alpha(\\cdot)$ is Tarjan's inverse of Ackermann's function.\n                \\item[Reference] \\cite{Gabow1977}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning trees in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and an integer $k$.\n                \\item[Output] The all spanning trees in $G$ in order.\n                \\item[Complexity] $O(N|V|)$ total time and $O(N + |E|)$ space, $N$ is the number of spanning trees in $G$.\n                \\item[Reference] \\cite{Gabow1977}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning trees in an undirected graph}\n            \\begin{description}\n                \\item[Input] An undirected graph $G = (V, E)$.\n                \\item[Output] All spanning trees in $G$.\n                \\item[Complexity] $O(|V| + |E| + |V|N)$ total time and $O(|V| + |E|)$ space.\n                \\item[Comment] $N$ is the number of spanning trees in $G$.\n                \\item[Reference] \\cite{Gabow1978}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning trees in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$.\n                \\item[Output] All spanning trees in $G$.\n                \\item[Complexity] $O(|V| + |E| + |E|N)$ total time and $O(|V| + |E|)$ space.\n                \\item[Comment] $N$ is the number of spanning trees in $G$.\n                \\item[Reference] \\cite{Gabow1978}\n            \\end{description}\n        \\subsubsection{Enumeration of the $k$ smallest weight spanning trees in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and an integer $k$.\n                \\item[Output] The $k$ smallest weight spanning trees in $G$.\n                \\item[Complexity] $O(k|E| + \\min (|V|^2 ,|E|\\log \\log |V|))$ total time and $O(k + |E|)$ space\n                \\item[Reference] \\cite{Katoh1981}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning trees in an undirected graph}\n            \\begin{description}\n                \\item[Input] An undirected graph $G$.\n                \\item[Output] All spanning trees in $G$.\n                \\item[Comment] They analized Char's enumeration algorithm.\n                \\item[Reference] \\cite{Jayakumar1984}\n            \\end{description}\n        \\subsubsection{Enumeration of the $k$ smallest weight spanning trees in a graph in increasing order}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and an integer $k$.\n                \\item[Output] The $k$ smallest weight spanning trees in $G$.\n                \\item[Complexity] $O(|E|\\log\\log_{(2+|E|\/|V|)} n + k^2\\sqrt{|E|})$ total time and $O(|E| + k\\sqrt{|E|})$ space.\n                \\item[Reference] \\cite{Frederickson1985}\n            \\end{description}\n        \\subsubsection{Enumeration of the $k$ smallest weight spanning trees in a planar graph in increasing order}\n            \\begin{description}\n                \\item[Input] A planar graph $G = (V, E)$ and an integer $k$.\n                \\item[Output] The $k$ smallest weight spanning trees in $G$.\n                \\item[Complexity] $O(|V| + k^2(\\log |V|)^2)$ total time and $O(|V| + k(\\log |V|)^2)$ space.\n                \\item[Reference] \\cite{Frederickson1985}\n            \\end{description}\n        \\subsubsection{Enumeration of all undirected minimum spanning trees in an undirected graph}\n            \\begin{description}\n                \\item[Input] An undirected graph $G = (V, E)$.\n                \\item[Output] All undirected minimum spanning trees in $G$.\n                \\item[Complexity] $O(|E|\\log \\beta(|E|, |V|))$ total time.\n                \\item[Comment] $\\beta(|E|, |V|) = \\min\\{i | \\log^{(i)}|V| \\le |E|\/|N|\\}$.\n                \\item[Reference] \\cite{Gabow1986a}\n            \\end{description}\n        \\subsubsection{Enumeration of all directed minimum spanning trees in an directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$.\n                \\item[Output] All directed minimum spanning trees in $G$.\n                \\item[Complexity] $O(|V|\\log \\beta(|E|, |V|))$ total time.\n                \\item[Comment] $\\beta(|E|, |V|) = \\min\\{i | \\log^{(i)}|V| \\le |E|\/|N|\\}$.\n                \\item[Reference] \\cite{Gabow1986a}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning tree in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All spanning tree of $G$.\n                \\item[Complexity] $O(|V|+|E|+N)$ total time and $O(|V||E|)$ space.\n                \\item[Comment] $N$ is the number of spanning trees in $G$.\n                \\item[Reference] \\cite{Kapoor1991}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning tree in an weighted graph}\n            \\begin{description}\n                \\item[Input] An weighted graph $G = (V, E)$.\n                \\item[Output] All spanning tree of $G$ in increasing order of weight.\n                \\item[Complexity] $O(N\\log |V|+|V||E|)$ total time and $O(N + |V|^2|E|)$ space.\n                \\item[Comment] $N$ is the number of spanning trees in $G$.\n                \\item[Reference] \\cite{Kapoor1991}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning tree in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$.\n                \\item[Output] All spanning tree of $G$.\n                \\item[Complexity] $O(N|V|+|V|^3)$ total time and $O(|V|^2)$ space.\n                \\item[Comment] $N$ is the number of spanning trees in $G$.\n                \\item[Reference] \\cite{Kapoor1991}\n            \\end{description}\n        \\subsubsection{Enumeration of the $k$ best spanning trees in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and an integer $k$.\n                \\item[Output] The $k$ best spanning trees of $G$.\n                \\item[Complexity] $O(m\\log\\beta(|E|, |V|) + k^2)$ total time.\n                \\item[Comment] $\\beta(|E|, |V|) = \\min \\{\\log^{(i)}|V| \\le |E|\/|V|\\}$.\n                \\item[Reference] \\cite{Eppstein1992}\n            \\end{description}\n        \\subsubsection{Enumeration of the $k$ best spanning trees in a planar graph}\n            \\begin{description}\n                \\item[Input] A planar graph $G = (V, E)$ and an integer $k$.\n                \\item[Output] The $k$ best spanning trees of $G$.\n                \\item[Complexity] $O(n + k^2)$ total time\n                \\item[Reference] \\cite{Eppstein1992}\n            \\end{description}\n        \\subsubsection{Generation of the $k$-th minimum spanning tree in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and an integer $k$.\n                \\item[Output] The $k$-th minimum spanning tree of $G$.\n                \\item[Complexity] $O((|V||E|)^{k-1})$ time.\n                \\item[Reference] \\cite{Mayr1992}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning trees}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All spanning trees in $G$.\n                \\item[Complexity] $O(N + |V| + |E|)$ total time and $O(|V||E|)$ space.\n                \\item[Comment] $N$ is the number of spanning trees in $G$.\n                \\item[Reference] \\cite{Shioura1995}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning trees in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$.\n                \\item[Output] All spanning trees in $G$.\n                \\item[Complexity] INCORRECT: $O(N \\log |V| + |V|^2\\alpha(V, V) + |V||E|)$\n                \\item[Comment] $\\alpha$: the inverse Ackermann's function.  [KR2000] gives this result is wrong.\n                \\item[Reference] \\cite{Hariharan1995}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning trees in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$.\n                \\item[Output] All spanning trees in $G$.\n                \\item[Complexity] $O(|E|+ND(|V|, |E|))$ total time and $O(|E|+DS(|V|, |E|))$ space.\n                \\item[Comment] $D(|V|, |E|)$ and $DS(|V|, |E|)$ are the time and space complexities of the data structure for updating the minimum spanning tree in an undirected graph with $|V|$ vertices and $|E|$ edges. Here $N$ denotes the number of directed spanning trees in $G$.\n                \\item[Reference] \\cite{Uno1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning trees in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G=(V,E)$.\n                \\item[Output] All spanning trees included in $G$.\n                \\item[Complexity] $O(N+|V|+|E|)$ total time and $O(|V|+|E|)$ space.\n                \\item[Comment] $N$ = number of spanning trees in $G$.\n                \\item[Reference] \\cite{Shioura1997}\n            \\end{description}\n        \\subsubsection{Enumeration of the $k$ smallest weight spanning trees in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and an integer $k$.\n                \\item[Output] The $k$ smallest weight spanning trees in $G$.\n                \\item[Complexity] $O(m \\log \\log* n + k \\min(n, k)^{1\/2})$ total time, or a randomized version taking $O(m + k \\min(n, k)^{1\/2})$ total time.\n                \\item[Reference] \\cite{Eppstein1997}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning trees in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All spanning trees in $G$.\n                \\item[Complexity] $O(|V| + |E| + \\tau)$ time and $O(|V| + |E|)$ space (depth first manner) or $O(\\tau|V| + |E|)$ space (breadth first manner).\n                \\item[Comment] By using breadth first manner, the proposed algorithm can be used in a parallel computer.\n                \\item[Reference] \\cite{Matsui1997}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning trees in a graph in non-decreasing order}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All spanning trees in $G$ in non-decreasing order.\n                \\item[Complexity] $O(|V| + |E| + \\tau)$ time and $O(\\tau|V| + |E|)$ space.\n                \\item[Comment] Using breadth first manner.\n                \\item[Reference] \\cite{Matsui1997}\n            \\end{description}\n        \\subsubsection{Enumeration of all directed spanning trees in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$.\n                \\item[Output] All directed spanning trees in $G$.\n                \\item[Complexity] $O(|E| \\log |V| + |V| + N \\log^2 |V|)$ total time and $O(|E| + |V|)$ space.\n                \\item[Comment] $N$ is the number of directed spanning trees.\n                \\item[Reference] \\cite{Uno1998a}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning trees of an weighted graph in order of increasing cost}\n            \\begin{description}\n                \\item[Input] An weighted graph $G = (V, E)$.\n                \\item[Output] All spanning trees of $G$ in order of increasing cost.\n                \\item[Complexity] $O(N|E|\\log |E| + N^2)$ total time and $O(N|E|)$ space.\n                \\item[Comment] $N$ is the number of spanning trees of $G$.\n                \\item[Reference] \\cite{Sorensen2005}\n            \\end{description}\n        \\subsubsection{Enumeration of all the minimum spanning trees in a graph}\n            \\begin{description}\n                \\item[Input] An weighted graph $G = (V, E)$.\n                \\item[Output] All the minimum spanning trees in $G$.\n                \\item[Complexity] $O(N|E|\\log|V|)$ total time and $O(|E|)$ space.\n                \\item[Comment] $N$ is the number of the minimum spanning trees in $G$.\n                \\item[Reference] \\cite{Yamada2010}\n            \\end{description}\n     \\subsection{Steiner tree}\n        \\subsubsection{Enumeration of all Steiner $W$-trees in a connected graph}\n            \\begin{description}\n                \\item[Input] A connected graph $G = (V, E)$, a vertex set $W \\subseteq V$ such that $|W| = k$, for a fixed integer $k$.\n                \\item[Output] Enumeration of all Steiner $W$-trees in $G$.\n                \\item[Complexity] $O(|V|^2(|V|+|E|) + |V|^{k-2} + N|V|)$ total time with $O(|V|^{k-2} + |V|^2(|V| + |E|))$ space.\n                \\item[Comment] A connected subgraph $T$ of $G$ is a Steiner $W$-tree if $W \\subseteq V(T)$ and $|E(T)$ is minimum.\n                \\item[Reference] \\cite{Dourado2014}\n            \\end{description}\n     \\subsection{Subforest}\n        \\subsubsection{Enumeration of all $k$-trees in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$. \n                \\item[Output] All $k$-trees in $G$. \n                \\item[Comment] A $k$-tree is a forest with $k$ connected components. \n                \\item[Reference] \\cite{Hakimi1964}\n            \\end{description}\n     \\subsection{Subgraph}\n        \\subsubsection{Enumeration of all connected induced subgraph of a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All connected induced subgraph of $G$.\n                \\item[Complexity] $O(|V||E|N)$ total time and $O(|V| + |E|)$ space.\n                \\item[Comment] $N$ is the number of solutions.