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+{"text":"\\section{Introduction}\n\nThere are a great number of applications for differential geometry and\nmathematical physics. This applications can be use in many areas in this\ncentury. One of the most important applications of differential geometry is\non geodesics. A geodesic is the shortest route between two points. Geodesics\ncan be found with the help of the Euler-Lagrange and Hamilton equations.\nAlso, the information about them can be seen in many mechanical and geometry\nbooks. It is well known that differential geometry provides a suitable field\nfor studying Lagrangians and Hamiltonians of classical mechanics\\ and field\ntheory. So, the dynamic equations for moving bodies were obtained for\nLagrangian and Hamiltonian mechanics by many authors and are illustrated as\nfollows:\n\n\\textbf{I.} \\textbf{Lagrange Dynamics Equations} \\cite{klein,deleon,abraham}\n\\textbf{\\ }Let $M$ be an $n$-dimensional manifold and $TM$ its tangent\nbundle with canonical projection $\\tau _{M}:TM\\rightarrow M$. $TM$ is called\nthe phase space of velocities of the base manifold $M$. Let $L:TM\\rightarrow\nR$ be a differentiable function on $TM$\\ and is called the \\textbf\nLagrangian function}. We consider closed 2-form on $TM$\n\\begin{equation}\n\\Phi _{L}=-dd_{J}L.  \\label{0.1}\n\\end{equation\n(if $J^{2}=-I,$ $J$\\ is a complex structure or if $J^{2}=I,$ $J$ is a\npara-complex structure and $Tr(J)=0$). Consider the equatio\n\\begin{equation}\n\\mathbf{i}_{\\xi }\\Phi _{L}=dE_{L}.  \\label{1}\n\\end{equation\nWhere the semispray $\\xi $ is a vector field. We know that $E_{L}=V\\left(\nL\\right) -L$\\ \\ is an energy function and $V=J(\\xi )$\\ a Liouville vector\nfield. Here $dE_{L}$ denotes the differential of $E_{L}$. It is well-known\nthat (\\ref{1}) under a certain condition on $\\xi $ is the intrinsical\nexpression of the Euler-Lagrange equations of motion. This equation is named\n\\ as \\textbf{Lagrange dynamical equation}. The triple $(TM,\\Phi _{L},\\xi )$\nis known as \\textbf{Lagrangian system} on the tangent bundle $TM$. The\noperations run on (\\ref{1}) for\\ any coordinate system $(q^{i}(t),p_{i}(t))\n. Infinite dimension \\textbf{Lagrangian's\\ equation }is obtained the form\nbelow\n\\begin{equation}\n\\begin{array}{l}\n\\frac{dq^{i}}{dt}=\\dot{q}^{i}\\ ,\\ \\frac{d}{dt}\\left( \\frac{\\partial L}\n\\partial \\dot{q}^{i}}\\right) =\\frac{\\partial L}{\\partial q^{i}}\\ ,\\\ni=1,...,n\n\\end{array}\n\\label{2}\n\\end{equation\n\\textbf{II.} \\textbf{Hamilton Dynamics Equations }\\cite{deleon,abraham\n\\textbf{: }Let $M$ be the base manifold and its cotangent manifold $T^{\\ast\n}M$. By a symplectic form we mean a 2-form $\\Phi $ on $T^{\\ast }M$ such that:\n\n\\textbf{(i)} $\\Phi $ is closed , that is, $d\\Phi =0;$ \\textbf{(ii)} for each \n$z\\in T^{\\ast }M$ , $\\Phi _{z}:T_{z}T^{\\ast }M\\times T_{z}T^{\\ast\n}M\\rightarrow \n\\mathbb{R}\n\\ $is weakly non$\\deg $enerate. If $\\Phi _{z}$ in (ii) is non$\\deg $enerate,\nwe speak of a \\textbf{strong symplectic form}. If (ii) is dropped we refer\nto $\\Phi $\\ as a presymplectic form\\textbf{. }Now\\textbf{\\ }let $(T^{\\ast\n}M,\\Phi )$ us take as a symplectic manifold. A vector field $Z_{H}:T^{\\ast\n}M\\rightarrow TT^{\\ast }M$ is called \\textbf{Hamiltonian vector field}\\ if\nthere is a $C^{1}$ \\textbf{Hamiltonian function}\\ $H:T^{\\ast }M\\rightarrow \n\\mathbb{R}\n$ such that \\textbf{Hamilton dynamical equation} is determined b\n\\begin{equation}\n\\mathbf{i}_{Z_{H}}\\Phi =dH.  \\label{3}\n\\end{equation\nWe say that $Z_{H}$ is locally Hamiltonian vector field if $\\Phi $\\ is\nclosed. Where $\\Phi $ shows the canonical symplectic form so that $\\Phi\n=-d\\lambda ,$ $\\lambda =J^{\\ast }(\\omega ),$ such that $J^{\\ast }$ a dual of \n$J,$ $\\omega $ a 1-form on $T^{\\ast }M$. The triple $(T^{\\ast }M,\\Phi\n,Z_{H}) $ is named \\textbf{Hamiltonian system }which it is defined on the\ncotangent bundle $T^{\\ast }M.$ From the local expression of $Z_{H}$ we see\nthat $(q^{i}(t),p_{i}(t))$ is an integral curve of $Z_{H}$ iff \\textbf\nHamilton's equations }are expressed as follows\n\\begin{equation}\n\\dot{q}^{i}=\\frac{\\partial H}{\\partial p_{i}}\\ ,\\ \\dot{p}_{i}=-\\frac\n\\partial H}{\\partial q^{i}}.  \\label{4}\n\\end{equation\n\\textbf{Considering information the above, in a lot of articles and books,\nit is possible to show how differential geometric methods are applied in\nLagrangian's and Hamiltonian's mechanics in the below. Some works in\nparacomplex geometry are used for mathematical models. }\n\n\\textit{Cruceanu}, \\textit{Fortuny} and \\textit{Geada} have presented\nparacomplex geometry which is related to algebra of paracomplex number and\nthe study of the structures on differentiable manifolds called paracomplex\nstructures. Furthermore, they have considered a compatible neutral\npseudo-Riemannian metric, the para-Hermitian and para-Kahler structures, and\ntheir variants \\textbf{\\cite{cruceanu}. }Kaneyuki and Kozai have introduced\na class of affine symmetric spaces, which are called para-Hermitian\nsymmetric spaces, a paracomplex analogue Hermitian symmetric space \\textbf\n\\cite{kaneyuki}. }In the study of para-Kahlerian manifolds, \\textit{Tekkoyun}\nhas introduced paracomplex analogues of the Euler-Lagrange and Hamilton\nequations. Furthermore, the geometric results on the related mechanic\nsystems have been presented\\textbf{\\ \\cite{tekkoyun1}. }\\textit{Etayo} an\n\\textit{\\ Santamaria} studied connections attached to non-integrable almost\nbiparacomplex manifolds. Manifolds endowed with three foliations pairwise\ntransversal are called $3$-webs. Similarly, they can be algebraically\ndefined as biparacomplex or complex product manifolds, i.e., manifolds\nendowed with three tensor fields of type $(1,1),$ $F,$ $P$ and $J=FoP$,\nwhere the two first are product and the third one is complex, and they\nmutually anti-commute. In this case, it is well known that there exists a\nunique torsion-free connection parallelizing the structure. A para-K\\\"{a\nhlerian manifold $M$\\ is said to be endowed with an almost\nbi-para-Lagrangian structure (a bi-para-Lagrangian manifold) if $M$\\ has two\ntransversal Lagrangian distributions (involutive transversal Lagrangian\ndistributions) $D_{1}$\\ and $D_{2}$\\ \\cite{etayo}. \\textit{Carinena} and \n\\textit{Ibort} obtained the Lax equations which are associated with a\ndynamical endowed with a bi-Lagrangian connection and a closed two-form \n\\Omega $ parallel along dynamical field $\\Gamma $. The case of Lagrangian\ndynamical systems is analysed and the nonnoether constants of motion found\nby Hojman and Harleston are recovered as being associated to a reduced Lax\nequation. Completely integrable dynamical systems have been shown to be a\nparticular case of these systems by their \\textbf{\\cite{carinena}. }\\textit\nGordejuela} and \\textit{Santamaria} have proved that the canonical\nconnection of a bi-Lagrangian manifold introduced which was by Hess is a\nLevi-Civita connection by showing that a bi-Lagrangian manifold (i.e. a\nsymplectic manifold endowed with two transversal Lagrangian foliations) is\nendowed with a canonical semi-Riemannian metric \\textbf{\\cite{etayo2}. \n\\textit{Kanai} has been concerned with closed $C^{\\infty }$ Riemannian\nmanifolds of negative curvature whose geodesic flows have $C^{\\infty }$\nstable and unstable foliations. In particular, we have shown that the\ngeodesic flow of such a manifold is isomorphic to that of a certain closed\nRiemannian manifold of constant negative curvature if the dimension of the\nmanifold is greater than two and if the sectional curvature lies between \n\\frac{-9}{4}$ and $-1$ strictly \\textbf{\\cite{kanai}. }Since they have shown\nfundamental physical properties in turbulence (conservation laws, wall laws,\nKolmogorov energy spectrum,...), symmetries are used to analyse common\nturbulence models. A class of symmetry preserving turbulence models has been\nproposed. This class has been refined in such a way that the models respect\nthe second law of thermodynamics. Moreover, an example of model belonging to\nthe class has been numerically tested by \\textit{Razafindralandy} and \n\\textit{Hamdouni} \\textbf{\\ \\cite{dina}. }A base-equation method has been\nimplemented to actualize the hereditary algebra of the Korteweg--de Vries\n(KdV) hierarchy and the N-soliton manifold is reconstructed. The novelty of\nour approach is the fact that it can in a rather natural way, predict other\nnonlinear evolution equations which admit local conservation laws.\nSignificantly enough, base functions themselves are found to provide a basis\nto regard the KdV-like equations as higher order degenerate bi-Lagrangian\nsystems by \\textit{Chakrabarti} and \\textit{Talukdar}\\textbf{\\ \\cit\n{chakrabarti}. }Bi-para-complex analogue of Lagrangian and Hamiltonian\nsystems has been introduced on Lagrangian distributions by \\textit{Tekkoyun}\nand \\textit{Sari}. Additionally, the geometric and physical results related\nto bi-para-dynamical systems have also been presented by them \\textbf{\\cit\n{tekkoyun2010}. }Authors introduced generalized-quaternionic K\\\"{a}hler\nanalogue of Lagrangian and Hamiltonian mechanical systems. Moreover , the\ngeometrical-physical results which are related to generalized-quaternionic \n\\\"{a}hler mechanical systems have also been also given by Tekkoyun and Yayl\n\\textbf{\\ \\cite{tekkoyun2011}.}\n\nIn the above studies; although para-complex mechanical systems were analyzed\nsuccessfully in relatively broad area of science, they have not dealt with\nbi-para-complex conformal mechanical systems on the bi-Lagrangian conformal\nmanifold. In this study, therefore, equations related to bi-para-conformal\nmechanical systems on the bi-Lagrangian conformal manifold used in obtaining\ngeometric quantization have been presented.\n\n\\section{Preliminaries}\n\nIn this study, all the manifolds and geometric objects are $C^{\\infty }$\\\nand the Einstein summation convention $\\left( \\sum x_{i}=x_{i}\\right) $ is\nin use. Also, $A$, $F(TM)$, $\\chi (TM)$\\ and $\\Lambda ^{1}(TM)$\\ denote the\nset of para-complex numbers, the set of para-complex functions on $TM$, the\nset of para-complex vector fields on $TM$\\ and the set of para-complex\n1-forms on $TM$, respectively. The definitions and geometric structures on\nthe differential manifold $M$\\ given in \\cite{cruceanu} may be extended to \nTM$\\ as follows:\n\n\\section{Conformal Geometry}\n\nA conformal map is a function which preserves angles. Conformal maps can be\ndefined between domains in higher dimensional Euclidean spaces, and more\ngenerally on a Riemann or semi-Riemann manifold. A conformal manifold is a\ndifferentiable manifold equipped with an equivalence class of (pseudo)\nRiemann metric tensors, in which two metrics $g^{\\prime }$ and $g$ are\nequivalent if and only i\n\\begin{equation}\ng^{\\prime }=\\lambda ^{2}g  \\label{4.1}\n\\end{equation\nwhere $\\lambda >0$ is a smooth positive function. An equivalence class of\nsuch metrics is known as a conformal metric or conformal class \\cite{wiki}.\nTwo Riemann metrics $g$ and $g^{\\prime }$ on $M$ \\ are said to be equivalent\nif and only i\n\\begin{equation}\ng^{\\prime }=e^{\\lambda }g  \\label{7}\n\\end{equation\nwhere $\\lambda $ is a smooth function on $M$. The equation given by (\\ref{7\n) is called a \\textbf{Conformal Structure }\\cite{folland}.\n\n\\section{Bi-Para-Complex Geometry}\n\nLet $M$ be a differentiable manifold. An almost bi-para-complex structure on \n$M$ is denoted by two tensor fields $F$ and $P$ of type (1,1) giving \nF^{2}=P^{2}=1$, $F\\circ P+P\\circ F=0$ \\cite{etayo}. It is seen that $P\\circ\nF $ is an almost complex structure. If the matrix-structure defined by the\nalmost bi-para-complex structure is integrable then for every point $p\\in M$\nthere exists an open neighbourhood $U$ of $p$ and local coordinates \n(U;x^{_{i}},y^{i})$ such tha\n\\begin{eqnarray}\nF(\\partial \/\\partial x^{i}) &=&\\partial \/\\partial y^{i},F(\\partial \/\\partial\ny^{i})=\\partial \/\\partial x^{i},  \\label{2.0} \\\\\nP(\\partial \/\\partial x^{i}) &=&\\partial \/\\partial x^{i},\\text{ }P(\\partial\n\/\\partial y^{i})=-\\partial \/\\partial y^{i},\\text{ }\\forall i=\\overline{1,n} \n\\notag\n\\end{eqnarray\n\\cite{etayo1}. The existence of these kind of local coordinates on $M$\npermits to construct holomorphic local coordinates, $(U;z^{k}),$ \nz^{k}=x^{k}+\\mathbf{i}y^{k}$, $\\mathbf{i}^{2}=-1,$ $k=\\overline{1,n},$ or\npara-holomorphic local coordinates, $(U;z^{k}),$ $z^{k}=x^{k}+\\mathbf{j\ny^{k},k=\\overline{1,n},$ $\\mathbf{j}^{2}=1$ \\cite{gadea, newlander}. \n(M,g,J) $ is a para-K\\\"{a}hlerian manifold that always has two transversal\ndistributions defined by the eigen-spaces associated to $+1$ and $-1$\neigenvalues of $J$. Besides, the mentioned distributions are involutive\nLagrangian distributions if somebody thinks of the symplectic form $\\Phi $\ndefined by \n\\begin{equation*}\n\\Phi (X,Y)=g(JX,Y),\\forall X,Y\\in \\chi (M).\n\\end{equation*\nConsider $(x^{i},y^{i})$ to be a real coordinate system on a neighborhood $U$\nof any point $p$ of $M.$ Also let $\\{\\frac{\\partial }{\\partial x^{i}},\\frac\n\\partial }{\\partial y^{i}}\\}$ and $\\{dx^{i},dy^{i}\\}$ be natural bases over \nR$ of the tangent space $T_{p}(M)$ and the cotangent space $T_{p}^{\\ast }(M)$\nof $M,$ respectively. Then the below equalities may be written b\n\\begin{equation}\nJ(\\frac{\\partial }{\\partial x^{i}})=\\frac{\\partial }{\\partial y^{i}\n,\\,\\,\\,\\,\\,\\,\\,J(\\frac{\\partial }{\\partial y^{i}})=\\frac{\\partial }{\\partial\nx^{i}}.  \\label{2.1}\n\\end{equation\nLet $\\ z^{i}=\\ x^{i}+\\mathbf{j}\\ y^{i},$ $\\mathbf{j}^{2}=1,$ also be a\npara-complex local coordinate system on $M.$ So the vector fields will be\nshown\n\\begin{equation}\n\\frac{\\partial }{\\partial z^{i}}=\\frac{1}{2}\\{\\frac{\\partial }{\\partial x^{i\n}-\\mathbf{j}\\frac{\\partial }{\\partial y^{i}}\\},\\,\\,\\,\\frac{\\partial }\n\\partial \\overline{z}^{i}}=\\frac{1}{2}\\{\\frac{\\partial }{\\partial x^{i}}\n\\mathbf{j}\\frac{\\partial }{\\partial y^{i}}\\}.  \\label{2.2}\n\\end{equation\nwhich represent the bases of $M$. Also, the dual covector fields are \n\\begin{equation}\ndz^{i}=dx^{i}+\\mathbf{j}dy^{i},\\,\\,\\,\\,\\,d\\overline{z}^{i}=dx^{i}-\\mathbf{j\ndy^{i}  \\label{2.22}\n\\end{equation\nwhich represent the cobases of $M$. Then the following expression can be\nwritte\n\\begin{equation}\nJ(\\frac{\\partial }{\\partial z^{i}})=\\mathbf{-j}\\frac{\\partial }{\\partial \n\\overline{z}^{i}},\\,\\,\\,\\,\\,\\,J(\\frac{\\partial }{\\partial \\overline{z}^{i}})\n\\mathbf{j}\\frac{\\partial }{\\partial z^{i}}.  \\label{2.3}\n\\end{equation\nThe dual endomorphism $J^{\\ast }$ of $T_{p}^{\\ast }(M)$ at any point $p$ of\nthe manifold $M$ satisfies that $J^{\\ast 2}=I,$ and is denoted b\n\\begin{equation}\nJ^{\\ast }(dz^{i})=\\mathbf{-j}d\\overline{z}^{i},\\,\\,\\,\\,\\,\\,J^{\\ast }(\n\\overline{z}^{i})=\\mathbf{j}dz^{i}.  \\label{2.4}\n\\end{equation\nLet $V^{A}$ be a commutative group $(V,+)$ endowed with a structure of\nunitary module over the ring $A.$ Let $V^{R}$ denote the group $(V,+)$\nendowed with the structure of real vector space inherited from the\nrestriction of scalars to $R\\mathbf{.}$ The vector space $V^{R}$ will then\nbe called the real vector space associated to $V^{A}.$ Settin\n\\begin{equation}\nJ(u)=ju,\\,\\,\\,\\,\\,\\,P^{+}(u)=e^{+}u,\\,\\,\\,P^{-}(u)=e^{-}u\\,\\,,\\,\\,\\,u\\in\nV^{A},  \\label{2.5}\n\\end{equation\nthe equalitie\n\\begin{equation}\n\\begin{array}{l}\nJ^{2}=1_{V},\\,\\,\\,\\,P^{+2}=P^{+},\\,\\,\\,\\,P^{-2}=P^{-},\\,\\,\\,\\,\\,\\,P^{+}\\circ\n\\,\\,\\,P^{-}=P^{-}\\circ P^{+}=0 \\\\ \nP^{+}+P^{-}=1_{V},\\,\\,\\,P^{+}-\\,\\,\\,P^{-}=J, \\\\ \nP^{-}=(1\/2)(1_{V}-J),\\,\\,\\,\\,\\,\\,P^{+}=(1\/2)(1_{V}+J), \\\\ \nj^{2}=1,\\,\\,\\,\\,e^{+2}=e^{+},\\,\\,\\,\\,e^{-2}=e^{-},\\,\\,\\,\\,\\,\\,e^{+}\\circ\n\\,\\,\\,e^{-}=e^{-}\\circ e^{+}=0,\\, \\\\ \n\\,\\,e^{+}+e^{-}=1,\\,\\,\\,e^{+}-\\,\\,\\,e^{-}=j, \\\\ \ne^{-}=(1\/2)(1-j),\\,e^{+}=(1\/2)(1+j)\n\\end{array}\n\\label{2.6}\n\\end{equation\ncan be found. Also, we calculated whic\n\\begin{equation}\n\\begin{array}{l}\nP^{\\mp }\\left( \\frac{\\partial }{\\partial z^{i}}\\right) =-e^{\\mp }\\frac\n\\partial }{\\partial \\overline{z}^{i}}\\text{ \\ \\ \\ \\ },\\text{ \\ \\ \\ \\ }P^{\\mp\n}\\left( \\frac{\\partial }{\\partial \\overline{z}^{i}}\\right) =e^{\\mp }\\frac\n\\partial }{\\partial z^{i}}, \\\\ \nP^{\\ast \\mp }\\left( dz^{i}\\right) =-e^{\\mp }d\\overline{z}^{i}\\text{ \\ \\ \\ \\ \n,\\text{ \\ \\ \\ \\ }P^{\\ast \\mp }\\left( d\\overline{z}^{i}\\right) =e^{\\mp\n}dz^{i}\n\\end{array}\n\\label{2.8}\n\\end{equation\nIf the conformal manifold $(M,g,J=P^{+}-P^{-})$ satisfies the following\nconditions simultaneously then the conformal manifold is an almost\npara-conformal\\textbf{\\ }Hermitian manifold. The first expression can be\ngiven as follows\n\\begin{equation}\ng(X,Y)+g(X,Y)=0\\Leftrightarrow g(X,Y)=0,\\,\\,\\,\\,\\forall X,Y\\in \\chi (D_{1}),\n\\label{2.9}\n\\end{equation\nsince $P^{+}$ and $P^{-}$ are the projections over $D_{1}$ and $D_{2}$\nrespectively. Then $(P^{+}-P^{-})(X)=P^{+}X-P^{-}X=P^{+}X=X,$ \n(P^{+}-P^{-})(Y)=P^{+}Y-P^{-}Y=P^{+}Y=Y.$ Similarly the second expression\ncan be shown as follows\n\\begin{equation}\ng(X,Y)+g(X,Y)=0\\Leftrightarrow g(X,Y)=0,\\,\\,\\,\\,\\forall X,Y\\in \\chi (D_{2}).\n\\label{2.10}\n\\end{equation\nLet $X=X_{1}+X_{2},Y=Y_{1}+Y_{2}$ be vector fields on $M$ such that \nX_{1},Y_{1}\\in D_{1}$ and $X_{2},Y_{2}\\in D_{2}.$ The\n\\begin{equation}\n\\begin{array}{c}\ng(JX,Y)=g(JX_{1}+JX_{2},Y_{1}+Y_{2})=g(X_{1}-X_{2},Y_{1}+Y_{2}) \\\\ \n=g(X_{1},Y_{1})-g(X_{2},Y_{1})+g(X_{1},Y_{2})-g(X_{2},Y_{2}) \\\\ \n=-g(X_{2},Y_{1})+g(X_{1},Y_{2}), \\\\ \ng(X,JY)=g(X_{1}+X_{2},JY_{1}+JY_{2})=g(X_{1}+X_{2},Y_{1}-Y_{2}) \\\\ \n=g(X_{1},Y_{1})+g(X_{2},Y_{1})-g(X_{1},Y_{2})-g(X_{2},Y_{2}) \\\\ \n=g(X_{2},Y_{1})-g(X_{1},Y_{2})\n\\end{array}\n\\label{2.101}\n\\end{equation\nand hence \ng(JX,Y)+g(X,JY)=-g(X_{2},Y_{1})+g(X_{1},Y_{2})+g(X_{2},Y_{1})-g(X_{1},Y_{2})=0, \n$ for all vector fields $X,Y$ on $M$. If the conditions (\\ref{2.9}) and (\\re\n{2.10}) are true then $D_{1}$ and $D_{2}$ are Lagrangian distributions in\nterms of the 2- form $\\Phi (X,Y)=g(JX,Y).$ Therefore, if the almost\npara-complex structure $J$\\ is integrable then $(M,g,J)$ is a para-conformal\nK\\\"{a}hlerian manifold, or equivalently, $(M,\\Phi ,D_{1},D_{2})$ is a\nbi-Lagrangian conformal manifold. \\cite{gilkey,weyl,kim}. Where $W_{\\pm }$\nis a conformal para-complex structure to be similar to an integrable almost\n(para)-complex $P^{\\mp }$ given in (\\ref{2.8}). Similarly $W_{\\pm }^{\\ast }$\nare the dual of $W_{\\pm }$ structures. So, we adapt the following equations\nusing (\\ref{7})\n\\begin{equation}\n\\begin{array}{l}\nW^{\\mp }\\left( \\frac{\\partial }{\\partial z^{i}}\\right) =-e^{\\mp }e^{\\lambda \n\\frac{\\partial }{\\partial \\overline{z}^{i}}\\text{ \\ \\ \\ \\ },\\text{ \\ \\ \\ \\ \nW^{\\mp }\\left( \\frac{\\partial }{\\partial \\overline{z}^{i}}\\right) =e^{\\mp\n}e^{-\\lambda }\\frac{\\partial }{\\partial z^{i}}, \\\\ \nW^{\\ast \\mp }\\left( dz_{i}\\right) =-e^{\\mp }e^{\\lambda }d\\overline{z}_{i\n\\text{\\ \\ \\ \\ },\\text{ \\ \\ \\ \\ }W^{\\ast \\mp }\\left( dz_{i}\\right) =e^{\\mp\n}e^{-\\lambda }d\\overline{z}_{i}\n\\end{array}\n\\label{2.11}\n\\end{equation}\n\n\\section{Conformal Bi-Para Euler-Lagrangians}\n\nHere, conformal bi-para-Euler-Lagrange equations and a conformal\nbi-para-mechanical system will be obtained under the consideration of the\nbasis $\\{e^{+},e^{-}\\}$ on the bi-Lagrangian conformal manifold $(M,\\Phi\n,D_{1},D_{2})$. Let $(W^{+},W^{-})$ be an almost bi-para-complex conformal\nstructure on $(M,\\Phi ,D_{1},D_{2})$, and $(z^{i},\\overline{z}^{i})$ be its\npara-complex coordinates. Let the vector field $\\xi $ be a semispray given b\n\\begin{equation}\n\\begin{array}{l}\n\\xi =e^{+}(\\xi ^{i+}\\frac{\\partial }{\\partial z^{i}}+\\overline{\\xi }^{i+\n\\frac{\\partial }{\\partial \\overline{z}^{i}})+e^{-}(\\xi ^{i-}\\frac{\\partial }\n\\partial z^{i}}+\\overline{\\xi }^{i-}\\frac{\\partial }{\\partial \\overline{z\n^{i}}); \\\\ \nz^{i}=\\,z^{i+}e^{+}+z^{i-}e^{-};\\,\\overset{.}{\\,z}^{i}=\\overset{.}{\\,z\n^{i+}e^{+}+\\overset{.}{z}^{i-}e^{-}=\\xi ^{i+}e^{+}+\\xi ^{i-}e^{-}; \\\\ \n\\overline{z}^{i}=\\,\\overline{z}^{i+}e^{+}+\\overline{z}^{i-}e^{-};\\,\\overset{\n}{\\,\\overline{z}}^{i}=\\overset{.}{\\,\\overline{z}}^{i+}e^{+}+\\overset{.}\n\\overline{z}}^{i-}e^{-}=\\overline{\\xi }^{i+}e^{+}+\\overline{\\xi }^{i-}e^{-}\n\\end{array}\n\\label{3.1}\n\\end{equation\nwhere the dot indicates the derivative with respect to time $t$. The vector\nfield denoted by $V=(W^{+}-W^{-})(\\xi )$ and given b\n\\begin{equation}\n(W^{+}-W^{-})(\\xi )=e^{+}(-e^{\\lambda }\\xi ^{i+}\\frac{\\partial }{\\partial\nz^{i}}+e^{-\\lambda }\\overline{\\xi }^{i+}\\frac{\\partial }{\\partial \\overline{\n}^{i}})-e^{-}(-e^{\\lambda }\\xi ^{i-}\\frac{\\partial }{\\partial z^{i}\n+e^{-\\lambda }\\overline{\\xi }^{i-}\\frac{\\partial }{\\partial \\overline{z}^{i}\n)  \\label{3.2}\n\\end{equation\nis called conformal \\textit{bi}-\\textit{para} \\textit{Liouville vector field}\non the bi-Lagrangian conformal manifold. The maps given by $T,$ \nP:M\\rightarrow A$ such that $T=\\frac{1}{2}m_{i}(\\overline{z}^{i})^{2}=\\frac{\n}{2}m_{i}(\\overset{.