\n                \\item[Reference] \\cite{Avis1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all connected common maximal subgraphs in two graphs}\n            \\begin{description}\n                \\item[Input] Two graphs $G$ and $G'$.\n                \\item[Output] All connected common maximal subgraphs in $G$ and $G'$.\n                \\item[Reference] \\cite{Koch2001}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal spanning graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$, $S \\subseteq V$, and requirements $r(u, v)$ for all $(u, v) \\in V \\times V$.\n                \\item[Output] All minimal spanning graph $H$ of $G$ satisfying $\\lambda^S_H \\ge r(u, v) \\; \\forall (u, v) \\in V \\times V$.\n                \\item[Complexity] Incremental polynomial time.\n                \\item[Comment] The $S$-connectivity $\\lambda^S_G(u, v)$ of $(u, v)$ in $G$ is the maximum number of $uv$-paths such that no two of them have an edge or a node in $S \\setminus \\{u, v\\}$ in common. This complexity holds for edge-connectivity.\n                \\item[Reference] \\cite{Nutov2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-outconnected minimal spanning graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$, a vertex $s \\in V$, and an integer $k$.\n                \\item[Output] All minimal $k$-outconneted from $s$ spanning subgraph of $G$.\n                \\item[Complexity] Incremental polynomial time.\n                \\item[Comment] A graph is $k$-outconnected from $s$ if it contains $k$ internally-disjoint $st$-paths for every $t \\in V$. This complexity holds for both vertex and edge-connectivity.\n                \\item[Reference] \\cite{Nutov2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-outconnected minimal spanning graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$, a vertex $s \\in V$, and an integer $k$.\n                \\item[Output] All minimal $k$-outconneted from $s$ spanning subgraph of $G$.\n                \\item[Complexity] Incremental polynomial time.\n                \\item[Comment] A graph is $k$-outconnected from $s$ if it contains $k$ internally-disjoint $st$-paths for every $t \\in V$. This complexity holds for both vertex and edge-connectivity.\n                \\item[Reference] \\cite{Nutov2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-connected minimal spanning graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$ and an integer $k$.\n                \\item[Output] All minimal $k$-connected spanning subgraph of $G$.\n                \\item[Complexity] Incremental polynomial time.\n                \\item[Comment] This complexity holds for both vertex and edge-connectivity.\n                \\item[Reference] \\cite{Nutov2009}\n            \\end{description}\n     \\subsection{Subtree}\n        \\subsubsection{Enumeration of all subtrees in an input tree}\n            \\begin{description}\n                \\item[Input] A tree $T=(V, E)$.\n                \\item[Output] All subtrees included in $T$.\n                \\item[Complexity] $O(|V|)$ delay and $O(|V|)$ space.\n                \\item[Reference] \\cite{Ruskey1981}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-noded subtrees in a tree}\n            \\begin{description}\n                \\item[Input] A tree $T$ and an integer $k$.\n                \\item[Output] All $k$-noded subtrees in $T$.\n                \\item[Reference] \\cite{Hikita1983}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-subtrees in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G=(V, E)$ and a positive integer $k$.\n                \\item[Output] All $k$-subtrees included in $G$.\n                \\item[Complexity] $O(sk)$ total time, $O(k)$ amortized time per solution, and $O(|E|)$ space.\n                \\item[Comment] $s$ = number of $k$-subtrees in $G$, a $k$-subtree means a connected, acyclic, and edge induced subgraph with $k$ vertices.\n                \\item[Reference] \\cite{Ferreira2011}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-subtrees in an input tree}\n            \\begin{description}\n                \\item[Input] A tree $T=(V, E)$ and an integer $k$.\n                \\item[Output] All $k$-subtrees included in $T$.\n                \\item[Complexity] $O(1)$ delay and $O(|V|)$ space after $O(|V|)$ time preprocessing.\n                \\item[Comment] A $k$-subtree is a connected, acyclic, and edge induced subgraph with $k$ vertices.\n                \\item[Reference] \\cite{Wasa2012b}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-cardinarity subtrees of a tree with $w$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $k$ and a tree $T$ with $w$ element, where $k \\le w$.\n                \\item[Output] All subtrees with $k$ vertices of $T$.\n                \\item[Complexity] $O(Nw^5)$ total time.\n                \\item[Comment] $N$ is the number of ideals.\n                \\item[Reference] \\cite{Wild2013}\n            \\end{description}\n        \\subsubsection{Enumeration of all induced subtrees in a $k$-degenerate graph}\n            \\begin{description}\n                \\item[Input] A $k$-degenerate graph $G = (V, E)$.\n                \\item[Output] All induced subtrees in $G$.\n                \\item[Complexity] $O(k)$ amortized time per solution with $O(|V| + |E|)$ space and preprocessing time.\n                \\item[Reference] \\cite{Wasa2010}\n            \\end{description}\n     \\subsection{Tour}\n        \\subsubsection{Enumeration of $k$ best solutions to the Chinese postman problem solutions}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] $K$ best solutions to the Chinese postman problem.\n                \\item[Complexity] $O(S( n,m ) + K( n + m + \\log k + nT( n + m,m ) ) )$ where $S( s,t )$ denotes the time complexity of an algorithm for ordinary Chinese postman problems and $T( s,t )$ denotes the time complexity of a post-optimal algorithm for non-bipartite matching problems defined on a graph with s vertices and t edges.\n                \\item[Reference] \\cite{Saruwatari1993}\n            \\end{description}\n     \\subsection{Tree}\n        \\subsubsection{Enumeration of all binary trees with fixed number leaves in lexicographically}\n            \\begin{description}\n                \\item[Input] An integer $n$.  \n                \\item[Output] All binary trees with $n$ leaves in lexicographically. \n                \\item[Complexity] $O(1)$ time per binary tree. \n                \\item[Reference] \\cite{Ruskey1977}\n            \\end{description}\n        \\subsubsection{Enumeration of all $t$-ary trees with fixed number leaves in lexicographically}\n            \\begin{description}\n                \\item[Input] An integer $n$.  \n                \\item[Output] All $t$-ary trees with $n$ leaves in lexicographically. \n                \\item[Complexity] $O(t)$ time per binary tree. \n                \\item[Reference] \\cite{Ruskey1978a}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-ary trees with $n$ vertices}\n            \\begin{description}\n                \\item[Input] Integers $k$ and $n$. \n                \\item[Output] All $k$-ary trees with $n$ vertices\n                \\item[Complexity] $O(1)$ time per solution\n                \\item[Reference] \\cite{Trojanowski1978}\n            \\end{description}\n        \\subsubsection{Enumeration of all binary trees with $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$. \n                \\item[Output] All binary trees with $n$ vertices. \n                \\item[Reference] \\cite{Rotem1978a}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-ary trees with $n$ vertices}\n            \\begin{description}\n                \\item[Input] Integers $k$ and $n$. \n                \\item[Output] All $k$-ary trees with $n$ vertices\n                \\item[Reference] \\cite{Zaks1979}\n            \\end{description}\n        \\subsubsection{Enumeration of all rooted trees with $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All rooted trees with $n$ vertices.\n                \\item[Complexity] $O(1)$ amortized time per solution.\n                \\item[Reference] \\cite{Beyer1980}\n            \\end{description}\n        \\subsubsection{Enumeration of all binary trees with $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All binary trees with $n$ vertices.\n                \\item[Reference] \\cite{Proskurowski1980}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-ary trees in lexicographically}\n            \\begin{description}\n                \\item[Input] An integer $n$. \n                \\item[Output] All $k$-ary trees with $n$ internal vertices in lexicographically\n                \\item[Complexity] $O(1- (k-1)^{k-1}\/k^k)^{-1}$ time per solution. This limit is $4\/3$ for the binary case. \n                \\item[Reference] \\cite{Zaks1980}\n            \\end{description}\n        \\subsubsection{Enumeration of all regular $k$-ary trees with $n$ ndoes}\n            \\begin{description}\n                \\item[Input] Integers $k$ and $n$.\n                \\item[Output] All regular $k$-ary trees with $n$ ndoes.\n                \\item[Reference] \\cite{Zaks1982}\n            \\end{description}\n        \\subsubsection{Enumeration of all binary trees with $n$ vertices in the lexicographic ordering}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All binary trees with $n$ vertices in the lexicographic ordering\n                \\item[Complexity] $O(1)$ time per solution on average.\n                \\item[Reference] \\cite{Zerling1985}\n            \\end{description}\n        \\subsubsection{Enumeration of all binary trees with $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All binary trees with $n$ vertices.\n                \\item[Complexity] $O(n)$ time per solution.\n                \\item[Reference] \\cite{Proskurowski1985}\n            \\end{description}\n        \\subsubsection{Enumeration of all ordered trees with $n$ internal vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All ordered trees with $n$ internal vertices.\n                \\item[Reference] \\cite{Er1985}\n            \\end{description}\n        \\subsubsection{Enumeration of all free trees with $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All free trees with $n$ vertices.\n                \\item[Complexity] $O(1)$ time per solution.\n                \\item[Reference] \\cite{Wright1986}\n            \\end{description}\n        \\subsubsection{Enumeration of all $t$-ary trees with $n$ vertices}\n            \\begin{description}\n                \\item[Input] Integers $t$ and $n$.\n                \\item[Output] All $t$-ary trees with $n$ vertices.\n                \\item[Reference] \\cite{Er1987}\n            \\end{description}\n        \\subsubsection{Enumeration of all binary trees with $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All binary trees with $n$ vertices.\n                \\item[Complexity] $O(1)$ time per solution.\n                \\item[Reference] \\cite{Lucas1987}\n            \\end{description}\n        \\subsubsection{Enumeration of all trees with $n$ vertices and $m$ leaves}\n            \\begin{description}\n                \\item[Input] Integers $n$ and $m$.\n                \\item[Output] All trees with $n$ vertices and $m$ leaves\n                \\item[Reference] \\cite{Pallo1987}\n            \\end{description}\n        \\subsubsection{Enumeration of all $t$-ary trees in A-order}\n            \\begin{description}\n                \\item[Input] Integers $t$ and $n$.\n                \\item[Output] All $t$-ary trees with $n$ vertices in A-order.\n                \\item[Complexity] $O(1)$ amortized time per solution.\n                \\item[Reference] \\cite{VanBaronaigien1988}\n            \\end{description}\n        \\subsubsection{Enumeration of all binary trees with $n$ leaves}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All binary trees with $n$ leaves.\n                \\item[Complexity] $O(1)$ amortized time per solution.\n                \\item[Comment] A strong Gray code can be listed in constant average time per solution.\n                \\item[Reference] \\cite{Ruskey1990}\n            \\end{description}\n        \\subsubsection{Enumeration of all binary trees with $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All binary trees with $n$ vertices.\n                \\item[Complexity] $O(1)$ time per solution.\n                \\item[Comment] A loopless generation algorithm is an algorithm where the amount of computation to go from one object to the next is $O(1)$.\n                \\item[Reference] \\cite{VanBaronaigien1991}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-ary trees in natural order}\n            \\begin{description}\n                \\item[Input] Two integers $k$ and $n$.