}{z}^{i})^{2},$ $P=m_{i}gh$ are called \\textit{the\nkinetic energy} and \\textit{the potential energy of the system,}\nrespectively.\\textit{\\ }Here\\textit{\\ }$m_{i},g$ and $h$ stand for mass of a\nmechanical system having $n_{i}$ particle, the gravity acceleration and\ndistance to the origin of a mechanical system on the bi-Lagrangian conformal\nmanifold,\n\nrespectively. Then $L:M\\rightarrow A$ is a map that satisfies the conditions;\n\n\\textbf{i)} $L=T-P$ is a conformal \\textit{bi-para} \\textit{Lagrangian\nfunction,}\n\n\\textbf{ii)} the function given by $E_{L}=V(L)-L$ is \\textit{a conformal\nbi-para energy function}.\n\nThe operator $i_{(W^{+}-W^{-})}$ induced by $W^{+}-W^{-}$ and shown b\n\\begin{equation}\ni_{W^{+}-W^{-}}\\omega (Z_{1},Z_{2},...,Z_{r})=\\sum_{i=1}^{r}\\omega\n(Z_{1},...,(W^{+}-W^{-})Z_{i},...,Z_{r})  \\label{3.3}\n\\end{equation\nis said to be \\textit{vertical derivation, }where $\\omega \\in \\wedge\n^{r}{}M, $ $Z_{i}\\in \\chi (M).$ The \\textit{vertical differentiation} \nd_{(P^{+}-P^{-})}$ is defined b\n\\begin{equation}\nd_{(W^{+}-W^{-})}=[i_{(W^{+}-W^{-})},d]=i_{(W^{+}-W^{-})}d-di_{(W^{+}-W^{-})}\n\\label{3.4}\n\\end{equation\nwhere $d$ is the usual exterior derivation. For an almost para-complex\nstructure $W^{+}-W^{-}$, the closed para-conformal K\\\"{a}hlerian form is the\nclosed 2-form given by $\\Phi _{L}=-dd_{_{(W^{+}-W^{-})}}L$ such tha\n\\begin{equation}\nd_{\\left( W^{+}-W^{-}\\right) }L=\\mathbf{e}^{+}\\left( -e^{\\lambda }\\frac\n\\partial L}{\\partial \\overline{z}^{i}}dz^{i}+e^{-\\lambda }\\frac{\\partial L}\n\\partial z^{i}}d\\overline{z}^{i}\\right) -\\mathbf{e}^{-}\\left( -e^{\\lambda \n\\frac{\\partial L}{\\partial \\overline{z}^{i}}dz^{i}+e^{-\\lambda }\\frac\n\\partial L}{\\partial z^{i}}d\\overline{z}^{i}\\right) :\\mathcal{F\n(M)\\rightarrow \\wedge ^{1}{}M  \\label{3.5}\n\\end{equation\nLet $\\xi $ be the second order differential equations given by equation (\\re\n{3.1}) an\n\\begin{equation}\n\\begin{array}{l}\ni_{\\xi }\\Phi _{L}=\\Phi _{L}(\\xi ) \\\\ \n=-\\mathbf{e}^{+}\\xi ^{i+}e^{\\lambda }\\frac{\\partial \\lambda }{\\partial z^{j}\n\\frac{\\partial L}{\\partial \\overline{z}^{i}}dz^{j}+\\mathbf{e}^{+}\\xi\n^{i+}e^{\\lambda }\\frac{\\partial \\lambda }{\\partial z^{j}}\\frac{\\partial L}\n\\partial \\overline{z}^{i}}dz^{i}-\\mathbf{e}^{+}\\xi ^{i+}e^{\\lambda }\\frac\n\\partial ^{2}L}{\\partial z^{j}\\partial \\overline{z}^{i}}dz^{j}+\\mathbf{e\n^{+}\\xi ^{i+}e^{\\lambda }\\frac{\\partial ^{2}L}{\\partial z^{j}\\partial \n\\overline{z}^{i}}dz^{i} \\\\ \n-\\mathbf{e}^{+}\\overline{\\xi }^{i+}e^{-\\lambda }\\frac{\\partial \\lambda }\n\\partial z^{j}}\\frac{\\partial L}{\\partial z^{i}}dz^{i}+\\mathbf{e}^{+\n\\overline{\\xi }^{i+}e^{-\\lambda }\\frac{\\partial ^{2}L}{\\partial\nz^{j}\\partial z^{i}}dz^{i}+\\mathbf{e}^{+}\\overline{\\xi }^{i+}e^{\\lambda \n\\frac{\\partial \\lambda }{\\partial \\overline{z}^{j}}\\frac{\\partial L}\n\\partial \\overline{z}^{i}}dz^{i}+\\mathbf{e}^{+}\\overline{\\xi \n^{i+}e^{\\lambda }\\frac{\\partial ^{2}L}{\\partial \\overline{z}^{j}\\partial \n\\overline{z}^{i}}dz^{i} \\\\ \n+\\mathbf{e}^{-}\\xi ^{i-}e^{\\lambda }\\frac{\\partial \\lambda }{\\partial z^{j}\n\\frac{\\partial L}{\\partial \\overline{z}^{i}}dz^{j}-\\mathbf{e}^{-}\\xi\n^{i-}e^{\\lambda }\\frac{\\partial \\lambda }{\\partial z^{j}}\\frac{\\partial L}\n\\partial \\overline{z}^{i}}dz^{i}+\\mathbf{e}^{-}\\xi ^{i-}e^{\\lambda }\\frac\n\\partial ^{2}L}{\\partial z^{j}\\partial \\overline{z}^{i}}dz^{j}-\\mathbf{e\n^{-}\\xi ^{i-}e^{\\lambda }\\frac{\\partial ^{2}L}{\\partial z^{j}\\partial \n\\overline{z}^{i}}dz^{i} \\\\ \n+\\mathbf{e}^{-}\\overline{\\xi }^{i-}e^{-\\lambda }\\frac{\\partial \\lambda }\n\\partial z^{j}}\\frac{\\partial L}{\\partial z^{i}}dz^{i}-\\mathbf{e}^{-\n\\overline{\\xi }^{i-}e^{-\\lambda }\\frac{\\partial ^{2}L}{\\partial\nz^{j}\\partial z^{i}}dz^{i}-\\mathbf{e}^{-}\\overline{\\xi }^{i-}e^{\\lambda \n\\frac{\\partial \\lambda }{\\partial \\overline{z}^{j}}\\frac{\\partial L}\n\\partial \\overline{z}^{i}}dz^{i}-\\mathbf{e}^{-}\\overline{\\xi \n^{i-}e^{\\lambda }\\frac{\\partial ^{2}L}{\\partial \\overline{z}^{j}\\partial \n\\overline{z}^{i}}dz^{i} \\\\ \n\\mathbf{e}^{+}\\xi ^{i+}e^{-\\lambda }\\frac{\\partial \\lambda }{\\partial z^{j}\n\\frac{\\partial L}{\\partial z^{i}}d\\overline{z}^{i}-\\mathbf{e}^{+}\\xi\n^{i+}e^{-\\lambda }\\frac{\\partial ^{2}L}{\\partial z^{j}\\partial z^{i}}\n\\overline{z}^{i}-\\mathbf{e}^{+}\\xi ^{i+}e^{\\lambda }\\frac{\\partial \\lambda }\n\\partial \\overline{z}^{j}}\\frac{\\partial L}{\\partial \\overline{z}^{i}}\n\\overline{z}^{j}-\\mathbf{e}^{+}\\xi ^{i+}e^{\\lambda }\\frac{\\partial ^{2}L}\n\\partial \\overline{z}^{j}\\partial \\overline{z}^{i}}d\\overline{z}^{j} \\\\ \n-\\mathbf{e}^{+}\\overline{\\xi }^{i+}e^{-\\lambda }\\frac{\\partial \\lambda }\n\\partial \\overline{z}^{j}}\\frac{\\partial L}{\\partial z^{i}}d\\overline{z}^{i}\n\\mathbf{e}^{+}\\overline{\\xi }^{i+}e^{-\\lambda }\\frac{\\partial \\lambda }\n\\partial \\overline{z}^{j}}\\frac{\\partial L}{\\partial z^{i}}d\\overline{z}^{j}\n\\mathbf{e}^{+}\\overline{\\xi }^{i+}e^{-\\lambda }\\frac{\\partial ^{2}L}\n\\partial \\overline{z}^{j}\\partial z^{i}}d\\overline{z}^{i}-\\mathbf{e}^{+\n\\overline{\\xi }^{i+}e^{-\\lambda }\\frac{\\partial ^{2}L}{\\partial \\overline{z\n^{j}\\partial z^{i}}d\\overline{z}^{j} \\\\ \n-\\mathbf{e}^{-}\\xi ^{i-}e^{-\\lambda }\\frac{\\partial \\lambda }{\\partial z^{j}\n\\frac{\\partial L}{\\partial z^{i}}d\\overline{z}^{i}+\\mathbf{e}^{-}\\xi\n^{i-}e^{-\\lambda }\\frac{\\partial ^{2}L}{\\partial z^{j}\\partial z^{i}}\n\\overline{z}^{i}+\\mathbf{e}^{-}\\xi ^{i-}e^{\\lambda }\\frac{\\partial \\lambda }\n\\partial \\overline{z}^{j}}\\frac{\\partial L}{\\partial \\overline{z}^{i}}\n\\overline{z}^{j}+\\mathbf{e}^{-}\\xi ^{i-}e^{\\lambda }\\frac{\\partial ^{2}L}\n\\partial \\overline{z}^{j}\\partial \\overline{z}^{i}}d\\overline{z}^{j} \\\\ \n+\\mathbf{e}^{-}\\overline{\\xi }^{i-}e^{-\\lambda }\\frac{\\partial \\lambda }\n\\partial \\overline{z}^{j}}\\frac{\\partial L}{\\partial z^{i}}d\\overline{z}^{j}\n\\mathbf{e}^{-}\\overline{\\xi }^{i-}e^{-\\lambda }\\frac{\\partial \\lambda }\n\\partial \\overline{z}^{j}}\\frac{\\partial L}{\\partial z^{i}}d\\overline{z}^{j}\n\\mathbf{e}^{-}\\overline{\\xi }^{i-}e^{-\\lambda }\\frac{\\partial ^{2}L}\n\\partial \\overline{z}^{j}\\partial z^{i}}d\\overline{z}^{j}+\\mathbf{e}^{-\n\\overline{\\xi }^{i-}e^{-\\lambda }\\frac{\\partial ^{2}L}{\\partial \\overline{z\n^{j}\\partial z^{i}}d\\overline{z}^{j}\n\\end{array}\n\\label{3.7}\n\\end{equation\nSince the closed conformal para K\\\"{a}hlerian form $\\Phi _{L}$ on $M$ is in\na para-symplectic\\emph{\\ }structure, it is found tha\n\\begin{equation}\n\\begin{array}{l}\nE_{L}=\\mathbf{e}^{+}\\left( -\\xi ^{i+}e^{\\lambda }\\frac{\\partial L}{\\partial \n\\overline{z}^{i}}+\\overline{\\xi }^{i+}e^{-\\lambda }\\frac{\\partial L}\n\\partial z^{i}}\\right) -\\mathbf{e}^{-}\\left( -\\xi ^{i-}e^{\\lambda }\\frac\n\\partial L}{\\partial \\overline{z}^{i}}+\\overline{\\xi }^{i-}e^{-\\lambda \n\\frac{\\partial L}{\\partial z^{i}}\\right) -\n\\end{array}\n\\label{3.8}\n\\end{equation\nand thu\n\\begin{equation}\n\\begin{array}{l}\ndE_{L}=\\mathbf{e}^{+}\\left[ -\\xi ^{i+}e^{\\lambda }\\frac{\\partial \\lambda }\n\\partial z^{j}}\\frac{\\partial L}{\\partial \\overline{z}^{i}}dz^{j}-\\xi\n^{i+}e^{\\lambda }\\frac{\\partial ^{2}L}{\\partial z^{j}\\overline{z}^{i}}dz^{j}\n\\overline{\\xi }^{i+}e^{-\\lambda }\\frac{\\partial \\lambda }{\\partial z^{j}\n\\frac{\\partial L}{\\partial z^{i}}dz^{j}+\\overline{\\xi }^{i+}e^{-\\lambda \n\\frac{\\partial ^{2}L}{\\partial z^{j}\\partial z^{i}}dz^{j}\\right] \\\\ \n-\\mathbf{e}^{-}\\left[ -\\xi ^{i-}e^{\\lambda }\\frac{\\partial \\lambda }\n\\partial z^{j}}\\frac{\\partial L}{\\partial \\overline{z}^{i}}dz^{j}-\\xi\n^{i-}e^{\\lambda }\\frac{\\partial ^{2}L}{\\partial z^{j}\\partial \\overline{z\n^{i}}dz^{j}-\\overline{\\xi }^{i-}e^{-\\lambda }\\frac{\\partial \\lambda }\n\\partial z^{j}}\\frac{\\partial L}{\\partial z^{i}}dz^{j}+\\overline{\\xi \n^{i-}e^{-\\lambda }\\frac{\\partial ^{2}L}{\\partial z^{j}\\partial z^{i}}dz^{j\n\\right] -\\frac{\\partial L}{\\partial z^{j}}dz^{j} \\\\ \n+\\mathbf{e}^{+}\\left[ -\\xi ^{i+}e^{\\lambda }\\frac{\\partial \\lambda }\n\\partial \\overline{z}^{j}}\\frac{\\partial L}{\\partial \\overline{z}^{i}}\n\\overline{z}^{j}-\\xi ^{i+}e^{\\lambda }\\frac{\\partial ^{2}L}{\\partial \n\\overline{z}^{j}\\partial \\overline{z}^{i}}d\\overline{z}^{j}-\\overline{\\xi \n^{i+}e^{-\\lambda }\\frac{\\partial \\lambda }{\\partial \\overline{z}^{j}}\\frac\n\\partial L}{\\partial z^{i}}d\\overline{z}^{j}+\\overline{\\xi }^{i+}e^{-\\lambda\n}\\frac{\\partial ^{2}L}{\\partial \\overline{z}^{j}\\partial z^{i}}d\\overline{z\n^{j}\\right] \\\\ \n-\\mathbf{e}^{-}\\left[ -\\xi ^{i-}e^{\\lambda }\\frac{\\partial \\lambda }\n\\partial \\overline{z}^{j}}\\frac{\\partial L}{\\partial \\overline{z}^{i}}\n\\overline{z}^{j}-\\xi ^{i-}e^{\\lambda }\\frac{\\partial ^{2}L}{\\partial \n\\overline{z}^{j}\\partial \\overline{z}^{i}}d\\overline{z}^{j}-\\overline{\\xi \n^{i-}e^{-\\lambda }\\frac{\\partial \\lambda }{\\partial \\overline{z}^{j}}\\frac\n\\partial L}{\\partial z^{i}}d\\overline{z}^{j}+\\overline{\\xi }^{i-}e^{-\\lambda\n}\\frac{\\partial ^{2}L}{\\partial \\overline{z}^{j}\\partial z^{i}}d\\overline{z\n^{j}\\right] -\\frac{\\partial L}{\\partial \\overline{z}^{i}}d\\overline{z}^{j}\n\\end{array}\n\\label{3.9}\n\\end{equation\nUse of equation (\\ref{1}) gives\n\\begin{equation}\n\\begin{array}{l}\n\\mathbf{e}^{+}\\xi ^{i+}e^{\\lambda }\\frac{\\partial ^{2}L}{\\partial\nz^{j}\\partial \\overline{z}^{i}}+\\mathbf{e}^{+}\\overline{\\xi }^{i+}e^{\\lambda\n}\\frac{\\partial ^{2}L}{\\partial \\overline{z}^{j}\\partial \\overline{z}^{i}}\n\\mathbf{e}^{-}\\xi ^{i-}e^{\\lambda }\\frac{\\partial ^{2}L}{\\partial\nz^{j}\\partial \\overline{z}^{i}}-\\mathbf{e}^{-}\\overline{\\xi }^{i-}e^{\\lambda\n}\\frac{\\partial ^{2}L}{\\partial \\overline{z}^{j}\\partial \\overline{z}^{i}}\n\\\\ \n+\\mathbf{e}^{+}\\xi ^{i+}e^{\\lambda }\\frac{\\partial \\lambda }{\\partial z^{j}\n\\frac{\\partial L}{\\partial \\overline{z}^{i}}+\\mathbf{e}^{+}\\overline{\\xi \n^{i+}e^{\\lambda }\\frac{\\partial \\lambda }{\\partial \\overline{z}^{j}}\\frac\n\\partial L}{\\partial \\overline{z}^{i}}-\\mathbf{e}^{-}\\xi ^{i-}e^{\\lambda \n\\frac{\\partial \\lambda }{\\partial z^{j}}\\frac{\\partial L}{\\partial \\overline\nz}^{i}}-\\mathbf{e}^{-}\\overline{\\xi }^{i-}e^{\\lambda }\\frac{\\partial \\lambda \n}{\\partial \\overline{z}^{j}}\\frac{\\partial L}{\\partial \\overline{z}^{i}}\n\\frac{\\partial L}{\\partial z^{j}} \\\\ \n-\\mathbf{e}^{+}\\xi ^{i+}e^{-\\lambda }\\frac{\\partial ^{2}L}{\\partial\nz^{j}\\partial z^{i}}-\\mathbf{e}^{+}\\overline{\\xi }^{i+}e^{-\\lambda }\\frac\n\\partial ^{2}L}{\\partial \\overline{z}^{j}\\partial z^{i}}+\\mathbf{e}^{-}\\xi\n^{i-}e^{-\\lambda }\\frac{\\partial ^{2}L}{\\partial z^{j}\\partial z^{i}}\n\\mathbf{e}^{-}\\overline{\\xi }^{i-}e^{-\\lambda }\\frac{\\partial ^{2}L}\n\\partial \\overline{z}^{j}\\partial z^{i}} \\\\ \n\\mathbf{e}^{+}\\xi ^{i+}e^{-\\lambda }\\frac{\\partial \\lambda }{\\partial z^{j}\n\\frac{\\partial L}{\\partial z^{i}}+e^{+}\\overline{\\xi }^{i+}e^{-\\lambda \n\\frac{\\partial \\lambda }{\\partial \\overline{z}^{j}}\\frac{\\partial L}\n\\partial z^{i}}-\\mathbf{e}^{-}\\xi ^{i-}e^{-\\lambda }\\frac{\\partial \\lambda }\n\\partial z^{j}}\\frac{\\partial L}{\\partial z^{i}}-e^{-}\\overline{\\xi \n^{i-}e^{-\\lambda }\\frac{\\partial \\lambda }{\\partial \\overline{z}^{j}}\\frac\n\\partial L}{\\partial z^{i}}+\\frac{\\partial L}{\\partial \\overline{z}^{j}}=0\n\\end{array}\n\\label{3.10}\n\\end{equation\nIf a curve denoted by $\\alpha :A\\rightarrow M$ is considered to be an\nintegral curve of $\\xi ,$ $\\xi (L)=\\frac{\\partial L}{\\partial t}$ $,$ then\nthe following equation is obtained\n\\begin{equation}\n\\begin{array}{l}\n\\left( \\mathbf{e}^{+}-\\mathbf{e}^{-}\\right) e^{\\lambda }\\left[ \\mathbf{e\n^{+}\\xi ^{i+}\\frac{\\partial }{\\partial z^{j}}+\\mathbf{e}^{+}\\overline{\\xi \n^{i+}\\frac{\\partial }{\\partial \\overline{z}^{j}}+\\mathbf{e}^{-}\\xi ^{i-\n\\frac{\\partial }{\\partial z^{j}}+\\mathbf{e}^{-}\\overline{\\xi }^{i-}\\frac\n\\partial }{\\partial \\overline{z}^{j}}\\right] \\left( \\frac{\\partial L}\n\\partial \\overline{z}^{j}}\\right) \\\\ \n+\\left( \\mathbf{e}^{+}-\\mathbf{e}^{-}\\right) e^{\\lambda }\\left[ \\mathbf{e\n^{+}\\xi ^{i+}\\frac{\\partial }{\\partial z^{j}}+\\mathbf{e}^{+}\\overline{\\xi \n^{i+}\\frac{\\partial }{\\partial \\overline{z}^{j}}+\\mathbf{e}^{-}\\xi ^{i-\n\\frac{\\partial }{\\partial z^{j}}+\\mathbf{e}^{-}\\overline{\\xi }^{i-}\\frac\n\\partial }{\\partial \\overline{z}^{j}}\\right] \\left( \\lambda \\right) \\left( \n\\frac{\\partial L}{\\partial \\overline{z}^{j}}\\right) +\\frac{\\partial L}\n\\partial z^{j}} \\\\ \n-\\left( \\mathbf{e}^{+}-\\mathbf{e}^{-}\\right) e^{-\\lambda }\\left[ \\mathbf{e\n^{+}\\xi ^{i+}\\frac{\\partial }{\\partial z^{j}}+\\mathbf{e}^{+}\\overline{\\xi \n^{i+}\\frac{\\partial }{\\partial \\overline{z}^{j}}+\\mathbf{e}^{-}\\xi ^{i-\n\\frac{\\partial }{\\partial z^{j}}+\\mathbf{e}^{-}\\overline{\\xi }^{i-}\\frac\n\\partial }{\\partial \\overline{z}^{j}}\\right] \\frac{\\partial L}{\\partial z^{j\n} \\\\ \n\\left( \\mathbf{e}^{+}-\\mathbf{e}^{-}\\right) e^{-\\lambda }\\left[ \\mathbf{e\n^{+}\\xi ^{i+}\\frac{\\partial }{\\partial z^{j}}+\\mathbf{e}^{+}\\overline{\\xi \n^{i+}\\frac{\\partial }{\\partial \\overline{z}^{j}}+\\mathbf{e}^{-}\\xi ^{i-\n\\frac{\\partial }{\\partial z^{j}}+\\mathbf{e}^{-}\\overline{\\xi }^{i-}\\frac\n\\partial }{\\partial \\overline{z}^{j}}\\right] \\left( \\lambda \\right) \\frac\n\\partial L}{\\partial z^{j}}+\\frac{\\partial L}{\\partial \\overline{z}^{j}}=\n\\end{array}\n\\label{3.11}\n\\end{equation\no\n\\begin{eqnarray}\n\\left( \\mathbf{e}^{+}-\\mathbf{e}^{-}\\right) e^{\\lambda }\\xi \\left( \\frac\n\\partial L}{\\partial \\overline{z}^{i}}\\right) +\\left( \\mathbf{e}^{+}-\\mathbf\ne}^{-}\\right) e^{\\lambda }\\xi \\left( \\lambda \\right) \\left( \\frac{\\partial \n}{\\partial \\overline{z}^{i}}\\right) +\\frac{\\partial L}{\\partial z^{i}} &=&0,\n\\label{3.12} \\\\\n-\\left( \\mathbf{e}^{+}-\\mathbf{e}^{-}\\right) e^{-\\lambda }\\xi \\left( \\frac\n\\partial L}{\\partial z^{i}}\\right) +\\left( \\mathbf{e}^{+}-\\mathbf{e\n^{-}\\right) e^{-\\lambda }\\xi \\left( \\lambda \\right) \\frac{\\partial L}\n\\partial z^{i}}+\\frac{\\partial L}{\\partial \\overline{z}^{i}} &=&0.  \\notag\n\\end{eqnarray\nThen the following equations are found\n\\begin{equation}\n\\left( \\mathbf{e}^{+}-\\mathbf{e}^{-}\\right) \\frac{\\partial }{\\partial t\n\\left( e^{\\lambda }\\frac{\\partial L}{\\partial \\overline{z}^{i}}\\right) \n\\frac{\\partial L}{\\partial z^{i}}=0\\text{ \\ , \\ }\\left( \\mathbf{e}^{+}\n\\mathbf{e}^{-}\\right) \\frac{\\partial }{\\partial t}\\left( e^{-\\lambda }\\frac\n\\partial L}{\\partial z^{i}}\\right) -\\frac{\\partial L}{\\partial \\overline{z\n^{i}}=0.  \\label{3.13}\n\\end{equation\nThus equations\\textbf{\\ }(\\ref{3.13}) are seen to be \\textbf{conformal\nbi-para Euler-Lagrange equations} on the distributions $D_{1}$ and $D_{2},$\nand then the triple $(M,\\Phi _{L},\\xi )$ is seen to be a \\textbf{conformal\nbi-para mechanical system}\\textit{\\ }with taking into account the basis \n\\{e^{+},e^{-}\\}$ on the bi-Lagrangian conformal manifold $(M,\\Phi\n,D_{1},D_{2})$.\n\n\\section{Conformal Bi-Para Hamiltonians}\n\nIn the part, conformal bi-para Hamilton equations and a conformal bi-para\nHamiltonian mechanical system on the bi Lagrangian conformal manifold \n(M,\\Phi ,D_{1},D_{2})$ will be derived. Let $(z_{i},\\overline{z}_{i})$ be\nits para-complex coordinates. Let $\\{\\frac{\\partial }{\\partial z_{i}},\\frac\n\\partial }{\\partial \\overline{z}_{i}}\\}$ and $\\{dz_{i},d\\overline{z}_{i}\\},$\nbe bases and cobases of $T_{p}(M)$ and $T_{p}^{\\ast }(M)$ of $M,$\nrespectively. Let us assume that an almost bi-para-complex conformal\nstructure, a bi-para-conformal Liouville form and a bi-para-complex\nconformal 1-form on the distributions ${}D_{1}$ and $D_{2}$ are shown by \nW^{\\ast +}-W^{\\ast -}$, $\\lambda $ and $\\omega $, respectively. Then, we\nusing \\cite{miron} and (\\ref{2.11}):\n\n\\begin{equation}\n\\begin{array}{c}\n\\omega =\\frac{1}{2}[(z_{i}dz_{i}+\\overline{z}_{i}d\\overline{z\n_{i})e^{+}+(e^{2\\lambda }z_{i}dz_{i}+e^{2\\lambda }\\overline{z}_{i}d\\overline\nz}_{i})e^{-}], \\\\ \n\\lambda =(W^{\\ast +}-W^{\\ast -})(\\omega )=\\frac{1}{2}[(\\mathbf{-\ne^{+}e^{\\lambda }z_{i}d\\overline{z}_{i}+e^{+}e^{\\lambda }\\overline{z\n_{i}dz_{i})] \\\\ \n-\\frac{1}{2}(\\mathbf{-}e^{-}e^{\\lambda }z_{i}d\\overline{z\n_{i}+e^{-}e^{\\lambda }\\overline{z}_{i}dz_{i})]\n\\end{array}\n\\label{4.111}\n\\end{equation\nIt is well known that if $\\Phi $ is a closed para-K\\\"{a}hlerian form on the\nbi-Lagrangian conformal manifold, then $\\Phi $ is also a para-symplectic\nstructure on ${}$the bi-Lagrangian conformal manifold. Given a\nbi-para-conformal Hamiltonian vector field $Z_{H}$ fixed with the\nbi-para-conformal Hamiltonian energy\\textit{\\ }$H$ that is \n\\begin{equation}\nZ_{H}=(Z_{i}\\frac{\\partial }{\\partial z_{i}}+\\overline{Z}_{i}\\frac{\\partial \n}{\\partial \\overline{z}_{i}})e^{+}+(Z_{i}\\frac{\\partial }{\\partial z_{i}}\n\\overline{Z}_{i}\\frac{\\partial }{\\partial \\overline{z}_{i}})e^{-}.\n\\label{4.2}\n\\end{equation\nThen closed 2-form i\n\\begin{eqnarray}\n\\Phi &=&-d\\lambda =e^{+}\\overline{Z}_{i}dz_{i}-e^{-}\\overline{Z}_{i}dz_{i}\n\\frac{1}{2}\\left[ \n\\begin{array}{c}\ne^{+}e^{\\lambda }\\frac{\\partial \\lambda }{\\partial z_{i}}\\bar{Z\n_{i}z_{i}dz_{i}+e^{+}e^{\\lambda }\\frac{\\partial \\lambda }{\\partial \\overline\nz}_{i}}\\overline{Z}_{i}\\overline{z}_{i}dz_{i} \\\\ \n-e^{-}e^{\\lambda }\\frac{\\partial \\lambda }{\\partial z_{i}}\\overline{Z\n_{i}z_{i}dz_{i}-e^{-}e^{\\lambda }\\frac{\\partial \\lambda }{\\partial \\overline\nz}_{i}}\\overline{Z}_{i}\\overline{z}_{i}dz_{i\n\\end{array\n\\right]  \\label{4.3} \\\\\n&&-e^{+}Z_{i}d\\overline{z}_{i}+e^{-}Z_{i}d\\overline{z}_{i}-\\frac{1}{2}\\left[ \n\\begin{array}{c}\n\\mathbf{-}e^{+}e^{\\lambda }\\frac{\\partial \\lambda }{\\partial z_{i}\nZ_{i}z_{i}d\\overline{z}_{i}-e^{+}e^{\\lambda }\\frac{\\partial \\lambda }\n\\partial \\overline{z}_{i}}Z_{i}\\overline{z}_{i}d\\overline{z}_{i} \\\\ \n+e^{-}e^{-}e^{\\lambda }\\frac{\\partial \\lambda }{\\partial z_{i}}Z_{i}z_{i}\n\\overline{z}_{i}+e^{-}e^{\\lambda }\\frac{\\partial \\lambda }{\\partial \n\\overline{z}_{i}}Z_{i}\\overline{z}_{i}d\\overline{z}_{i\n\\end{array\n\\right] .  \\notag\n\\end{eqnarray\nAnd then it follow\n\\begin{equation}\n\\begin{array}{c}\ni_{Z_{H}}\\Phi =\\Phi (Z_{H}) \\\\ \n=\\bar{Z}_{i}e^{+}\\left[ 1-\\frac{1}{2}e^{\\lambda }\\left( z_{i}\\frac{\\partial\n\\lambda }{\\partial z_{i}}+\\overline{z}_{i}\\frac{\\partial \\lambda }{\\partial \n\\overline{z}_{i}}\\right) \\right] dz_{i}-Z_{i}e^{+}\\left[ 1+\\frac{1}{2\ne^{\\lambda }\\left( z_{i}\\frac{\\partial \\lambda }{\\partial z_{i}}+\\overline{z\n_{i}\\frac{\\partial \\lambda }{\\partial \\overline{z}_{i}}\\right) \\right] \n\\overline{z}_{i} \\\\ \n-\\bar{Z}_{i}e^{-}\\left[ 1-\\frac{1}{2}e^{\\lambda }\\left( z_{i}\\frac{\\partial\n\\lambda }{\\partial z_{i}}+\\overline{z}_{i}\\frac{\\partial \\lambda }{\\partial \n\\overline{z}_{i}}\\right) \\right] dz_{i}+Z_{i}e^{-}\\left[ 1+\\frac{1}{2\ne^{\\lambda }\\left( z_{i}\\frac{\\partial \\lambda }{\\partial z_{i}}+\\overline{z\n_{i}\\frac{\\partial \\lambda }{\\partial \\overline{z}_{i}}\\right) \\right] \n\\overline{z}_{i\n\\end{array}\n\\label{4.4}\n\\end{equation\nOn the other hand, the differential of the bi-para-conformal Hamiltonian\nenergy $H$\\textit{\\ }is calculated as follows\n\\begin{equation}\ndH=(\\frac{\\partial H}{\\partial z_{i}}dz_{i}+\\frac{\\partial H}{\\partial \n\\overline{z}_{i}}d\\overline{z}_{i})e^{+}+(\\frac{\\partial H}{\\partial z_{i}\ndz_{i}+\\frac{\\partial H}{\\partial \\overline{z}_{i}}d\\overline{z}_{i})e^{-}.\n\\label{4.5}\n\\end{equation\nBy means of equation\\textbf{\\ }(\\ref{3}), using equation (\\ref{4.4}) and \n\\ref{4.5}), the conformal bi-para Hamiltonian vector field is seen to b\n\\begin{equation}\nZ_{H}=(Z_{i}\\frac{\\partial }{\\partial z_{i}}+\\overline{Z}_{i}\\frac{\\partial \n}{\\partial \\overline{z}_{i}})e^{+}+(Z_{i}\\frac{\\partial }{\\partial z_{i}}\n\\overline{Z}_{i}\\frac{\\partial }{\\partial \\overline{z}_{i}})e^{-}.\n\\label{4.6}\n\\end{equation\nIf a curve $\\alpha :I\\subset A\\rightarrow M$ is an integral curve of the\nconformal bi-para Hamiltonian vector field $Z_{H}$, i.e., $Z_{H}(\\alpha (t))\n\\dot{\\alpha}(t)$ $,\\,\\,t\\in I.$ In the local coordinates, we get $\\alpha\n(t)=(z_{i}(t),\\overline{z}_{i}(t))$ an\n\\begin{equation}\n\\dot{\\alpha}(t)=(\\frac{dz_{i}}{dt}\\frac{\\partial }{\\partial z_{i}}+\\frac{\n\\overline{z}_{i}}{dt}\\frac{\\partial }{\\partial \\overline{z}_{i}})e^{+}+\n\\frac{dz_{i}}{dt}\\frac{\\partial }{\\partial z_{i}}+\\frac{d\\overline{z}_{i}}{d\n}\\frac{\\partial }{\\partial \\overline{z}_{i}})e^{-}.  \\label{4.10}\n\\end{equation\nTaking equations\\textbf{\\ }(\\ref{4.6}),\\textbf{\\ }(\\ref{4.10}), the\nfollowing equations are foun\n\\begin{equation}\n\\frac{dz_{i}}{dt}=\\frac{-(e^{+}-e^{-})}{\\left[ 1+\\frac{1}{2}e^{\\lambda\n}\\left( z_{i}\\frac{\\partial \\lambda }{\\partial z_{i}}+\\overline{z}_{i}\\frac\n\\partial \\lambda }{\\partial \\overline{z}_{i}}\\right) \\right] }\\frac{\\partial\nH}{\\partial \\overline{z}_{i}}\\text{ , }\\frac{d\\overline{z}_{i}}{dt}=\\frac\n(e^{+}-e^{-})}{\\left[ 1-\\frac{1}{2}e^{\\lambda }\\left( z_{i}\\frac{\\partial\n\\lambda }{\\partial z_{i}}+\\overline{z}_{i}\\frac{\\partial \\lambda }{\\partial \n\\overline{z}_{i}}\\right) \\right] }\\frac{\\partial H}{\\partial z_{i}}.\n\\label{4.12}\n\\end{equation\nHence, equations\\textbf{\\ }(\\ref{4.12}) are seen to be \\textbf{conformal\nbi-para Hamilton equations} on the bi-Lagrangian conformal manifold $(M,\\Phi\n,D_{1},D_{2}),$ and then the triple $(M,\\Phi ,Z_{H})$ is seen to be a \n\\textbf{conformal bi-para Hamiltonian mechanical system}\\textit{\\ }with the\nuse of basis $\\{e^{+},e^{-}\\}$ on $(M,\\Phi ,D_{1},D_{2})$.\n\n\\section{Conclusion}\n\nIt is seen in the above, formalisms of Lagrangian and Hamiltonian mechanics\nhad intrinsically been described by taking into account the basis \n\\{e^{+},e^{-}\\}$\\ on the bi-Lagrangian conformal manifold $(M,\\Phi\n,D_{1},D_{2})$. Conformal bi-para Lagrangian and bi-para Hamiltonian models\nhave arisen to be very important tools since they present a simple method to\ndescribe the model for bi-para-conformal mechanical systems. So, the\nequations derived in (\\ref{3.13}) and (\\ref{4.12}) are only considered to be\na first step to realize how bi-para-complex conformal geometry has been used\nin solving problems in different physically spaces. For further research,\nbi-para-complex conformal Lagrangian and Hamiltonian vector fields derived\nhere are suggested to deal with problems in different fields of physics. In\nthe literature, the equations, which explains the linear orbits of the\nobjects, were presented. This study explained the non-linear orbits of the\nobjects in the space by the help of revised equations using Weyl theorem.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{The Context: Tiny Robots}\n\nIn the recent years, there has been a wide interest in\nthe cooperative behavior of tiny robots.\nIn particular, many distributed coordination protocols have been devised\nfor a wide range of models and for a wide range of problems,\nlike convergence, gathering, pattern formation,\nflocking, etc.\nAt the same time, researchers have also started characterizing\nthe scenarios in which such problems cannot be solved, deriving\nimpossibility results.\n\n\\subsection{Our Motivation: Even Simpler Robots}\n\nAn interesting question regards the minimal cognitive\ncapabilities that such tiny robots need to have for completing a particular task.\nIn particular, researchers have initiated the study of ``weak robots''\\cite{FlocchiniPSW99}.\nWeak robots are \\textit{anonymous} (they do not have any identifier),\n\\textit{autonomous} (they work independently),\n\\textit{homogeneous} (they behave the same in the same situation),\nand \\textit{silent} (they also do not communicate with each other).\n\nWeak robots are usually assumed to have their own local view, represented as a Cartesian\ncoordinate system with origin and unit length and axes. The orientation of axes, or the \\textit{chirality} (relative order of the orientation of axes or handedness),\nis not common among the robots.\nThe robots move in a sequence of three consecutive actions, \\textit{Look-Compute-Move}:\nthey observe the positions of other robots in their local coordinate system and the observation step returns a set of points to the observing robot.\nThe robots cannot distinguish if there are multiple robots at the same position,\ni.e., they do not have the capability of \\textit{multiplicity detection}.\nImportantly, the robots are \\textit{oblivious} and cannot maintain state between rounds\n(essentially moving steps).\nThe computation they perform are always based on the data they have collected in the \\emph{current} observation step; in the next round they again collect the data.\nSuch weak robots are therefore interesting\nfrom a self-stabilizing perspective: as robots do not rely on memory,\nan adversary cannot manipulate the memory either.\nIndeed, researchers have demonstrated that weak robots\nare sufficient to solve a wide range of problems.\n\nWe in this paper aim to relax the assumptions on the tiny robots\nfurther. In particular, to the best of our knowledge, all prior literature\nassumes that robots can observe the positions of other robots in their local view. This enables them to calculate the distance between any pair of robots.\nThis seems to be a very strong assumption, and accordingly,\nwe in this paper initiate the study of even weaker robots which\ncannot locate other robots positions in their local view, preventing them from measuring distances. We define these kind of robots as \\textit{monoculus robots}.\n\n\nIn particular, we\ninitiate to explore two naturally weaker models for monoculus robots with less cognitive capabilities\n\\begin{enumerate}\n\\item \\emph{Locality Detection} ($\\mathcal{LD}$): The robots can distinguish whether a neighbor robot is at a distance more than a predefined value $c$ or not.\n\\item \\emph{Orthogonal Line Agreement} ($\\mathcal{OLA}$): The robots agree on a pair of orthogonal lines (but not necessarily the orientation of the lines).\n\\end{enumerate}\n\n\n\\subsection{The Challenge: Convergence}\n\nWe focus on the fundamental convergence problem for monoculus robots\nand show that the problem is already non-trivial in this setting.\n\nIn particular, many naive strategies lead to non-monotonic\nbehaviors. For example, strategies where boundary robots\n(robots located on the convex hull) move toward the\n``median'' robot they see, may actually \\emph{increase}\nthe area of the convex hull in the next round,\ncounteracting convergence as shown in Fig.~\\ref{fig:anglebisector}$(a)$.\nA similar counterexample exists for a strategy where\nrobots move in the direction of the angle bisector as shown in Fig.~\\ref{fig:anglebisector} ($b$).\n\\noindent\n\\begin{figure}[H]\\centering\n\\includegraphics[width=\\linewidth]{figanglebisector.pdf}\n\\caption{ The 4 boundary robots are moving $(a)$ towards the median robot $(b)$ along the angle bisector. The discs are the old positions and circles are the new positions. The old convex hull is drawn in solid line, the new convex hull is dashed. The arrows denote the direction of moving.  }\\label{fig:anglebisector}\n\\end{figure}\nBut not only enforcing convex hull invariants is challenging,\nalso the fact that visibility is restricted and we cannot detect\nmultiplicity:\nWe in this paper assume that robots are not transparent, and\naccordingly, a robot does not see whether and how many robots\nmay be hidden behind a visible robot. As robots are also not able\nto perform multiplicity detection (i.e., determine how many robots\nare collocated at a certain point), strategies such as ``move toward\nthe center of gravity'' (the direction in which most robots are located),\nare not possible.\n\n\n\\subsection{Our Contributions}\n\nThis paper studies distributed convergence problems\nfor anonymous, autonomous, oblivious, non-transparent, monoculus, point robots under a most general asynchronous scheduling model\nand makes the following contributions.\n\\begin{enumerate}\n\\item We initiate the study of a new kind of robot,\n the \\emph{monoculus robot} which cannot measure distances.\n The robot comes in two natural flavors, and we introduce the\n Locality Detection ($\\mathcal{LD}$)\nand the Orthogonal Line Agreement ($\\mathcal{OLA}$) model accordingly.\n\\item We present and formally analyze deterministic and self-stabilizing distributed convergence algorithms for both\n$\\mathcal{LD}$ and $\\mathcal{OLA}$.\n\\item We show our assumptions in $\\mathcal{LD}$ and $\\mathcal{OLA}$ are minimal in the sense that\nrobot convergence is not possible for monoculus robots without any additional capability.\n\\item We report on the performance of our algorithms through simulation.\n\\item We show that our approach can be generalized to higher dimensions and, with a small extension, supports termination.\n\\end{enumerate}\n\n\\subsection{Paper Organization}\n\nThe remainder of this paper is organized as follows.\nSection~\\ref{sec:preli} introduces the necessary\nbackground and preliminaries.\nSection~\\ref{sec:algo} introduces two algorithms for convergence.\nSection~\\ref{sec:impossibilty} presents an impossibility result which shows the minimality of our assumptions.\nWe report on simulation results in\nSection~\\ref{sec:simul} and discuss extensions in\nSection~\\ref{sec:discussion}.\nIn Section~\\ref{sec:relwork}, we review related work,\nbefore we conclude in Section~\\ref{sec:conclusion}.\n\n\\section{Preliminaries}\\label{sec:preli}\n\n\n\\subsection{Model}\n\nWe consider anonymous, autonomous, homogeneous, oblivious, non-transparent robots with unlimited visibility, unless the view is obstructed by another robot:\nSince the robots are non-transparent, any robot can see at most one robot in any direction.\nAs usual, the robots in each round execute a sequence of \\textit{Look-Compute-Move} steps:\nFirst, the robot observes other robots (\\textit{Look} step); second, on the basis of the observed information, it executes an algorithm\nwhich computes a direction which the robot must move towards\n(\\textit{Compute} step);\nthe robot then moves in this direction (\\textit{Move} step),\nfor a fixed distance $b$ (the step size).\nThe robots are silent, cannot detect multiplicity points, and can\npass over each other (no collision occurs).\n\nIn this paper, we introduce monoculus robots:\n\\begin{definition}(Monoculus Robot)\nA robot is called \\emph{monoculus} if it is anonymous, autonomous, oblivious, homogeneous, and silent. We assume the robot is a non-transparent point\nrobot, has unlimited visibility, and can neither determine the position of other robots nor detect multiplicty.\n\\end{definition}\n\nWe consider the most general CORDA or ASYNC scheduling model\nknown from weak robots~\\cite{FlocchiniPSW99} as well as\nthe ATOM or Semi-Synchronous (SSYNC) model~\\cite{suzu}.\nThese models define the activation schedule of the robots:\nthe SSYNC model considers instantaneous computation and movement, i.e.,\nthe robots cannot observe other robots in motion,\nwhile in the ASYNC model any robot can look at any time. In SSYNC the time is divided into global rounds and a subset of the robots are activated in each round\nwhich finish their \\textit{Look-Compute-Move} within that round.\nIn case of ASYNC, there is no global notion of time.\nThe Fully-synchronous (FSYNC) is a special case of SSYNC, in which all the robots are activated in each round.\nThe algorithms presented in this paper work in both the ASYNC and the\nSSYNC setting. For the sake of generality, we present our proofs\nin terms of the ASYNC model.\n\n\\subsection{Notation and Terminology}\n\n A configuration $C$ is a multiset containing all the robot positions in 2D.\n At any time $t$ the configuration (the mapping of robots in the plane)\n is denoted by $C_t$. The convex hull of configuration $C_t$ is denoted as $CH_t$.\n\\emph{Convergence} is achieved when the distance between any pair of robots is less than a predefined value $c$ (and subsequently does not violate this anymore).\nOur multi-robot system is vulnerable to adversarial manipulation,\nhowever the algorithms presented in this paper are self-stabilizing~\\cite{dolev2000self}\n and robust to state manipulations.\nSince the robots are oblivious, they only depend on the \\emph{current state}: if the state is perturbed,  the algorithms are still able to converge in a self-stabilizing manner \\cite{Gilbert2009}.\n\n\n\n\\section{Convergence Algorithms}\\label{sec:algo}\n\nWe now present distributed robot convergence algorithms for both our models,\n$\\mathcal{LD}$ and $\\mathcal{OLA}$.\n\n\\subsection{Convergence for $\\mathcal{LD}$}\n\nIn this section we consider the convergence problem for the monoculus robots in\nthe $\\mathcal{LD}$ model. Our claims hold for any $c\\geq 2b$.\nAlgorithm~\\ref{algo:convergelocality}\ndistinguishes between two cases: (1) If the robot only sees one other robot,\nit infers that the current configuration must be a line (of 2 or more robots),\nand that this robot must be on the border of this line;\nin this case,  the boundary robots always move inside (usual step size $b$).\n(2) Otherwise, a robot moves towards  any visible, non-local robot (distance\nat least $c$),\nfor a $b$ distance (the step size).\n\nOur proof unfolds in a number of lemmas followed by a theorem.\nFirst, Lemma~\\ref{lem:4bdistance} shows that it is impossible to have a pair of robots with distance larger than $2c$ in the converged situation. Lemma~\\ref{lem:chsubset} shows that our algorithm ensures a monotonically decreasing convex hull size.  Lemma~\\ref{lem:finitedecrement} then proves that the decrement in perimeter for each movement is greater than a constant (the convex hull decrement is strictly monotonic). Combining all the three lemmas, we obtain the correctness proof of the algorithm.\nIn the following, we call two robots \\emph{neighboring} if they see each other (line of sight is not obstructed by another robot).\n\n\\begin{algorithm}\n\\DontPrintSemicolon\n\\caption{\\textsc{ConvergeLocality}}\\label{algo:convergelocality}\n\\SetKwInOut{Input}{Input}\\SetKwInOut{Output}{Output}\n\\Input{Any arbitrary configuration}\n\\Output{All robots are inside a circle of radius $c$}\n\\eIf{only one robot is visible}{Move distance $b$ towards that robot}\n{\\eIf{there is at least one robot farther than $c$}{Move distance $b$ towards any one of the robots with distance more than $c$}{Do not move\\tcp*{All neighbor robots are within a distance $c$}}}\n\\end{algorithm}\n\n\\begin{lemma}\\label{lem:4bdistance}\nIf there exists a pair of robots at distance more than $2c$ in a non-linear configuration, then there exists a pair of neighboring robots at distance more than $c$.\n\\end{lemma}\n\\begin{proof}\nProof by contradiction. If there is a pair of robots with distance more than $2c$, then for them not to move, there are at least two robots on the line joining them positioned such that each pair has a distance less than $c$. Since the robots are non-transparent, the end robots cannot look beyond their neighbors to know that there is a robot at a distance more than $c$.\nIn Fig.~\\ref{fig:4bdistance}, $r_1$ and $r_4$ are $2c$ apart. So $r_2$ and $r_3$ block the view. Since it is a non-linear configuration, say robot $r_5$ is not on the line joining $r_1$ and $r_4$. $l$ is the perpendicular bisector of $\\overline{r_1r_4}$. If $r_5$ is on the left side, then it is more than $c$ distance away from $r_4$ and vice versa. If there is another robot on $\\overline{r_4r_5}$, then consider that as the new robot in a non-linear position, and we can argue similarly. Hence there would at least be a single robot similar to $r_5$ in a non-linear configuration for which the distance is more than $c$.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[height=0.2\\linewidth]{fig4bdistance.pdf}\n\\caption{A non-linear configuration with a pair of robots at a distance $2c$}\\label{fig:4bdistance}\n\\end{figure}\n\\qed\n\\end{proof}\n\n\\begin{lemma}\\label{lem:chsubset}\nFor any time $t' > t$ before convergence, $CH_{t'}\\subseteq CH_{t}$.\n\\end{lemma}\n\\begin{proof}\nThe proof follows from a simple observation. Consider any robot $r_i$. If $r_i$ decides to move towards some robot, say $r_j$, then it is at least $c$ distance away. Even if $r_j$ is on the boundary, $r_i$ cannot cross the boundary. If $r_i$ is already on the boundary, then it always moves on the perimeter or inside the convex hull. Hence the convex hull gradually decreases.\n\n If all the robots are on a straight line, then the boundary robots move monotonically closer in each step. The distance between the end robots is a monotonically decreasing sequence until it reaches $c$.\n\\qed\n\\end{proof}\n\n\\begin{lemma}\\label{lem:finitedecrement}\nIn finite time the decrement in the perimeter of the convex hull is at least $ b \\left(1- \\sqrt{\\frac{1}{2}\\left(1+\\cos\\left(\\frac{2\\pi}{n}\\right)\\right)}\\right)$.\n\\end{lemma}\n\\begin{proof}\nThe sum of internal angles of a $k$-sided convex polygon is $(k-2)\\pi$. So there exists a robot $r$ at a corner $A$ (ref. Fig.~\\ref{fig:decrement}) of the convex hull such that the internal angle is less than $(1-\\frac{2}{n})\\pi$, where $n$ is the total number of robots.\nLet $B$ and $C$ be the points where the  circle centered at $A$ with radius $b\/2$ intersects the convex hull.\nAny robot lying outside the circle will not move inside the circle according to Algorithm~\\ref{algo:convergelocality}.\nAll the robots inside the circle will eventually move out once they are activated.\nAfter all the robots are activated at least once, the decrement in perimeter is at least $AB + AC - BC$. From cosine rule,\n$$AB + AC - BC = \\frac{b}{2}+\\frac{b}{2} -\\sqrt{\\left(\\frac{b}{2}\\right)^2+\\left(\\frac{b}{2}\\right)^2-  2\\frac{b}{2}\\frac{b}{2}\\cos\\left(\\pi -\\frac{2\\pi}{n}\\right)}  $$\n$$ = b \\left(1- \\sqrt{\\frac{1}{2}\\left(1+\\cos\\left(\\frac{2\\pi}{n}\\right)\\right)}\\right)$$\n\\begin{figure}[H]\\centering\n\\includegraphics[height=0.3\\linewidth]{exampleforfinitedecrement.pdf}\n\\caption{Once activated the robots $r$ and $r'$ will move outside the solid circle with radius $b\/2$. The robot $r''$ moves a distance $b$ towards $r'$ because distance between them is more than $2b$ and stops at $D$.}\\label{fig:decrement}\n\\end{figure}\n\\qed\n\\end{proof}\n\n\\begin{theorem}(Correctness)\nAlgorithm~\\ref{algo:convergelocality} terminates when all the robots are within a $c$ radius disc.\n\\end{theorem}\n\\begin{proof}\nFrom Lemmas~\\ref{lem:chsubset}\nand~\\ref{lem:finitedecrement} we know that the convex hull never\nincreases, and eventually a robot will be activated which strictly\ndecreases the hull.\nAccording to Lemma~\\ref{lem:4bdistance}, eventually there will not be a pair of robots with more than $2c$ distance.\nNote that the distance between any two points in a disc of radius $c$ is less than or equal to $2c$. Hence the robots will converge within a disc of radius $c$.\n\\qed\\end{proof}\n\n\n\\subsection{Convergence for $\\mathcal{OLA}$}\n\nIn this section we consider monoculus robots in the $\\mathcal{OLA}$ model.\nOur algorithm will distinguish between\n\\textit{boundary-}, \\textit{corner-} and \\textit{inner-robots}, defined\nin the canonical way. We note that robots can determine their type:\nFrom the Fig.~\\ref{fig:orthogonalline}, we can observe that for $r_2$, all the robots lie below the horizontal line. That means, one side of the horizontal line is empty\nand therefore $r_2$ can figure out that it is a boundary robot. Similarly all $r_i$, $i\\in \\{3,4,5,6,7,8\\}$ are boundary robots. Whereas, for $r_1$, both horizontal and vertical lines have one of the sides empty, hence $r_1$ is a corner robot. Other robots are all inner robots.\n Consequently, we define \\textit{boundary robots} to be those, which have exactly one side of one of the orthogonal lines empty.\n\n  Algorithm~\\ref{algo:convergequadrant} (\\textsc{ConvergeQuadrant}) can be described as follows.\n  A rectangle can be constructed with lines parallel to the orthogonal lines passing through boundary robots such that, all the robots are inside this rectangle.\n  In Fig.~\\ref{fig:orthogonalline}, each boundary robot always moves inside the rectangle perpendicular to the boundary and the inside robots do not move.\n  Note that the corner robot $r_1$ has two possible directions to move. So it moves toward any robot in that common quadrant.\n  Gradually the distance between opposite boundaries becomes smaller and smaller and the robots converge. In case all the robots are on a line which is parallel to either of the orthogonal lines, then the robots will find that both sides of the line are empty.\n  In that case they should not move. But the robots on either end of the line would only see one robot. So they would move along the line towards that robot.\n\n\\begin{figure}[!h]\\centering\n\\includegraphics[height=0.4\\linewidth]{figdirectionlesscompass.pdf}\n\\caption{Movement direction of the boundary robots}\\label{fig:orthogonalline}\n\\end{figure}\n\n\\begin{algorithm}[!h]\n\\DontPrintSemicolon\n\\caption{\\textsc{ConvergeQuadrant}}\\label{algo:convergequadrant}\n\\SetKwInOut{Input}{Input}\\SetKwInOut{Output}{Output}\n\\Input{Any arbitrary configuration and robot $r$}\n\\Output{All robots are inside a square with side $2b$ }\n\\uIf{only one robot is visible}{Move towards that robot}\n\\uElseIf{$r$ is a boundary robot}{Move perpendicular to the boundary to the side with robots}\n\\uElseIf{$r$ is a corner robot}{Move towards any robot in the non-empty quadrant}\n\\Else{Do not move \\tcp*{It is an inside robot}}\n\\end{algorithm}\n\n\\begin{theorem}(Correctness)\nAlgorithm~\\ref{algo:convergequadrant} moves all the robots inside some $2b$-sided square in finite time.