\n                \\item[Output] All $k$-ary trees with $n$ vertices.\n                \\item[Reference] \\cite{Er1992}\n            \\end{description}\n        \\subsubsection{Enumeration of all binary trees with $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All binary trees with $n$ vertices.\n                \\item[Complexity] $O(1)$ delay.\n                \\item[Reference] \\cite{Lucas1993}\n            \\end{description}\n        \\subsubsection{Enumeration of all binary trees with $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All binary trees with $n$ vertices.\n                \\item[Reference] \\cite{Bapiraju1994}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-ary tree}\n            \\begin{description}\n                \\item[Input] An integer $k$.\n                \\item[Output] All $k$-ary trees.\n                \\item[Complexity] $O(1)$ delay.\n                \\item[Reference] \\cite{Korsh1994}\n            \\end{description}\n        \\subsubsection{Enumeration of all binary trees}\n            \\begin{description}\n                \\item[Output] All binary trees.\n                \\item[Complexity] $O(1)$ time per tree.\n                \\item[Reference] \\cite{Xiang1997}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-ary trees with $n$ vertices}\n            \\begin{description}\n                \\item[Input] Two integers $k$ and $n$.\n                \\item[Output] All $k$-ary trees with $n$ vertices.\n                \\item[Complexity] $O(1)$ delay.\n                \\item[Comment] Shifts and loopless algorithm.\n                \\item[Reference] \\cite{Korsh1998}\n            \\end{description}\n        \\subsubsection{Enumeration of all rooted plane trees with at most $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All rooted plane trees with at most $n$ vertices.\n                \\item[Complexity] $O(1)$ time per tree with $O(n)$ space.\n                \\item[Comment] A \\textit{rooted plane tree} is a rooted tree with a left-to-right ordering specified for the children of each vertex.\n                \\item[Reference] \\cite{Nakano2002}\n            \\end{description}\n        \\subsubsection{Enumeration of all rooted plane trees with exactly $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All rooted plane trees with exactly $n$ vertices.\n                \\item[Complexity] $O(1)$ time per tree with $O(n)$ space.\n                \\item[Comment] A \\textit{rooted plane tree} is a rooted tree with a left-to-right ordering specified for the children of each vertex.\n                \\item[Reference] \\cite{Nakano2002}\n            \\end{description}\n        \\subsubsection{Enumeration of all rooted plane trees with at most $n$ vertices and the maximum degree $D$}\n            \\begin{description}\n                \\item[Input] Integers $n$ and $D$.\n                \\item[Output] All rooted plane trees with at most $n$ vertices and the maximum degree $D$.\n                \\item[Complexity] $O(1)$ time per tree with $O(n)$ space.\n                \\item[Comment] A \\textit{rooted plane tree} is a rooted tree with a left-to-right ordering specified for the children of each vertex.\n                \\item[Reference] \\cite{Nakano2002}\n            \\end{description}\n        \\subsubsection{Enumeration of all rooted plane trees with exactly $n$ vertices and exactly $c$ leaves}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All rooted plane trees with exactly $n$ vertices and exactly $c$ leaves .\n                \\item[Complexity] $O(n-c)$ time per tree with $O(n)$ space.\n                \\item[Comment] A \\textit{rooted plane tree} is a rooted tree with a left-to-right ordering specified for the children of each vertex.\n                \\item[Reference] \\cite{Nakano2002}\n            \\end{description}\n        \\subsubsection{Enumeration of all plane trees with exactly $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All plane trees with exactly $n$ vertices.\n                \\item[Complexity] $O(n^3)$ time per tree with $O(n)$ space.\n                \\item[Comment] A \\textit{plane tree} is a tree with a left-to-right ordering specified for the children of each vertex.\n                \\item[Reference] \\cite{Nakano2002}\n            \\end{description}\n        \\subsubsection{Enumeration of all rooted trees with at most $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All rooted tree with at most $n$ vertices.\n                \\item[Complexity] $O(1)$ time per tree and $O(n)$ space.\n                \\item[Reference] \\cite{Nakano2003}\n            \\end{description}\n        \\subsubsection{Enumeration of all $n$-trees}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All $n$-trees.\n                \\item[Complexity] $O(n^4N)$ total time.\n                \\item[Comment] Reverse search. $N$ is the number of solutions.\n                \\item[Reference] \\cite{Aringhieri2003}\n            \\end{description}\n        \\subsubsection{Enumeration of all trees with $n$ vertices and $d$ diameter}\n            \\begin{description}\n                \\item[Input] Integers $n$ and $d$.\n                \\item[Output] All trees with $n$ vertices and $d$ diameter.\n                \\item[Complexity] $O(1)$ time per tree with $O(n)$ space.\n                \\item[Comment] By using the algorithm for each $d = 2, \\dots, n-1$, all trees can be enumerated.\n                \\item[Reference] \\cite{Nakano2004}\n            \\end{description}\n        \\subsubsection{Enumeration of all $c$-tree with at most $v$ vertices and diameter $d$}\n            \\begin{description}\n                \\item[Input] Integers $n$ and $d$\n                \\item[Output] All $c$-tree with at most $v$ vertices and diameter $d$.\n                \\item[Complexity] $O(1)$ time per tree.\n                \\item[Comment] A tree is a $c$-tree if each vertex has a color $c \\in \\{c_1, \\dots, c_m\\}$.\n                \\item[Reference] \\cite{Nakano2005a}\n            \\end{description}\n        \\subsubsection{Enumeration of all nonisomorphic rooted plane trees with $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All nonisomorphic rooted plane trees with $n$ vertices.\n                \\item[Complexity] Constant amortized time per solution.\n                \\item[Reference] \\cite{Sawada2006}\n            \\end{description}\n        \\subsubsection{Enumeration of all nonisomorphic free plane trees with $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All nonisomorphic free plane trees with $n$ vertices.\n                \\item[Complexity] Constant amortized time per solution.\n                \\item[Reference] \\cite{Sawada2006}\n            \\end{description}\n        \\subsubsection{Enumeration of all multitrees satisfying given constraints}\n            \\begin{description}\n                \\item[Input] A set $\\Sigma$ of labels, a function $val: \\Sigma \\to \\mathbb{Z}^+$, and a feature vector $g$ of level $K$.\n                \\item[Output] All $(\\Sigma, val)$-labeled multitrees $T$ such that $f_K(T) = g$ and $deg(v;T) = val(\\ell(v))$ for all vertices $v \\in T$.\n                \\item[Comment] This algorithm is for chemical graphs.\n                \\item[Reference] \\cite{Ishida2008}\n            \\end{description}\n        \\subsubsection{Enumeration of all ordered trees with $n$ vertices and $k$ leaves}\n            \\begin{description}\n                \\item[Input] Integers $n$ and $k$.\n                \\item[Output] All ordered trees with $n$ vertices and $k$ leaves.\n                \\item[Complexity] $O(1)$ delay and $O(n)$ space.\n                \\item[Reference] \\cite{Yamanaka2009a}\n            \\end{description}\n        \\subsubsection{Enumeration of all trees with specified degree sequence}\n            \\begin{description}\n                \\item[Input] A degree sequence $D$.\n                \\item[Output] All trees with $D$.\n                \\item[Complexity] $O(1)$ time per tree.\n                \\item[Reference] \\cite{Nakano2009}\n            \\end{description}\n     \\subsection{Triangle}\n        \\subsubsection{Enumeration of all minimal triangle graphs with a fixed number vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.  \n                \\item[Output] All minimal triangle graphs with $n$ vertices. \n                \\item[Complexity]  \n                \\item[Reference] \\cite{Bowen1967b}\n            \\end{description}\n        \\subsubsection{Enumeration of all triangles in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All triangles in $G$.\n                \\item[Complexity] $O(\\alpha(G)|E|)$ total time and linear space.\n                \\item[Comment] $\\alpha(G)$ is the minimum number of edge-disjoint spanning forests into which $G$ can be decomposed.  If $G$ is planar, then the time complexity becomes $O(|V|)$.\n                \\item[Reference] \\cite{Chiba1985}\n            \\end{description}\n     \\subsection{Triangulation}\n        \\subsubsection{Enumeration of all triangulations of 2-sphere}\n            \\begin{description}\n                \\item[Input] A 2-sphere $G$.  \n                \\item[Output] All triangulations of $G$. \n                \\item[Complexity]  \n                \\item[Reference] \\cite{Bowen1967a}\n            \\end{description}\n        \\subsubsection{Enumeration of all $r$-rooted $2$-connected triangulations of a planar graph}\n            \\begin{description}\n                \\item[Input] A planar graph $G = (V, E)$ and an integer $r$.\n                \\item[Output] All $r$-rooted $2$-connected triangulations of $G$.\n                \\item[Complexity] $O(|V|^2N)$ total time and $O(|V|)$ space.\n                \\item[Comment] $N$ is the number of solutions.\n                \\item[Reference] \\cite{Avis1996b}\n            \\end{description}\n        \\subsubsection{Enumeration of all $r$-rooted $3$-connected triangulations of a planar graph}\n            \\begin{description}\n                \\item[Input] A planar graph $G = (V, E)$ and an integer $r$.\n                \\item[Output] All $r$-rooted $3$-connected triangulations of $G$.\n                \\item[Complexity] $O(|V|^2N)$ total time and $O(|V|)$ space.\n                \\item[Comment] $N$ is the number of solutions.\n                \\item[Reference] \\cite{Avis1996b}\n            \\end{description}\n        \\subsubsection{Enumeration of all based plane triangulations with $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All based plane triangulations.\n                \\item[Complexity] $O(1)$ time per based plane triangulation with $O(n)$ space.\n                \\item[Comment] A \\textit{based plane triangulation} is a plane triangulation with one designated edge on the outer face. The algorithm does not output entire solution but output the difference from the previous solution.\n                \\item[Reference] \\cite{Nakano2004a}\n            \\end{description}\n        \\subsubsection{Enumeration of all biconnected based plane triangulations with $n$ vertices and $r$ vertices on the outer face}\n            \\begin{description}\n                \\item[Input] Integers $n$ and $r$.\n                \\item[Output] All biconnected based plane triangulations with $n$ vertices and $r$ vertices on the outer face.\n                \\item[Complexity] $O(1)$ time per based plane triangulation with $O(n)$ space.\n                \\item[Comment] A \\textit{based plane triangulation} is a plane triangulation with one designated edge on the outer face. The algorithm does not output entire solution but output the difference from the previous solution.\n                \\item[Reference] \\cite{Nakano2004a}\n            \\end{description}\n        \\subsubsection{Enumeration of all biconnected plane triangulations with $n$ vertices and $r$ vertices on the outer face}\n            \\begin{description}\n                \\item[Input] Integers $n$ and $r$.\n                \\item[Output] All biconnected based plane triangulations with $n$ vertices and $r$ vertices on the outer face.\n                \\item[Complexity] $O(r^2n)$ time per based plane triangulation with $O(n)$ space.\n                \\item[Reference] \\cite{Nakano2004a}\n            \\end{description}\n        \\subsubsection{Enumeration of all rooted triconnected plane triangulations with at most $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All triconnected rooted plane triangulations with at most $n$ vertices.\n                \\item[Complexity] $O(1)$ time per tree and $O(n)$ space.\n                \\item[Comment] A \\textit{rooted} plane triangulation is a plane triangulation with one designated vertex on the outer face.\n                \\item[Reference] \\cite{Nakano2001}\n            \\end{description}\n        \\subsubsection{Enumeration of all rooted triconnected plane triangulations with exactly $n$ vertices and $r$ leaves}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All triconnected rooted plane triangulations with exactly $n$ vertices and $r$ leaves.\n                \\item[Complexity] $O(r)$ time per tree and $O(n)$ space.\n                \\item[Comment] A \\textit{rooted} plane triangulation is a plane triangulation with one designated vertex on the outer face.\n                \\item[Reference] \\cite{Nakano2001}\n            \\end{description}\n        \\subsubsection{Enumeration of all triconnected plane triangulations with exactly $n$ vertices and $r$ leaves}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All triconnected plane triangulations with exactly $n$ vertices and $r$ leaves.\n                \\item[Complexity] $O(r^n)$ time per triangulation and $O(n)$ space.\n                \\item[Reference] \\cite{Nakano2001}\n            \\end{description}\n        \\subsubsection{Enumeration of all triangulations}\n            \\begin{description}\n                \\item[Input] A set $S$ of $n$ points in the general position in the plane.\n                \\item[Output] all the triangulations whose vertex set is $S$ and edge set includes the convex hull of $S$.\n                \\item[Complexity] $O(\\log\\log n)$ time per triangulation and linear space.\n                \\item[Comment] Whether there is the algorithm that outputs all triangulations in constant time delay?\n                \\item[Reference] \\cite{Bespamyatnikh2002}\n            \\end{description}\n        \\subsubsection{Enumeration of all biconnected plane triangulations with $n$ vertices and $r$ vertices on the outer faces}\n            \\begin{description}\n                \\item[Input] Integers $n$ and $r$.\n                \\item[Output] All biconnected plane triangulations with $n$ vertices and $r$ vertices on the outer faces.\n                \\item[Complexity] $O(rn)$ time per triangulation and $O(n)$ space.\n                \\item[Reference] \\cite{Nakano2004b}\n            \\end{description}\n        \\subsubsection{Enumeration of all triangulations of a triconnected plane graph of $n$ vertices}\n            \\begin{description}\n                \\item[Input] A triconnected planar graph $G$ with $n$ vertices.\n                \\item[Output] All triangulations of $G$.\n                \\item[Complexity] $O(1)$ time per triangulation and $O(n)$ space.\n                \\item[Reference] \\cite{Parvez2009}\n            \\end{description}\n     \\subsection{Vertex cover}\n        \\subsubsection{Enumeration of all minimal vertex covers in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All minimal vertex covers of size up to $k$ in $G$.\n                \\item[Complexity] $O^*(1.6181^k)$ total time.\n                \\item[Comment] This algorithm also lists some non-minimal vertex covers. This algorithm uses compact representation technique.\n                \\item[Reference] \\cite{Fernau2006}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal vertex covers of size at most $k$ in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All minimal vertex covers of size at most $k$ in $G$.\n                \\item[Complexity] $O(|E| + k^2 2^k)$ total time.\n                \\item[Reference] \\cite{Damaschke2006}\n            \\end{description}\n\\section{Hypergraph}\n     \\subsection{Acyclic subhypergraph}\n        \\subsubsection{Enumeration of all maximal $\\alpha$-acyclic subhypergraphs in a hypergraph}\n            \\begin{description}\n                \\item[Input] A hypergraph $H = (V, \\mathcal{E})$.\n                \\item[Output] All maximal $\\alpha$-acyclic subhypergraphs in $H$.\n                \\item[Complexity] $O(|\\mathcal{E}|^2(|V|+|\\mathcal{E}|))$ delay and $O(|\\mathcal{E}|)$space.\n                \\item[Comment] The name of their algorithm is $\\mathtt{GenMAS}$. This algorithm uses the algorithm $\\mathtt{FindMAS}$ that outputs a maximal $\\alpha$-acyclic subhypergraph.\n                \\item[Reference] \\cite{Daigo2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all Berge acyclic subhypergraphs in a hypergraph}\n            \\begin{description}\n                \\item[Input] A hypergraph $\\mathcal{H}$.\n                \\item[Output] All Berge acyclic subhypergraphs in $\\mathcal{H}$.\n                \\item[Complexity] $O(rd\\tau(m))$ time per subhypergraph.\n                \\item[Comment] $r$ and $d$ are the rank and the degree of $\\mathcal{H}$ and $\\tau(m) = O((\\log\\log m)^2 \/ \\log \\log \\log (m))$.\n                \\item[Reference] \\cite{Wasa2013}\n            \\end{description}\n     \\subsection{Independent set}\n        \\subsubsection{Enumeration of all maximal independent set of a hypergraph of bounded dimension}\n            \\begin{description}\n                \\item[Input] A hypergraph $H$ of bounded dimension.\n                \\item[Output] All maximal independent set of a hypergraph of bounded dimension. \n                \\item[Comment] The proposed algorithm runs in parallel.\n                \\item[Reference] \\cite{Boros2000a}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal independent sets in a $c$-conformal hypergraph}\n            \\begin{description}\n                \\item[Input] A $c$-conformal hypergraph $\\mathcal{H} \\in A(k, r)$, where $c \\le$ constant and $k + r \\le c$.\n                \\item[Output] All maximal independent sets in $\\mathcal{H}$.\n                \\item[Complexity] Incremental polynomial time.\n                \\item[Comment] $A(k, r)$ is the class of hyperedges with $(k, r)$-bounded intersections, i.e. in which the intersection of any $k$ distinct hyperedges has size at most $r$.\n                \\item[Reference] \\cite{Boros2004}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal independent sets in a hypergraph of bounded intersections}\n            \\begin{description}\n                \\item[Input] A hypergraph $\\mathcal{H} \\in A(k, r)$, where $k + r \\le$ constant.\n                \\item[Output] All maximal independent sets in $\\mathcal{H}$.\n                \\item[Complexity] Incremental polynomial time with polynomial space.\n                \\item[Comment] $A(k, r)$ is the class of hyperedges with $(k, r)$-bounded intersections, i.e. in which the intersection of any $k$ distinct hyperedges has size at most $r$.\n                \\item[Reference] \\cite{Boros2004}\n            \\end{description}\n     \\subsection{Transversal}\n        \\subsubsection{Enumeration of all minimal transversal of a hypergraph}\n            \\begin{description}\n                \\item[Input] A hypergraph $\\mathcal{H}$.\n                \\item[Output] All minimal transversal of $\\mathcal{H}$.\n                \\item[Complexity] $(n + N)^{O(\\log n)}$ total time with $O(n \\log n)$ words.\n                \\item[Comment] $n = \\sum_{X \\in \\mathcal{H}}|X|$ and $N$ is the number of solutions.\n                \\item[Reference] \\cite{Tamaki2000}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal transversal in a hypergraph}\n            \\begin{description}\n                \\item[Input] A hypergraph $\\mathcal{H}\\in A(k, r)$.\n                \\item[Output] All minimal transversal in $\\mathcal{H}$.\n                \\item[Complexity] $O(n^{k+r+1}|\\mathcal{H}^d|^{r+1})$ total time and $O(N^{r+1})$ total space.\n                \\item[Comment] $A(k, r)$ is the class of hyperedges with $(k, r)$-bounded intersections, i.e. in which the intersection of any $k$ distinct hyperedges has size at most $r$.  Minimal transversals of hypergraphs in some restricted classes can be enumerating in polynomial delay and space.\n                \\item[Reference] \\cite{Boros2004}\n            \\end{description}\n\\section{Matroid}\n     \\subsection{Basis}\n        \\subsubsection{Enumeration of all common bases in two matroids}\n            \\begin{description}\n                \\item[Input] Two matroids  $M_1 = (E, \\beta_1)$ and $M_2 = (E, \\beta_2)$.\n                \\item[Output] All $B \\in \\beta_1 \\cap \\beta_2$.\n                \\item[Complexity] $O(|E| ( |E|^2 + t)\\lambda)$ total time and $O(|E|^2)$ space.\n                \\item[Comment] $\\lambda$ is the number of common bases and $t$ is time complexity of one pivot operation.\n                \\item[Reference] \\cite{Fukuda1995}\n            \\end{description}\n        \\subsubsection{Enumeration of all basis of a matroid}\n            \\begin{description}\n                \\item[Input] A matroid $M$ on the ground set $P$ with rank $m$.\n                \\item[Output] All basis of $M$.\n                \\item[Complexity] $O((m|P| + t(Piv))N)$ total time and space complexity independent of $N$.\n                \\item[Comment] $N$ is the number of solutions. $t(Piv)$ is the time necessary to do one pivot operation.\n                \\item[Reference] \\cite{Avis1996}\n            \\end{description}\n     \\subsection{Spanning}\n        \\subsubsection{Enumeration of all minimal spanning and connected subsets in a matroid}\n            \\begin{description}\n                \\item[Input] A matroid $M$.\n                \\item[Output] All minimal spanning and connected subsets in $M$.\n                \\item[Complexity] Incremental quasi-polynomial time.\n                \\item[Comment] $f(x)$ is a quasi-polynomial if $f(x) \\in O(2^{polylog (n)})$.\n                \\item[Reference] \\cite{Khachiyan2006a}\n            \\end{description}\n     \\subsection{Subset}\n        \\subsubsection{Enumeration of all maximal subset}\n            \\begin{description}\n                \\item[Input] A binary matroid $M$ on ground set $S$ and $B = \\{b_1, b_2\\} \\subseteq S$.\n                \\item[Output] All maximal subsets $X$ of $A := S \\setminus B$ that span neither $b_1$ nor $b_2$.\n                \\item[Complexity] Incremental polynomial time.\n                \\item[Reference] \\cite{Khachiyan2008a}\n            \\end{description}\n\\section{Order}\n     \\subsection{Ideal}\n        \\subsubsection{Enumeration of all $k$-cardinarity ideals of a $w$-element poset}\n            \\begin{description}\n                \\item[Input] An integer $k$ and a poset $P$ with $w$ element, where $k \\le w$.\n                \\item[Output] All $k$-cardinarity ideals of $P$.\n                \\item[Complexity] $O(Nw^3)$ total time.\n                \\item[Comment] $N$ is the number of ideals.\n                \\item[Reference] \\cite{Wild2013}\n            \\end{description}\n\\section{Other}\n     \\subsection{Assignment}\n        \\subsubsection{Enumeration of all assignment}\n            \\begin{description}\n                \\item[Input] An integer $n$ and $n \\times n$ cost matrix $C = (c_{ij})$.\n                \\item[Output] All assignments that minimizes $\\sum^{i=n}_{i=1}\\sum^{j=n}_{j=1} c_{ij}x_{ij}$ subject to $\\sum^{i=n}_{i=1} x_{ij} = 1 \\; (j = 1, \\dots, n)$, $\\sum^{j=n}_{j=1} x_{ij} = 1\\; (i = 1, \\dots, n)$, and $x_{ij} \\ge 0$.\n                \\item[Reference] \\cite{Murty1968}\n            \\end{description}\n     \\subsection{Full disjunction}\n        \\subsubsection{Enumeration of all full disjunction in an acyclic set}\n            \\begin{description}\n                \\item[Input] An acyclic set of relation $R$ with $N$ tuples.\n                \\item[Output] All full disjunctions of $R$.\n                \\item[Complexity] $O(N)$ delay.\n                \\item[Reference] \\cite{Cohen2006}\n            \\end{description}\n     \\subsection{Matrix}\n        \\subsubsection{Enumeration of all minimal sets of at most $k$ rows the deletion of which leaves a PP matrix}\n            \\begin{description}\n                \\item[Input] A binary $n \\times m$ matrix $B$ and a positive integer $k$, where $n > 4k$.\n                \\item[Output] All minimal sets of at most $k$ rows the deletion of which leaves a PP matrix.\n                \\item[Complexity] $O(3^k nm)$ time.\n                \\item[Comment] A PP matrix is a perfect phylogeny matrix.\n                \\item[Reference] \\cite{Damaschke2006}\n            \\end{description}\n     \\subsection{Round-robin tournament score}\n        \\subsubsection{Enumeration of all round-robin tournament scores of $n$ players}\n            \\begin{description}\n                \\item[Input] An integer $n$. \n                \\item[Output] All round-robin tournament scores of $n$ players. \n                \\item[Reference] \\cite{Narayana1971}\n            \\end{description}\n\\section{Permutation}\n     \\subsection{Arrangements}\n        \\subsubsection{Enumeration of all arrangements with $n$ marks}\n            \\begin{description}\n                \\item[Input] An integer $n$. \n                \\item[Output] All arrangements with $n$ marks. \n                \\item[Complexity] $O(n!)