\n\\end{theorem}\n\\begin{proof}\nConsider the distance between the robots on the left and right boundary. The horizontal distance between them decreases each time either of them gets activated. The rightmost robot will move towards the left and the leftmost will move towards the right. The internal robots do not move. So in at most $n$ activation rounds of the boundary robot, the distance between two of the boundary nodes will decrease by at least $b$. Hence the distance is monotonically decreasing until $2b$.\nAfterwards, the total distance will never exceed $2b$ anymore.\n\nGiven there is a corner robot present in the configuration, that robot will move towards any robot in the non-empty quadrant. So, the movement of\nthe corner robot contributes to the decrement in distance in both directions.\nConsider robots inside the quadrant are presently very close to one of the boundaries and the corner robot moves towards that robot, then the decrement in one of the dimensions can be small (an $\\epsilon > 0$).\nConsider for example the configuration of a strip of width $b$,\nthen the corner robot becomes the adjacent corner in the next round;\nthis can happen only finitely many times.\nEach dimension converges within a distance $2b$, so in the converged state the shape of the converged area would be $2b$-sided square.\n\\qed\\end{proof}\n\n\\begin{remark}\nIf the robots have some sense of angular knowledge, the corner robots can always move in a $\\pi\/4$ angle, so the decrement in both dimension is significant, hence convergence time is less on average.\n\\end{remark}\n\n\n\\section{Impossibility and Optimality}\\label{sec:impossibilty}\n\nGiven these positive results, we now show that we cannot make the monoculus robots much weaker, otherwise we lose convergeability.\n\n\\begin{theorem}\nThere is no deterministic convergence algorithm for monoculus robots without any additional capability.\n\\end{theorem}\n\\begin{proof}\nWe prove the theorem using a symmetry argument.\nConsider the two configurations $C_1$ and $C_2$ in Fig.~\\ref{fig:indistinguishableconfig}.\nIn $C_1$, all the robots are equidistant from robot $r$, while in $C_2$, the robots are at different distances, however the relative angle of the robots is the same at $r$. Now considering the local view of robot $r$, it cannot distinguish between $C_1$ and $C_2$.\n Say a deterministic algorithm $\\phi$ decides a direction of movement for robot $r$ in configuration $C_1$.\n Since both $C_1$ and $C_2$ are the same from robot $r$'s perspective, the deterministic algorithm outputs the same direction of movement for both cases.\n \\begin{figure}[H]\n\\centering\n\\includegraphics[height=0.3\\linewidth]{figindistinguishableconfig.pdf}\n\\caption{Locally indistinguishable configurations with respect to $r$}\\label{fig:indistinguishableconfig}\n\\end{figure}\nNow consider the convex hull $CH_1$ and $CH_2$ of $C_1$ and $C_2$ respectively.\nThe robot $r$ moves a distance $b$ in one round.\nThe distance from any point inside $CH_1$ is more than $b$ but we can skew the convex hull in the direction of movement, so to make it like $CH_2$, where if the robot $r$ moves a distance $b$ it exits $CH_2$.\n Therefore there always exists a situation for any algorithm $\\phi$ such that the area of the convex hull increases. Hence it is impossible for the robots to converge.\n\n\\qed\\end{proof}\n\n\n\n\\section{Simulation}\\label{sec:simul}\n\nWe now complement our formal analysis with simulations, studying the average\ncase. We assume that robots are distributed uniformly at random\nin a square initially, that $b=1$\nand $c=2$, and\nwe consider FSYNC scheduling. As a baseline to evaluate performance, we consider the\noptimal convergence distance and time if the robots had capability to observe positions,\ni.e., they are \\emph{not} monoculus. Moreover, as a lower bound,\nwe compare to an algorithm which converges all robots to\nthe centroid, defined as follows:\n$$ \\{\\bar{x}, \\bar{y}\\} = \\left\\{\\cfrac{\\sum_{i=1}^nx_i}{n},\\cfrac{\\sum_{i=1}^ny_i}{n}\\right\\}$$\n\\noindent where $\\{x_i,y_i\\} \\forall i \\in \\{1,2,\\cdots,n\\}$ are the robots' coordinates.\n\nWe calculate distance $d_i$ from each robot to the centroid\nin the initial configuration. The optimal distance we have used as\nconvergence distance is the sum of distances from each robot to the unit disc\ncentered at the centroid. So the sum of the optimal convergence distances $d_{opt}$ is given by\n$$d_{opt} = \\sum_{i=1}^n (d_i -1), \\quad if \\, d_i>1 $$\nIn the simulation of Algorithm~\\ref{algo:convergelocality}, we define $d_{CL}$ as the cumulative number of steps taken by all the robots to converge (sometimes also known as the \\emph{work}). Now we define the performance ratio, $\\rho_{CL}$ as\n $$ \\rho_{CL} = \\cfrac{d_{CL}}{d_{opt}}$$\nSimilarly for Algorithm~\\ref{algo:convergequadrant} we define $d_{CQ}$ and $\\rho_{CQ}$. \\\\\n In Fig.~\\ref{fig:varrobot}, we plot the distribution of 100 iterations of simulation of Algorithm~\\ref{algo:convergelocality}, varying the number of robots for a fixed region of deployment.\nThe median increases if we increase the number of robots deployed in the same region.\n\nIn Fig.~\\ref{fig:varrobotCQ}, we plot the distribution of 100 iterations of simulation of Algorithm~\\ref{algo:convergequadrant} varying the number of robots for a fixed region of deployment.\nThe median increases if we increase the number of robots deployed in the same region.\nIn Fig.~\\ref{fig:varregionCQ}, we plot the distribution of 100 iterations of simulation of Algorithm~\\ref{algo:convergequadrant} for a fixed number of robots deployed in different regions. Here we can observe that the distribution does not vary much even if we change the region of deployment.\n\\begin{minipage}{\\linewidth}\n\\begin{minipage}{0.45\\linewidth}\n\\begin{figure}[H]\n\\includegraphics[width=\\linewidth]{VarRobotFixRange100.pdf}\n\\caption{Different number of robots in same region in $\\mathcal{LD}$}\\label{fig:varrobot}\n\\end{figure}\n\\end{minipage}\\hspace{0.1\\linewidth}\n\\begin{minipage}{0.45\\linewidth}\n\\begin{figure}[H]\n\\includegraphics[width=\\linewidth]{VarRangeFixRobot40.pdf}\n\\caption{Fixed number of robots deployed in different region in $\\mathcal{LD}$}\\label{fig:varrange}\n\\end{figure}\n\\end{minipage}\n\\begin{minipage}{0.45\\linewidth}\n\\begin{figure}[H]\n\\includegraphics[width=\\linewidth]{CQVarRobotFixRange100.pdf}\n\\caption{Different number of robots in same region in $\\mathcal{OLA}$}\\label{fig:varrobotCQ}\n\\end{figure}\n\\end{minipage}\\hspace{0.1\\linewidth}\n\\begin{minipage}{0.45\\linewidth}\n\\begin{figure}[H]\n\\includegraphics[width=\\linewidth]{CQVarRangeFixRobot40.pdf}\n\\caption{Fixed number of robots deployed in different region in $\\mathcal{OLA}$}\\label{fig:varregionCQ}\n\\end{figure}\n\\end{minipage}\n\\end{minipage}\\vspace{5mm}\nIn Fig.~\\ref{fig:varrange}, we plot the distribution of 100 iterations of simulation of Algorithm ~\\ref{algo:convergelocality} for a fixed number of robots deployed in different regions. Here we can observe that the distribution does not vary much even if we change the region of deployment.\n\n\n\\begin{figure}[!h]\\centering\n\\includegraphics[height=0.37\\linewidth]{CLvsCQdistance40robot.pdf}\n\\caption{ $\\rho_{CL}$ VS $\\rho_{CQ}$ for the same number of robots}\\label{fig:compareCLCQrobotdistance}\n\\end{figure}\n\\begin{figure}[!h]\\centering\n\\includegraphics[height=0.37\\linewidth]{CLvsCQdistance100region.pdf}\n\\caption{ $\\rho_{CL}$ VS $\\rho_{CQ}$ for the same size of region}\\label{fig:compareCLCQregiondistance}\n\\end{figure}\n\\begin{figure}[!h]\\centering\n\\includegraphics[height=0.37\\linewidth]{CLvsCQtime40robot.pdf}\n\\caption{ $\\tau_{CL}$ VS $\\tau_{CQ}$ for the same number of robots}\\label{fig:compareCLCQrobot}\n\\end{figure}\n\\begin{figure}[!h]\\centering\n\\includegraphics[height=0.37\\linewidth]{CLvsCQtime100region.pdf}\n\\caption{$\\tau_{CL}$ VS $\\tau_{CQ}$ for the same size of region}\\label{fig:compareCLCQregion}\n\\end{figure}\n\nFig.~\\ref{fig:compareCLCQrobotdistance}, and ~\\ref{fig:compareCLCQregiondistance} show the comparison between the performance ratio (PR) for distance. We can observe that Algorithm~\\ref{algo:convergequadrant} performs better. This is due to the fact that, in Algorithm~\\ref{algo:convergequadrant} only boundary robots move.\n\n Let $d_{max}$ be the distance of farthest robot from the centroid and  $t_{CL} $ be the number of synchronous rounds taken by Algorithm~\\ref{algo:convergelocality} for convergence. We define $\\tau_{CL}$ as follows\n$$\\tau_{CL} =\\cfrac{t_{CL}}{d_{max}}$$\nSimilarly for Algorithm~\\ref{algo:convergequadrant}, we define $t_{CQ}$ and $\\tau_{CQ}$. $\\tau_{CL}$ and $\\tau_{CQ}$ show performance ratio for convergence time of Algorithm~\\ref{algo:convergelocality} and ~\\ref{algo:convergequadrant} respectively.\n In Fig.~\\ref{fig:compareCLCQrobot} and ~\\ref{fig:compareCLCQregion}, we can observe that $\\tau_{CL}$ is very close to 1, so Algorithm~\\ref{algo:convergelocality} converges in almost the same number of synchronous rounds (proportional to distance covered, since step size $b=1$) as the maximum distance.\nWe can observer that Algorithm~\\ref{algo:convergequadrant} takes more time as the number of robots and the side length of square region increases.\n\\section{Discussion}\\label{sec:discussion}\n\nThis section shows that our approach supports some interesting\nextensions.\n\n\\subsection{Termination for $\\mathcal{OLA}$ Model}\n\nWhile we only focused on convergence and not termination so far,\nwe can show that with a small amount of memory,\ntermination is also possible in the $\\mathcal{OLA}$\nmodel.\nTo see this, assume that each robot has a 2-bit persistent memory in the $\\mathcal{OLA}$ model for each dimension, total 4-bits for two dimensions. \n Algorithm~\\ref{algo:convergequadrant} has been modified to Algorithm~\\ref{algo:convergequadranttermination} such that it can accommodate termination.\nAll the bits are initially set to 0. \nEach robot has its local coordinate system, which remains consistent over the execution of the algorithm. The four bits correspond to four boundaries in two dimensions, i.e., left, right, top and bottom.\n If a robot finds itself on one of the boundaries according to its local coordinate system, then it sets the corresponding bit of that boundary to 1. Once both bits corresponding to a dimension become 1, the robot stops moving in that dimension.