$ total time. \n                \\item[Reference] \\cite{Wells1961}\n            \\end{description}\n        \\subsubsection{Enumeration of all arrangements with $n$ marks}\n            \\begin{description}\n                \\item[Input] An integer $n$. \n                \\item[Output] All arrangements with $n$ marks. \n                \\item[Complexity] $O(n!)$ total time. \n                \\item[Reference] \\cite{Johnson1963}\n            \\end{description}\n     \\subsection{Ladder lottery}\n        \\subsubsection{Enumeration of all optimal ladder lotteries}\n            \\begin{description}\n                \\item[Input] A permutation.\n                \\item[Output] All optimal ladder lotteries with satisfying the permutation.\n                \\item[Complexity] $O(1)$ time per solution on average, $O(n^2)$ space, and $O(n^2)$ time preprocessing.\n                \\item[Comment] $n$ is the length of the input permutation.  We call a ladder lottery is an optimal when the number of horizontal lines in the ladder lottery is minimum. Ladder lotteries are also known as arrangements of pseudolines.\n                \\item[Reference] \\cite{Yamanaka2010}\n            \\end{description}\n        \\subsubsection{Enumeration of all ladder lotteries with $k$ bars}\n            \\begin{description}\n                \\item[Input] A permutation $\\pi$ and integer $k$.\n                \\item[Output] All ladder lotteries of $\\pi$ with $k$ bars.\n                \\item[Complexity] $O(1)$ time per ladder lottery.\n                \\item[Comment] Ladder lotteries are also known as \\textit{Amida kuji} in Japan.\n                \\item[Reference] \\cite{Yamanaka2014}\n            \\end{description}\n     \\subsection{Set}\n        \\subsubsection{Enumeration of all permutations of a set of elements}\n            \\begin{description}\n                \\item[Input] A set $S$. \n                \\item[Output] All permutations of $S$\n                \\item[Comment] He proposed the general algorithm for some combinatorial problems. \n                \\item[Reference] \\cite{Ehrlich1973a}\n            \\end{description}\n        \\subsubsection{Enumeration of all permutations of a set of elements}\n            \\begin{description}\n                \\item[Input] A set $S$. \n                \\item[Output] All permutations of $S$\n                \\item[Reference] \\cite{Dershowitz1975}\n            \\end{description}\n\\section{SAT}\n     \\subsection{Boolean CSP}\n        \\subsubsection{Enumeration of all models of $\\phi$ by non-decreasing weight}\n            \\begin{description}\n                \\item[Input] A $\\Gamma$-formula $\\phi$.\n                \\item[Output] All models of $\\phi$ by non-decreasing weight.\n                \\item[Complexity] If $\\Gamma$ is Horn or width-2 affine, there exists a polynomial delay algorithm.\n                \\item[Comment] Otherwise, such an algorithm does not exist unless $P \\neq NP$.\n                \\item[Reference] \\cite{Creignou2011}\n            \\end{description}\n\\section{Set}\n     \\subsection{Bitstring}\n        \\subsubsection{Enumeration of all bitstrings of length $n$ that contains exactly $k$ 1's}\n            \\begin{description}\n                \\item[Input] An even integer $n$ and odd integer $k$.\n                \\item[Output] All bitstrings of length $n$ that contains exactly $k$ 1's.\n                \\item[Complexity] $O(1)$ amortized time per solution.\n                \\item[Reference] \\cite{Ruskey1988}\n            \\end{description}\n     \\subsection{Ideals}\n        \\subsubsection{Enumeration of all ideals in a poset}\n            \\begin{description}\n                \\item[Input] A poset $\\mathcal{P}$.\n                \\item[Output] All ideals in $\\mathcal{p}$.\n                \\item[Complexity] $O(1)$ delay.\n                \\item[Reference] \\cite{Koda1993}\n            \\end{description}\n     \\subsection{Partition}\n        \\subsubsection{Enumeration of all paritions in natural order}\n            \\begin{description}\n                \\item[Input] An integer $n$. \n                \\item[Output] All partitions of $n$ in natural order\n                \\item[Reference] \\cite{McKay1970}\n            \\end{description}\n        \\subsubsection{Enumeration of all paritions with restriction}\n            \\begin{description}\n                \\item[Input] Integers $k$ and $n$. \n                \\item[Output] All partitions of $n$ whose the smallest part is greater than or equal to $k$. \n                \\item[Reference] \\cite{White1970}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-partitions of $n$}\n            \\begin{description}\n                \\item[Input] Integers $k$ and $n$. \n                \\item[Output] All $k$-partitions of $n$. \n                \\item[Reference] \\cite{Narayana1971}\n            \\end{description}\n        \\subsubsection{Enumeration of all paritions of an integer}\n            \\begin{description}\n                \\item[Input] An integer $n$.  \n                \\item[Output] All paritions of $n$. \n                \\item[Complexity]  \n                \\item[Reference] \\cite{Fenner1980}\n            \\end{description}\n        \\subsubsection{Enumeration of all partitions of a set}\n            \\begin{description}\n                \\item[Input] A set $S$.\n                \\item[Output] All partitions of $S$.\n                \\item[Complexity] $O(1)$ amortized time per solution.\n                \\item[Reference] \\cite{Semba1984}\n            \\end{description}\n        \\subsubsection{Enumeration of all partions of $n$ into integers of size at most $k$}\n            \\begin{description}\n                \\item[Input] Integers $n$ and $k$.\n                \\item[Output] All partions of $n$ into integers of size at most $k$.\n                \\item[Reference] \\cite{Savage1989}\n            \\end{description}\n        \\subsubsection{Enumeration of all partitions of a set into a fixed number of blocks}\n            \\begin{description}\n                \\item[Input] An integer $k$.\n                \\item[Output] All partitions of a set into $k$ blocks\n                \\item[Complexity] $O(1)$ amortized time per solution.\n                \\item[Reference] \\cite{Ruskey1993}\n            \\end{description}\n        \\subsubsection{Enumeration of all partitions of an integer $n$}\n            \\begin{description}\n                \\item[Input] Three integers $n$, $k$, and $\\sigma$.\n                \\item[Output] All partitions $P_\\sigma(n, k)$ of $n$ into parts of size at most $k$ in which parts are congruent to $1$ modulo $\\sigma$.\n                \\item[Complexity] $O(N)$ total time. E.g., $P_3(11, 8) = P_3(11, 7) = \\left\\{ \\{7, 4\\}, \\{7, 1, 1, 1, 1\\}, \\{4, 4, 1, 1, 1\\}, \\{4, 1, 1, 1, 1, 1, 1, 1\\}, \\{1, \\dots, 1\\}\\right\\}$.\n                \\item[Comment] $N$ is the number of partitions.\n                \\item[Reference] \\cite{Rasmussen1995}\n            \\end{description}\n        \\subsubsection{Enumeration of all partitions of an integer $n$}\n            \\begin{description}\n                \\item[Input] Two integers $n$ and $k$.\n                \\item[Output] All partitions $D(n, k)$ of $n$ into distinct parts of size at most $k$.\n                \\item[Complexity] $O(N)$ total time. E.g., $D(10, 5) = \\{\\{5, 4, 1\\}, \\{5, 3, 2\\}, \\{4, 3, 2, 1\\}\\}$ and $D(11, 4) = \\emptyset$.\n                \\item[Comment] $N$ is the number of partitions.\n                \\item[Reference] \\cite{Rasmussen1995}\n            \\end{description}\n        \\subsubsection{Enumeration of all partition of $\\{1, \\dots, n\\}$ into $k$ non-empty subsets}\n            \\begin{description}\n                \\item[Input] An integer $k$.\n                \\item[Output] All partition of $\\{1, \\dots, n\\}$ into $k$ non-empty subsets.\n                \\item[Complexity] $O(1)$ time per solution.\n                \\item[Comment] The number of such partitions is known as the Stirling number of the second kind.\n                \\item[Reference] \\cite{Kawano2005b}\n            \\end{description}\n        \\subsubsection{Enumeration of all integer partitions in (anti-)lexicographical order}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All integer partitions of $n$.\n                \\item[Complexity] $O(1)$ time per solution on average.\n                \\item[Reference] \\cite{Stojmenovic2007}\n            \\end{description}\n\\section{String}\n     \\subsection{Binary string}\n        \\subsubsection{Enumeration of all binary string with fixed number ones}\n            \\begin{description}\n                \\item[Input] Two integers $n$ and $k$, where $n \\ge k$.  \n                \\item[Output] All binary string with length $n$ and $k$ ones. \n                \\item[Complexity] $O(1)$ time per binary string. \n                \\item[Reference] \\cite{Bitner1976}\n            \\end{description}\n     \\subsection{Bracelet}\n        \\subsubsection{Enumeration of all $k$-ary bracelets}\n            \\begin{description}\n                \\item[Input] $n$: a length of a bracelet, $k$: a number of alphabet size.\n                \\item[Output] All $k$-ary bracelets.\n                \\item[Complexity] $O(1)$ amortized per output and $O(n)$ space.\n                \\item[Comment] A bracelet is the lexicographically smallest element of an equivalence class of $k$-ary strings under string rotation and reversal.\n                \\item[Reference] \\cite{Sawada2001}\n            \\end{description}\n     \\subsection{Lyndon word}\n        \\subsubsection{Enumeration of all $k$-ary Lyndon brackets of length $n$}\n            \\begin{description}\n                \\item[Input] Two integers $k$ and $n$.\n                \\item[Output] All $k$-ary Lyndon brackets of length $n$.\n                \\item[Complexity] $O(n)$ time per solution.\n                \\item[Reference] \\cite{Sawada2003}\n            \\end{description}\n     \\subsection{Necklace}\n        \\subsubsection{Enumeration of all $k$-color necklaces with $n$ beads}\n            \\begin{description}\n                \\item[Input] Integers $k$ and $n$. \n                \\item[Output] All $k$-color necklaces with $n$ beads\n                \\item[Reference] \\cite{Fredricksen1978a}\n            \\end{description}\n        \\subsubsection{Enumeration of all necklaces of length $n$ with two colors}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All necklaces of length $n$ with two colors.\n                \\item[Reference] \\cite{Fredricksen1986b}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-ary necklaces}\n            \\begin{description}\n                \\item[Input] $n$: a length of a necklace, $k$: a number of alphabet size.\n                \\item[Output] All $k$-ary necklaces with length $n$.\n                \\item[Complexity] $O(1)$ amortized per output and $O(n)$ space.\n                \\item[Comment] A $k$-ary necklace is an equivalence class of k-ary strings under rotation.\n                \\item[Reference] \\cite{Ruskey1992}\n            \\end{description}\n        \\subsubsection{Enumeration of all $n$-bit necklaces with fixed density $d$}\n            \\begin{description}\n                \\item[Input] Two integer $n$ and $d$.\n                \\item[Output] All $n$-bit necklaces with fixed density $d$.\n                \\item[Complexity] $O(nN)$ total time and $O(n)$ space.\n                \\item[Comment] $N$ is the number of solutions. A density of a $n$-bit necklace $T$ is $d$ if $T$ has $d$ ones.\n                \\item[Reference] \\cite{Wang1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-ary necklaces with fixed density}\n            \\begin{description}\n                \\item[Input] $n$: a length of a necklace, $k$: a number of alphabet size, $d$: a number of nonzero characters.\n                \\item[Output] All $k$-ary necklaces with fixed density $d$.\n                \\item[Complexity] $O(1)$ amortized per output and $O(n)$ space.\n                \\item[Comment] The set of $3$-ary necklace with $2$-density and $4$-length is $N_3(4,2) = \\{0011, 0012, 0021, 0022, 0101, 0102, 0202\\}$.\n                \\item[Reference] \\cite{Ruskey1999}\n            \\end{description}\n        \\subsubsection{Enumeration of all strings of some family}\n            \\begin{description}\n                \\item[Complexity] Constant amortized time per solution.\n                \\item[Comment] This algorithm can list not only all necklaces but also all strings in other some family with CAT.\n                \\item[Reference] \\cite{Cattell2000}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-ary necklaces with fixed content of length $n$}\n            \\begin{description}\n                \\item[Input] Two integer $k$ and $n$, and a content.\n                \\item[Output] All $k$-ary necklaces with fixed content of length $n$.\n                \\item[Complexity] Constant amortized time per solution.\n                \\item[Reference] \\cite{Sawada2003a}\n            \\end{description}\n     \\subsection{Parenthesis}\n        \\subsubsection{Enumeration of all well-formed parenthesis with length $2n$}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All well-formed parenthesis with length $2n$ in lexicographical ordering.\n                \\item[Reference] \\cite{Er1983}\n            \\end{description}\n        \\subsubsection{Enumeration of all well-formed parenthesis strings of size $n$}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All well-formed parenthesis strings of size $n$.\n                \\item[Complexity] $O(1)$ delay with $O(n)$ space, or $O(n)$ delay with $O(1)$ space.\n                \\item[Reference] \\cite{Walsh1998}\n            \\end{description}\n     \\subsection{Substring}\n        \\subsubsection{Enumeration of all $k$-ary strings of length $n$ that have no substring equal to $f$}\n            \\begin{description}\n                \\item[Input] A $k$-ary string $f$ with length $m$, and a positive integer $n$.\n                \\item[Output] All $k$-ary strings of length $n$ that have no substring equal to $f$.\n                \\item[Complexity] $O(1)$ time per string.\n                \\item[Reference] \\cite{Ruskey2000a}\n            \\end{description}\n        \\subsubsection{Enumeration of all circular $k$-ary strings of length $n$ that have no substring equal to $f$}\n            \\begin{description}\n                \\item[Input] A $k$-ary string $f$ with length $m$, and a positive integer $n$.\n                \\item[Output] All circular $k$-ary strings of length $n$ that have no substring equal to $f$.\n                \\item[Complexity] $O(1)$ time per string.\n                \\item[Reference] \\cite{Ruskey2000a}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-ary necklaces of length $n$ that have no substring equal to $f$}\n            \\begin{description}\n                \\item[Input] A $k$-ary string $f$ with length $m$, and a positive integer $n$.\n                \\item[Output] All $k$-ary necklaces of length $n$ that have no substring equal to $f$.\n                \\item[Complexity] $O(1)$ time per string.\n                \\item[Comment] $f$ is an aperiodic necklace.\n                \\item[Reference] \\cite{Ruskey2000a}\n            \\end{description}\n\\section{Survey}\n     \\subsection{Enumeration}\n        \\subsubsection{Enumeration of $k$-best enumeration}\n            \\begin{description}\n                \\item[Comment] This is the full version of the Springer Encyclopedia of Algorithms, 2014.\n                \\item[Reference] \\cite{Eppstein2014}\n            \\end{description}\n     \\subsection{Graph}\n        \\subsubsection{Alglrotihms for enumeration of all cycles in a given graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All cycles belonging to $G$. \n                \\item[Comment] 26 cycle enumeration algorithms are introduced. \n                \\item[Reference] \\cite{Mateti1976a}\n            \\end{description}\n        \\subsubsection{Enumeration of clique enumeration}\n            \\begin{description}\n                \\item[Comment] In Sec.3.\n                \\item[Reference] \\cite{Pardalos1994}\n            \\end{description}\n        \\subsubsection{Enumeration of gray code algorithms}\n            \\begin{description}\n                \\item[Reference] \\cite{Savage1997}\n            \\end{description}\n     \\subsection{Logic}\n        \\subsubsection{}\n            \\begin{description}\n                \\item[Comment] The author of this paper investigate the class of databases with constant delay the answers to a query.\n                \\item[Reference] \\cite{Segoufin2013}\n            \\end{description}\n\n\\section*{Acknowledgements}\nI am deeply grateful to \nKomei Fukuda, \nLeslie Goldberg, \nChristian Komusiewicz, \nAndrea Marino, \nYasuko Matsui, \nValia Mitsou, \nShin-ichi Nakano, \nRyuhei Uehara, \nand Katsuhisa Yamanaka for enumerating.\n\n\\printbibliography\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nEinstein challenged the Copenhagen interpretation of quantum mechanics by proposing with Podolsky and Rosen \\cite{EPR35}\nGedanken-Experiments, briefly called EPR experiments.\nThese experiments were to demonstrate that quantum mechanics is\nincomplete, having missed in its description of physical reality some elements of that reality.\n\nAbout 30 years after the EPR paper, Bell \\cite{BELL93} derived an inequality for functions of elements of physical reality,\nknown since as Bell's inequality, that in his opinion had to be obeyed by all of classical physics, meaning in essence by the\nframework of Einstein's relativity. Wigner transformed Bell's derivation into a set theoretical approach \\cite{WIGN70}.\n\nThe work of Bell and his followers had very important consequences for the views on the foundations of physics.\nA majority of physicists believe that either (i) any physical\ntheory using counterfactually definite functions and obeying Einstein's local realism (see \\ref{separation}) must obey Bell-type inequalities and\/or\n(ii) Wigner's derivation of Bell-type inequalities is only based on set theory (not involving counterfactual reasoning) and\nEinstein's local realism. There is a considerable body of serious mathematical-physical work that has raised objections against\nboth (i) and (ii) (see particularly \\cite{KHRE08b} and references therein as well as \\cite{ROSI14} and \\cite{GEUR98}). We have\nshown in \\cite{HESS16b} that (i) is false and it is the purpose of this paper to show that (ii) is also false.\n\nThe belief in (ii) arose mainly from the work of Wigner \\cite{WIGN70} and a popularized version of Wigner's work by\nd'Espagnat \\cite{ESPA79}. (d'Espagnat, however, also uses counterfactual reasoning that raises additional questions\n\\cite{HESS16b}.) The fact that many EPR-types of experiments have been published that violate Bell's and Wigner's inequalities,\nis therefore presented as proof against the validity of Einstein's local realism, because it appears inconceivable that set\ntheory is incorrect.\n\nThere exists, however, the following problem with this latter assessment. Set theory represents a mathematical framework based\non axioms that may be regarded as definitions. As such, set theory is not related to any measurements and experiments and,\ntherefore, also not to EPR experiments. It is thus necessary to find a connection between the sets of elements of physical reality\nthat the experiments provide and the sets of mathematical abstractions that Wigner uses in his proof. We analyze this\nconnection in great detail and show that Wigner's sets of mathematical abstractions contain a topological-combinatorial ordering\nthat is incompatible with the actual ordering of measurement-events in space and time (or space-time). We suggest that it is\nthis incompatibility that leads to the contradictions of Wigner's work with actual EPR measurements and not any failure of\nEinstein's local realism. An analogous situation was discussed by Poincar\\'e and Einstein, who resolved the experimental\ncontradictions to Euclidean geometry by pointing to the connection of Euclidean geometry with the physical reality, a connection\nachieved by the introduction of the concept of rigid bodies \\cite{EINS82}.\n\n\nAs we will show, there exists an analogy to the Einstein-Poincar\\'e discussions, because Wigner connected his mathematical\nsets to the actual measurements in a way that is inconsistent with the space-time physics of the actual EPR experiments. Key to\nour reasoning is the fact that the set of data of the actual measurements is necessarily involving space- and time- (or\nspace-time-) coordinates to designate the correlated pairs. These sets of space and time coordinates are inconsistent with\nWigner's sets and subsets that are ordered according to location in three-dimensional space and equipment settings in that space,\nbut do not involve measurement times.\n\nWe show in detail that Wigner's derivation of Bell type inequalities (that restrict the possible correlations exhibited by EPR\nmeasurements) does actually not use Einstein's local realism at all. Instead Wigner assumes in a hidden and unjustified way\ncertain topological combinatorial properties of his mathematical sets that are inconsistent with the macroscopic properties of\nthe measurement equipment in space and time as determined by clocks, protractors and meter-measures. The contradictions between\nWigner's (Bell-type) inequalities and well known experiments (see \\ref{exp}) must, therefore be blamed on Wigner's\ntopological-combinatorial assumptions and not on Einstein's local realism.\n\nTo prove this finding, we first analyze the elements of physical reality used in EPR experiments and form the sets of physical\nelements that must be dealt with in Wigner's mathematical proof (see section~\\ref{separation}). We then review the essential portion of\nWigner's mathematical proof and point to his unwarranted assumptions (see section~\\ref{add}). Finally, in section~\\ref{sec4}\nwe deduce from these latter\nassumptions the topological-combinatorial problems of Wigner's work and conclude that the well known line ``death by experiment\"\napplies to Wigner's theory not to Einstein's.\nOur conclusions are summarized in section~\\ref{sec5}.\n\n\n\n\\section{Einstein's realism, EPRB experiments and their resulting data} \\label{separation}\n\n\n\nWe consider in the following only a variation of EPR experiments as suggested by Bohm and, therefore called EPRB experiments\n\\cite{ASPE82b,WEIH98}. This type of experiment involves the measurement of the two single entities of correlated pairs\nof particles. Each single entity is measured at a different set of space-coordinates. These space-coordinates must, from the\nviewpoint of physics, describe the entire local measurement equipment for both measurement locations. For brevity and because\nimportant measurements have actually been performed there, we will denote the two measurement locations and corresponding space\ncoordinates just by ``Tenerife\" and ``La Palma\". The space-like separation of the measurements is a necessity to invoke\nEinstein's separation principle, which is an important factor of the EPR argument and focus of the EPR discussion. Einstein's\nseparation principle excludes all influences between the two measurement stations that are faster than the speed of light in\nvacuum.\nThis principle guarantees a certain independence of the two separated measurement stations and represents\nthe core of what is commonly called Einstein's local realism. A broader discussion and detailed definition of Einstein's local\nrealism has been given by Fine~\\cite{Fine96}.\n\nIt is, therefore, of utmost importance for the discussions of Bell and Wigner to exclude in the actual experiments\ninfluences from the other location that propagate with speed slower or equal to that light. How can one make sure that there exist no\nsuch other influences and in addition that one indeed measures correlated pairs? This goal has been achieved in \\cite{ASPE82b},\n\\cite{WEIH98} and other experiments.\nThese measurements relate to the Clauser-Horn-Shimony-Holt (CHSH) inequality.\nWe note here, however, that all of our\nreasoning applies, as will become evident below, also to this type of inequality. We just discuss the Bell inequality in our\nanalysis, because Wigner's proof referred only to this type of inequality.\n\n\\bigskip\n\\subsection{Switching instruments and pairing particles}\n\nKey to Bell's and Wigner's reasoning is the rapid switching of instruments on both Tenerife and La Palma, just before a\nmeasurement of the spin of an incoming particle is performed. In this way one ensures that the instrument settings on the other\nisland cannot influence a given measurement with the speed of light or lower speeds. What Bell and Wigner wished to find out was\nwhether Einstein's local realism is correct or incorrect. The importance of this fast switching is emphasized over and over by\nBell \\cite{BELL93} in his discussions, but no trace of it can be found in either Bell's or Wigner's formalism and algebra of\ntheir actual proofs. The reason for this obvious lack of correspondence between theory and experiment is the assumption of Bell\nand all his followers that the possible outcomes of pair-measurements are just a given for their mathematical abstractions\n(functions, sets) and the fact how the pairing experimentally occurred is not included in the theory.\n\nHowever, it is precisely this pairing and timing problem, where Einstein embedded the necessity of using a space-time system,\nhis space-time system, as the basis for his Gedanken-Experiments and the theoretical approach with Podolsky and Rosen,\n\\cite{EPR35}, \\cite{HESS15}. There also exists no other known way to pair and correlate space like separated measurement\noutcomes than by space and time measurements.%\n\\footnote{We do not wish to use quantum-physics arguments in our reasoning. However, it must be said that the nature of\nthe experiments and the possibility of quantum fluctuations excludes any other method of pairing and leaves only space and time\nmeasurements. Suppose, for example, that we just count on each island the number of measurements and then assume that those with\nequal number (deduced from the counting) will be selected as correlated pairs (post measurement, of course). Then a single\ncounting error (caused for example by a quantum fluctuation in the measurement equipment) will completely mess up the ordering\nand will lead to pairing of the wrong entities. One must, therefore, use synchronized clocks in the space-like separated\nmeasurement stations, as well as knowledge about the space-like separation in order to identify which detector clicks represent,\nwith high likelihood, measurements of a correlated pair.}\n\nThe timing of the measurements is, thus, a most important element of the physical reality, of the data, and we denote the\nclock-time of the $n$th measurement in Tenerife by $t_n$ and that in La Palma by $t_n'$. Note that subsequent measurement pairs\nare necessarily time-like separated because the switching of the macroscopic settings cannot be instantaneous.  To identify\nactual correlated pairs, some rule needs to be employed that selects the pairs. An example of such a rule would be that the time\ndifference of the measurements at the two different locations (islands) must be smaller than a certain value $W$. We emphasize\nthat through any such rule an instantaneous logical and physical connection is established between the measurements at the two\ndifferent locations that has nothing to do with instantaneous influences, much less with information transfer. Without such a\nconnection, no correlations between the space-like separated measurements can be established. Only with such a connection can\nintricate correlations due to physical law be revealed that now may involve also the many-body dynamics of the measurement\nequipment of both locations.\nIt also must be emphasized that the analysis of the raw data of three different\nEPRB experiments~\\cite{WEIH98,AGUE09,ADEN12,VIST12}\ndo show a significant $W$-dependence of the amount of violations of Bell-type inequalities (actually the CHSH inequality),\nfor values of $W$ that are much smaller than the average time between the detection of photons~\\cite{ZHAO08,RAED13a}.\n\nFrom what follows in this paper, it appears imperative to perform measurements that involve dependencies on the\ntiming and on the instrument settings in relation to that timing.\n\nWigner ignored the measurement times and corresponding connections when considering possible and actual outcomes in his theory\nand we will see that this negligence has serious consequences for his set theoretic calculations of correlations.\n\n\n\\subsection{Elements of physical reality: Notation and Notebook-entries} \\label{exp}\n\n\n\nActual EPRB measurements (e.g. \\cite{ASPE82b}, \\cite{WEIH98}) monitor, in addition to the clock-times of the detector events,\ntwo equipment orientations that determine the spin (polarization) of incoming particles. These registrations, for photons often\nobtained with polarizers and avalanche photodiodes, have been denoted by a variety of symbols such as $horizontal$ and\n$vertical$ etc. and Wigner uses $+$ and $-$ for any particles that he considers.\nWe denote the measurement result for the $n$th\nmeasurement in Tenerife by $s_n$ and that in La Palma by $s_n'$ each being either $-$ or $+$.\n\nAs mentioned, the atoms and molecules constituting the measurement equipment have many body interactions with the incoming\nparticles, from both the view of Einstein's physics and also from the most modern view of quantum physics. Nevertheless, both\nBell, Wigner and all their followers describe this equipment, as the Copenhagen school of quantum mechanics does, just by a\nthree-dimensional unit vector of space, indicating the direction of a polarizer or of a Stern-Gerlach magnet. We denote this unit\nvector for the nth experiment by ${\\bf j}_n$ in Tenerife and ${\\bf j}_n'$ in La Palma, respectively. Wigner assumed that this\nunit vector may assume precisely the same three directions $\\bf a, b, c$ in both Tenerife and La Palma and that these directions\nare randomly chosen on each island in the actual experiments. Using these assumptions and notational conventions we arrive at\nEPRB notebook-entry pairs of the kind:\n\\begin{equation} [s_n, {\\bf j}_n, t_n; s_n', {\\bf j}_n', t_n' ] ,\n\\label{25septn1}\n\\end{equation}\nwhere, as mentioned, both $s_n$ and $s_n'$ may (exclusively) assume the value of either $+$ or $-$ and both ${\\bf\nj}_n$ and ${\\bf j}_n'$ are randomly chosen from the vectors $\\bf a, b, c$. {\\it The parenthesis $[ \\cdot]$ indicates the\nmeasurement of a correlated pair} and the symbol ``;\" denotes the separation of entries for Tenerife and La Palma. The number 9\nof the time-selected pairs thus could be represented by the following actual notebook entry $[+, {\\bf a}, t_9; -, {\\bf b},\nt_9']$. As far as EPRB experiments are concerned such entries are collections of elements of physical reality (in the sense of\nMach) and represent the only physical reality that we need to consider to discuss Wigner's proof.\n\nAs mentioned, both Bell and Wigner use Einstein locality as a tool of reasoning: For carefully designed EPRB experiments they\nreason, that the measurement outcome $s_n$ does not depend on what vector ${\\bf j}_n'$ is chosen in La Palma and $s_n'$ does not\ndepend on ${\\bf j}_n$ in Tenerife.\n\nWigner discussed in his paper three pairs of measurements with the given settings $[{\\bf a, b}], [{\\bf a, c}]$ and $[{\\bf b,\nc}]$, exactly those that Bell had chosen for his inequality. Such a selection gives us the following six-tuple that we call\nhenceforth Bell's six-tuple:\n\n\\begin{equation} [s_k, {\\bf a}, t_k; s_k', {\\bf b}, t_k' ] \\text{ }[s_l, {\\bf a}, t_l; s_l', {\\bf\nc}, t_l' ] \\text{ }[s_m, {\\bf b}, t_m; s_m', {\\bf c}, t_m' ] .\\label{25septn2}\n\\end{equation}\nHere $k=1,\\ldots,M$, $l=M+1,\\ldots,2M$, and $m=2M+1,\\ldots,3M$\nwhere $M$ is a large number determined by the total number of\nmeasurements. The brackets $[ \\cdot]$ indicates again the measurement of a correlated pair.  If we assume that these pairs are derived from\nthe notebook entries (\\ref{25septn1}), then Einstein's local realism does not put any constraint on the possible values of the\npairs $s_k, s_k'$, $s_l, s_l'$ and $s_m, s_m'$. For given measurement times, they may assume any of the possible $2^6$\ncombinations of values of $+$ and $-$, because from both a physical and mathematical point of view there are at this point no restrictions to the possible outcomes except that they are two-valued.\n\nFor each pair, we are interested with Wigner in only two different possibilities when measuring on the individual particles:\neither equal outcomes or unequal outcomes.\nTherefore, for a string of $M$ measurements of pairs, we have $2^M$ different possible outcomes. If we have three such strings\nof pairs we have $2^{3M}$ possible different correlations of the pair outcomes. Wigner reduces the number of correlations that he\nconsiders essential significantly further, by counting only those that yield a different fraction of the final count of equal\nrelative to  unequal outcomes. Simple counting leads then to the conclusion that for each string of $M$ measurements of\npairs we have $M+1$ different possible outcomes that lead to a different fraction of equal and unequal outcomes\n(see Table~\\ref{tab1} for an illustration of the counting procedure).\nTherefore we obtain for the six-tuple (2) $(M+1)^3$, possible outcomes that Wigner counts as different.%\n\\footnote{The mathematical proof for\nthis fact is easily performed by writing out the possible different outcomes in matrix-form (first column all equal, last all\ndifferent) and using the method of complete induction assuming correctness for $M$ and proving the result for $M+1$. We have\npreviously given a very transparent proof by using the numbers $+1$ and $-1$ for the possible equal and different pair-outcomes\nrespectively [8].}\n\n\\begin{table}[ht]\n\\caption{\nFictive example of all possible notebook entries of an experiment illustrating Wigner's method of counting.\nThe ``e'' (``u'') in columns $C_1$, $C_2$, and $C_3$ indicate that a pair yields outcomes that are equal (unequal)\nfor $M=3$ different measurements with the same setting pair and the $2^3 = 8$ possibilities given.\nThe last column labeled ``e\/u'' gives the fraction of equal and unequal correlations $C_1$, $C_2$, and $C_3$.\nThe maximum number of different items in the last column is given by $M+1=4$ in this particular example.\n}\n\\setlength{\\tabcolsep}{10pt}\n\\begin{center}\n    \\begin{tabular}{ | c | c | c | c |}\n    \\hline\n $C_1$ & $C_2$ & $C_3$ & e\/u \\\\\n    \\hline\ne& e  &e  & 3\/0 \\\\\nu& u  &e  & 1\/2 \\\\\nu& u  &u  & 0\/3 \\\\\ne& u  &u  & 1\/2 \\\\\ne& e  &u  & 2\/1 \\\\\nu& e  &u  & 1\/2 \\\\\ne& u  &e  & 2\/1 \\\\\nu& e  &e  & 2\/1 \\\\\n    \\hline\n    \\end{tabular}\n\\end{center}\n\\label{tab1}\n\\end{table}\n\nAs we have mentioned, Einstein's local realism has not been used up to now. We show below that it also is not used in Wigner's\nfurther considerations that lead to his variation of Bell's inequality. Wigner's theory does thus derive, without reference to\nEinstein locality, how many different correlations of equal and different outcomes may occur for the $3M$\npairs of measurements. Nevertheless, Wigner's number of possible different correlations is significantly reduced below the above\nmentioned value of $(M+1)^3$ and is only proportional to $M^2$.\n\nWe ask, therefore, the question: What caused the reduction of the possible number of correlations in Wigner's proof? Our answer\nis that it was not any assumption regarding Einstein's local realism. Instead, the reduction of the number of possible\ncorrelations arises from Wigner's unwarranted mathematical assumption that is related to set theoretic probability theory.\nThe reduction of possible correlations is, therefore, artificial and does not apply to EPR experiments.\nAs mentioned, the celebrated line\n``death by experiment\" is correct but it applies to Wigner's theory and not, as reported in so many sensationalist articles, to Einstein's.\n\n\n\\section{Wigner's additional assumptions} \\label{add}\n\nWigner did not consider any dependence of the measurement outcomes on the measurement times and just assumed that the possible\nor actual outcomes $s_n$ and $s_n'$ involve automatically, so to speak per fiat, a correlated pair. The measurement locations\nand times have, however, topological-combinatorial consequences for the sets of elements of physical reality that are\nrepresented by Wigner's sets of mathematical abstractions. It is just these latter consequences that we discuss here.\n\\footnote{Bell's reasoning starts to deviate here from Wigner because of his introduction of counterfactually\ndefinite functions. We have dealt with this approach in connection with a many body dynamics of particle-equipment interactions\nin \\cite{HESS16b}.}\n\nWigner defines (p. 1007, left column last paragraph of \\cite{WIGN70}) ``domains of the space of the hidden variables\" and claims\n(in agreement with our thinking about possible outcomes outlined above) that ``we have only $2^6$ essentially different domains.\nThese can be characterized by symbols:\n \\begin{equation} (\\sigma_1, \\sigma_2, \\sigma_3; \\tau_1, \\tau_2, \\tau_3)\n , \\label{22octn2}\n\\end{equation}\nall $\\sigma$ and $\\tau$ assuming two possible values: $+$ or $-$, and the $\\sigma$ referring to the first, the\n$\\tau$ to the second, particle.\" Wigner's detailed explanations also point out that the domains characterized by\n$(\\sigma_1,\\sigma_2, \\sigma_3; \\tau_1, \\tau_2, \\tau_3)$\ncorrespond precisely to the respective equipment-settings\n$(\\bf a, b, c; a, b, c)$ which leads to the 9 possible pairings (see Eq.~(6) below)\nfrom which the possible pairs of Bell's six-tuple (2) and possible outcomes\n($s_k, s_k'$, $s_l, s_l'$ and $s_m, s_m'$ in our notation) ) can be selected.\n\nIn the next paragraph Wigner states his most crucial assumption by a definition: ``Let $(\\sigma_1, \\sigma_2, \\sigma_3; \\tau_1,\n\\tau_2, \\tau_3)$ denote henceforth the probability that the hidden parameters assume, for the singlet state of the two spins, a\nvalue lying in the domain which was denoted by this symbol\". As innocuous as this definition looks, it contains a very big\nassumption, the assumption of the existence of a joint probability for the six-tuples (\\ref{22octn2}) of possible outcomes of\nthe measurements, while the actual measurements are only performed in pairs.\n\nWigner's assumption about the joint probability of the domains of variables (parameters) that determine the possible outcomes\nimplies the existence of consistent joint probabilities for the possible and actual outcomes of measurements such as\n$(+, -, -; -, +, -)$. Note, however, that these possible outcomes are now assumed to be listed in six-tuples with the\nsetting-sequence of $\\bf a, b, c$ on each island, in spite of the fact that actually only pairs are measured. From a set\ntheoretic viewpoint we must ask ourselves: Which sets is Wigner discussing? Without doubt these must be sets of measurements of\ntime-correlated pairs.\n\nAs an aside, Bernard d'Espagnat \\cite{ESPA79} presented a counterfactual interpretation of Wigner's work by introducing\nmultiple hypothetical values that  every single particle ``possesses\" and that are assumed to exist in addition to the possible measurement outcomes\nfor particle pairs. We do not make these additional unwarranted assumptions. Our reasoning below does, however, also apply to d'Espagnat's work, with the addition of our findings about\ncounterfactual approaches in \\cite{HESS16b}.%\n\nWigner himself intended, as did Bell originally, to use exclusively elements of physical reality in the sense of Mach and we discuss his\nwork from this point of view only.\n\n\n\\section{Problems with Wigner's set-theoretical assumptions}\\label{sec4}\n\nAs an illustration of how far reaching Wigner's assumptions are, consider Bell's six-tuple (\\ref{25septn2}) which contains three\ncorrelated pairs and add to each of the pairs an arbitrary (artificial) third notebook entry for the setting that is not already\ncontained in the pair-outcomes: setting $\\bf c$ for the first pair, $\\bf b$ for the second and $\\bf a$ for the third.\nAs an example we obtain then for the first pair-column a triple-column:\\footnote{This is precisely what d'Espagnat did\nin  his variation on the theme of Wigner (see p162 of his essay in Scientific American \\cite{ESPA79})}\n\n\\begin{equation} [s_k, {\\bf a}, t_k; s_k', {\\bf b}, t_k' ] \\text{ }\\rightarrow\\text{ } (s_k, {\\bf a}, t_k ; s_k', {\\bf b}, t_k'\n\/ s_?,{\\bf c}, t_?) .\\label{24octn1}\n\\end{equation}\nThe $?$ as subscripts indicate the arbitrariness of choice of the additions.\nWe have separated these additions by $``\/\"$ instead of $``;\"$, because the latter indicates the separation of the measurements\non two islands and it is not determined from which island the additions originate. The three entries are also not all correlated\nin time, because $t_?$ cannot equal any of the other measurement times of the triple. Different settings must correspond to\ndifferent times in Einstein's physics. We did not use the symbol $[ \\cdot ]$ on the right side of (\\ref{24octn1}) but used\ninstead $( \\cdot )$, because the pairing by measurement times cannot be guaranteed for all of them. Together with the other two\npairs of the Bell six-tuple, we obtain thus three triples each with settings $\\bf a, b, c$. All the followers of Bell and\nWigner, not just d'Espagnat but also Leggett-Garg \\cite{LEGG85} and others, assume explicitly or implicitly that the joint\ntriple probabilities exist and are common to all three triples.\n\nHowever, it is a well known fact of probability theory, particularly the set theoretic probability theory of Kolmogorov, that\nthe existence of joint triple (and higher order) probabilities is not guaranteed \\cite{KHRE08b}, \\cite{HESS15}. In the above\nexample, each column of triples may, for example, have different joint triple probabilities and a joint triple probability that\nis common to all three columns of triples may not exist. The proof of the validity of Bell type inequalities requires an\nexistence proof of joint triple (quadruple etc.) probabilities from the underlying physics. Wigner just assumed these\nprobabilities to exist and, therefore, assumed what he had to prove.\n\nThis is an enormously important point that has been overlooked also by Bell and all his followers \\cite{BELL93}. If one claims\nthat Wigner's approach is a set theoretic approach, we need to use a set theoretic definition of probability measures. Such a\ndefinition indeed exists and is, as mentioned, given by the probability framework of Kolmogorov \\cite{FELL68}. That framework\nteaches us that, for a countable number of considered possible outcomes such as $(\\sigma_1, \\sigma_2, \\sigma_3; \\tau_1, \\tau_2,\n\\tau_3)$, one can define consistent joint probabilities if and only if the actual measurements can also be listed or taken in\nform of such six-tuples so that we can establish a one to one correlation of any of Wigner's $\\sigma, \\tau$ with the (possible)\nmeasurement outcomes $s_n$ and $s_n'$. Wigner has thus made a mathematical assumption of serious consequences without having any\nphysical or mathematical reason.\n\n\nSeveral different cases need to be considered to present the full proof that Wigner's approach must not be applied to EPRB\nexperiments. We just present one example that is typical and list the following six-tuple using Wigner's subsets related to\nsettings $\\bf a, b, c$ for both Tenerife and La Palma with a possible choice of measurement times included:\n\n\\begin{equation} (s_h, {\\bf a}, t_h \/\ns_i, {\\bf b}, t_i \/s_j, {\\bf c} ,t_j \\text{ } ; \\text{ }s_i', {\\bf a}, t_i' \/ s_j', {\\bf b}, t_j' \/ s_h', {\\bf c}, t_h').\n\\label{26septn1}\n\\end{equation}\n\nThe symbol $``\/\"$ separates the three Wigner subsets on one given island. The indexes $h, i, j$,\nrepresent natural numbers $1, 2, 3...$ that are now different from those used previously $(k, l, m)$ in the six-tuple\n(\\ref{25septn2}). The reason for using a different notation is that six-tuples (\\ref{25septn2}) and (\\ref{26septn1}) cannot be\ntransformed into each other without violating the space and time correlations of the actual pairing arising from the original\nmeasurements. The particular chosen pairing of six-tuple (\\ref{26septn1}) is determined by the measurement times and thus is\ngiven by $[{\\bf a; c}]$, $[{\\bf b; a}]$ and $[{\\bf c; b}]$, while the pairing of six-tuple (\\ref{25septn2}) is $[{\\bf a; b}]$,\n$[{\\bf a; c}]$ and $[{\\bf b; c}]$ and the pairing by measurement times must be different whenever the topology (location and setting) of the\nmeasurements of the pairs is different.\n\nWigner never included considerations of both measurement times and topology and used six-tuple (\\ref{26septn1}) to\nobtain the possible outcomes for all the following 9 possible pairings:\n\\begin{equation} ({\\bf a; a}), ({\\bf a; b}), [{\\bf a;\nc}], [{\\bf b; a}], ({\\bf b; b}), ({\\bf b; c}), ({\\bf c; a}), [{\\bf c; b}], ({\\bf c; c}). \\label{22octn1}\n\\end{equation}\nWe use again $[ \\cdot ]$ only for the guaranteed correlated pairs (through measurement times).\nFor all other pairs we use $( \\cdot )$.\nFor the pairings of Bell's six-tuple (\\ref{25septn2}) and using (\\ref{22octn2}) Wigner thus obtains:\n\n\\begin{equation} (\\sigma_1,\n{\\bf a}; \\tau_2, {\\bf b}) \\text{ }[\\sigma_1, {\\bf a}; \\tau_3, {\\bf c} ] \\text{ }(\\sigma_2, {\\bf b}; \\tau_3, {\\bf\nc}) .\\label{22octn3}\n\\end{equation}\n\nThe $\\sigma$ and $\\tau$ denote here a certain given outcome of either$+$ or $-$. For\nexample, we may have $\\sigma_1 = +$ , $\\tau_3 = -$, $\\tau_2 = +$ and $\\sigma_2 = -$. This innocently looking fact represents a\nbig restriction for the possible outcomes. The two pairs with settings $(\\bf a; b)$ and $[{\\bf a; c}]$ must now have the same\noutcome $\\sigma_1=+$ in Tenerife and the same is true for the two pairs $[{\\bf a; c}]$ and $({\\bf b; c})$ which now must have\nidentical outcome $\\tau_3= -$ in La Palma in spite of the fact that they must, in principle, correspond to two different pairs\nof actual measurement.\n\nWe consider now the possible correlations for the outcomes of the $M$ six-tuples of Bell (corresponding to $3M$ measurements of pairs) that lead to different ratios of equal and different outcomes of the pairs.\nThe number of these correlations is reduced from the original $(M+1)^3$ to only $2(M+1)^2$ different possibilities of correlations as can be seen from six-tuple (\\ref{22octn3}). If we also\nrequire that $\\tau_2 = \\sigma_2$ as Wigner did, because he argued that equal settings need to have equal outcomes, then we\nobtain only $(M+1)^2$ different possibilities of correlating equal $++,--$ and different $+-,-+$ outcomes for the pairs in\nBell's six-tuples (\\ref{25septn2}). This latter reduction is identical to the reduction obtained by Bell in his inequality.\n\nAs seen from the measurement times, only 3 of the pairings of (\\ref{22octn1}) do correspond to originally correlated pairs.\nNote that $(\\bf a; b)$ and $[\\bf b; a]$ cannot be treated on the same footing, although they have only interchanged settings. It\nis also very important to note that only one pair of Wigner's (\\ref{22octn3}) is actually correlated. Each of the equal settings\nchosen on the same island must appear with different measurement time in Bell's six-tuple (\\ref{25septn2}). In contrast, any\nsetting that appears twice on the same island must be associated with the same measurement and, therefore, be related to the\nsame measurement time because of Wigner's procedure involving six-tuple (\\ref{26septn1}) . {\\it However, it is physically\nimpossible to actually measure two different pairs which exhibit the same setting and the same measurement time at one given\nlocation}.\n\nThe ordering of the mathematical abstractions that form the subsets that Wigner chose are simply incompatible with\nthe ordering of the original measurements in space and time. The measurement time is the only guarantee for the correct pairing\nof the original measurements. These facts demonstrate that Wigner oversimplified complex topological-combinatorial factors that\nare vital for the outcomes of his considerations. Another important fact, for the incorrectness of Wigner's pairings that arise\nfrom his 9 possibilities (\\ref{22octn1}), is that the pairs with equal settings are not derived from originally correlated pairs\nand therefore may randomly assume all of the value-pairs $++$, $--$, $+-$ and $-+$. They contribute to the equal outcome\n$++, --$ count, that is so important for Wigner's reasoning, only half of what Wigner actually assumed.\n\nThus, Wigner reduced the number of possible correlations of EPRB measurements by assuming without justification the\nexistence of certain joint probabilities. He did not realize this fact and was, therefore, faced with the problem to explain\nthe contradictions with actual experiments if they occurred\n(and as indeed were found~\\cite{ASPE82b,WEIH98,AGUE09,ADEN12,VIST12}).\nBecause Wigner was convinced that he used only set theory and Einstein's local realism, he mentioned that a violation of Einstein's\nseparation principle would lead to ``$4^9$ domains .....of the nine measurements\" corresponding to Eq.~( \\ref{22octn1}). He thus\nintroduced violations of Einstein's realism to explain possible future disagreements of his theory with experiments and\nmeasurements, so to speak as a deus ex machina, that would resolve contradictions; not as essential part of his proof.\n\nWe note in passing that all of our (and Wigner's) reasoning that is related to the triple of settings $\\bf a, b, c$ and Bell's\ninequality can be repeated with the same findings for quadruples $\\bf a, b, c, d$ and the CHSH inequality.\n\n\\section{Conclusion}\\label{sec5}\n\nThese facts show that Wigner's procedure to derive Bell's inequality is set-theoretically neither general nor sound.\nWigner's work, taken in conjunction with experiments such as presented in \\cite{ASPE82b,WEIH98,AGUE09,ADEN12,VIST12} does not prove\nviolations of Einstein's realism. It proves only that Wigner's assumptions about the existence of certain joint probabilities\nare incorrect.  As discussed above, the essential part of Wigner's proof does not use Einstein's local realism at all.\nThus, we believe to have shown beyond any reasonable doubt that Wigner derived his\nreduction of possible correlations and his Bell type inequality from set theoretically unjustified assumptions about the\nexistence of joint probabilities.\n\n\\section*{Aknowledgement}\nThe authors thank Michael Revzen for bringing to our attention that the caption of Table I was lacking clarity.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}