\nConsider a robot $r$. Initially it was on the left boundary in its local coordinate system. Then it sets the first bit of the pair of bits corresponding to $x$-axis. It moves towards right. Once it reaches the right boundary, then it sets the second bit corresponding to $x$-axis to 1. Once both the bits are set to 1, it stops moving along the $x$-axis. Similar movement termination happens on the $y$-axis also. Once all the 4-bits are set to 1, the robot stops moving.\n\n\\begin{algorithm}[!h]\n\\caption{\\textsc{ConvergeQuadrantTermination}}\\label{algo:convergequadranttermination}\n\\SetKwInOut{Input}{Input}\\SetKwInOut{Output}{Output}\n\\DontPrintSemicolon\n\\Input{Any arbitrary configuration and robot $r$ with 4-bit memory}\n\\Output{All robots are inside a square with side $2b$ }\n\\eIf{the robot is on a boundary(ies)}{set the corresponding bit(s) to 1}{Do nothing \\tcp*{$r$ is an inside robot}}\n\\uIf{$r$ is a boundary robot and the bits corresponding to that dimension are not 1}{Move perpendicular to the boundary to the side with robots}\n\\uElseIf{$r$ is a corner robot}{\\eIf{Both bits corresponding to a dimension is 1}{Move in other dimension to the side with robots}{Move towards any robot in the non-empty quadrant}}\n\\Else{Do not move \\tcp*{$r$ is not on boundary OR all four bits are 1}}\n\\end{algorithm}\n\n\n\n\n\n\n\\subsection{Extension to $d$-Dimensions}\n\nBoth our algorithms can easily be extended to $d$-dimensions.\nFor the $\\mathcal{LD}$ model, the algorithm remains exactly the same.\nFor the proof of convergence, similar arguments as Lemma~\\ref{lem:finitedecrement} can be used in $d$ dimensions.\nWe can consider the convex hull in $d$-dimensions and the boundary robots of the convex hull always move inside. The size of convex hull reduces gradually and the robots converge.\n\nAnalogously for the $\\mathcal{OLA}$ model, the distance between two robots in the boundary of any dimension gradually decreases and the corner robots always move inside the $d$-dimensional cuboid. Hence it converges.\nHere the robot would require $2d$ number of bits for termination.\n\n\\section{Related Work}\\label{sec:relwork}\n\n The problems of gathering \\cite{suzuki1999distributed}, where all the robots gather at a single point,\n  convergence \\cite{cohen2006convergence}, where robots come very close to each other and\n Pattern formation \\cite{flochini1,suzuki1999distributed} have been studied intensively in the literature.\n\n   Flocchini et al. \\cite{FlocchiniPSW99} introduced the CORDA or Asynchronous (ASYNC) scheduling model for weak robots. Suzuki et al. \\cite{suzu} have introduced the ATOM or Semi-synchronous (SSYNC) model.\n       In \\cite{suzuki1999distributed}, impossibility of gathering for $n=2$ without assumptions on local coordinate system agreement for \\textit{SSYNC} and \\textit{ASYNC} is proved.\nAlso, for $n>2$ it is impossible to solve gathering without assumptions on either coordinate system agreement or multiplicity detection \\cite{Prencipe2007}. Cohen and Peleg \\cite{CohenP04} have proposed a center of gravity algorithm for convergence of two robots in ASYNC and any number of robots in SSYNC.\n\nTo the best of our knowledge in all the previous works, the mathematical models always assume that the robots can find out the location of other robots in their local coordinate system in the Look step. This in turn implies that the robots can measure the distance between any pair of robots albeit in their local coordinates. All the algorithms exploit this location information to create an invariant point or a robot where all the other robots gather. But in this paper we deprive the robots of the capability to determine the location of other robots. This leads to robots incapable of finding any kind of distance or angles.\n\nAny kind of pattern formation requires these robots to move to a particular point of the pattern. Since the monoculus robots cannot figure out locations, they cannot stop at a particular point. Hence any kind  of pattern formation algorithm described in the previous works which requires location information as input are obsolete. Gathering problem is nothing but the point formation problem \\cite{suzuki1999distributed}. Hence gathering is also not possible for the monoculus robots.\n\n\n\n\\section{Conclusion}\\label{sec:conclusion}\n\nThis paper introduced the notion of \\emph{monoculus robots}\nwhich cannot measure distance: a practically relevant generalization of\nexisting robot models. We have proved that the two basic models\nstill allow for convergence (and with a small memory,\neven termination), but with less capabilities,\nthis becomes impossible.\n\nThe $\\mathcal{LD}$ model converges in an almost optimal number of rounds, while the $\\mathcal{OLA}$ model takes more time.\nBut the cumulative number of steps is less for the $\\mathcal{OLA}$ model compared to the $\\mathcal{LD}$ model since only boundary robots move. Although we found in our simulations that the median and angle bisector strategies successfully converge, finding a proof accordingly remains  an\nopen question.\nWe see our work as a first step, and believe that the study of weaker robots opens an interesting field for future research.\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{A Rounding procedure using random walks}\n\\label{sec:algo}\nThe simple random walk based algorithm outlined in the introduction \ndoesn't take into account any of the constraints $V_j \n\\cdot x \\leq 1$ and therefore likely to violate them after some\nrandom walk steps. However, the probability that an $x_i$ reaches\n1 before it reaches 0 is equal to the ratio \n$\\frac{x'_i \/\\gamma}{1- x'_i + x'_i \/\\gamma}\n= x'_i$. \nThis is\na consequence of a more general property of martingales known as Doob's\noptional stopping theorem  (\\cite{feller1}).\n\\begin{theorem}\nLet $(\\Omega , \\Sigma, P)$ be a probability space and $\\{ F_i \\}$ be a \nfilteration of $\\Omega$, and $X = \\{ X_i \\}$ a martingale with respect to\n$\\{ F_i \\}$. Let $T$ be a stopping time such that $\\forall \\omega \\in \\Omega, \n\\ \\forall i , \\ \\ | X_i (\\omega ) | < K$ \nfor some positive integer $K$, and $T$ is almost surely bounded.\nThen $\\mathbb{E} [ X_T ] = \\mathbb{E} [ X_0 ]$.   \n\\label{stopping-theorem}\n\\end{theorem} \nIn the above application, the stopping times are $X_T = \\{-b , a \\}$ whichever\nis earlier. So $-b \\cdot \\Pr [ X = -b ] + a \\Pr [ X =a] = 0 \\cdot \\Pr[ X = 0]$.\nSince $\\Pr [X = -b] + \\Pr [ X=a ] = 1$ from the stopping criteria, \n$\\Pr [ X= a] = \\frac{b}{a+b}$. In our context, we start from $X = x'_i$\nand $b = \\frac{x'_i}{\\gamma}$ and $a = \\frac{1- x'_i}{\\gamma}$, so\nthe probability that $\\hat{x}_i = 1$ equals $\\frac{x'_i \/\\gamma}\n{(1 - x'_i + x'_i )\/\\gamma} = x'_i$. \nSo the distributions of the variables \nbeing absorbed at 0 or 1 are identical to the independent rounding. \nHowever, the random walk process has many intermediate steps that do not\nhave corresponding mappings in the $2^n$ possible configurations of the\nindependent rounding. Whether it makes this framework more powerful compared\nto one-step independent rounding is a difficult question but we feel that \nit could\ngive us a superior understanding of the process of independent rounding.   \n\nIn the algorithm presented in Figure \\ref{algo1}, instead of the simple\nBernoulli random walk step, we use a normal Gaussian random walk on\neach coordinate. \nWe run the basic algorithm for\n$T$ iterations such that the number of un-fixed variables in each \nconstraint of $C^T$ (the active set of constraints after $T$ iterations) \nis bounded by $\\log n$. \nThe value of $T$, more specifically $\\mathbb{E}[ T ]$, will be\ndetermined during the course of our analysis. \n\n Now we run the algorithm of \\cite{MT:10} (c.f. section \\ref{sec5}) \non the constraints in $C^T$ which\nare projections of the original constraints on the unfixed variables after\n$T$ steps.\n\\begin{figure}\n\\fbox{\\parbox{6.0in}{\n\\begin{center}\nAlgorithm {\\bf Iterative Randomized Rounding }\n\\end{center}\n{\\footnotesize\nInput : $x'_i \\ \\ 1 \\leq i \\leq n$ satisfying $Ax \\leq b \\cdot e^m$. $C^0$: set of constraints.\n\\\\\n$0 < \\gamma < \\delta < 1$ - the exact values are discussed in the analysis.\n\\\\\nOutput : $\\hat{x_i} \\in \\{0, 1\\}$ \\\\\n\nInitialize all variables as {\\it un-fixed}\nand set $X_0 = [x'_1 , x'_2 , \\ldots x'_n ] $.\\\\\n{\\bf Repeat} for iterations $i \\ \\  = 1, 2 \\ldots $\n\\begin{quote}\n\\begin{enumerate}\n\\item Generate a random vector $ R^{i} = U_i$ where $U_i$ is\na multidimensional gaussian r.v. restricted to the unfixed variables.\n\\item $X_{i+1} = X_i + \\gamma \\cdot R^i$.\n\\item {\\it Fix} a variable if it is\nless than $\\delta$ or greater than $1-\\delta$.\\\\ \nIf all variables are fixed then exit. \\\\\n\\{ * multiple variables may get fixed in a single iteration. * \\}\n\\item Update the set of constraints $C^t$ that contains at least one\nunabsorbed variable.\n\\end{enumerate}\n\n\\end{quote}\n{\\bf until} stopping condition ${\\cal S}$ (* \n$\\max_j \\{ \\mbox{ number of unfixed variables in } C_j \\}  \\leq \\log n $ *)\n\\\\[0.1in]\nRun Moser-Tardos \\cite{MT:10}  algorithm on $C^T$ on the unfixed \nvariables of $X_T$ according\ngiven in Figure \\ref{algo2} \\\\[0.1in]\nRound the {\\it fixed} variables to 0 or 1 whichever is closer \n and return this vector denoted by $\\hat{x}$.\n}\n}}\n\\caption{An iterative randomized rounding algorithm based on random walks}\n\\label{algo1}\n\\end{figure}\n\\subsection{The framework and some notations }\nIn the rounding algorithm, $X_t$ denotes\nthe random walk vector after $t$ steps. \nLet $U^d$ denote a\n $d$-dimensional Gaussian random variable\nand let $U_t$\ndenote the projection of $U^n$ on the current subspace corresponding to\nthe unfixed variables in the\n$t$th iteration. \nThe normal distribution is\ndenoted by ${\\cal N}(\\mu , \\sigma^2)$ where $\\mu$ is the\nmean and $\\sigma^2$ is the variance. \nUnless otherwise stated, we will\nrefer to the {\\it standard} normal distribution where $\\mu = 0$ and\n$\\sigma^2 = 1$. \n\nThe parameters $\\gamma, \\delta$ are chosen to ensure that the walk stays within\nthe feasible region. It suffices to have $\\gamma \\leq \\frac{\\delta}{\\log n}$\nfrom the pdf of the normal distribution if we are executing $O(\\frac{1}{\n\\gamma^2})$ steps (see \\cite{LM:12} for a rigorous proof). The value of\n$\\delta$ will be determined according to the approximation factor and the\nerror bound that we have in a specific application. In our analysis, \nthe the focus will be on \nvariables absorbed at 0 that will be rounded down. The decrease in\nthe objective value can be at most $n\\delta$ (recall the maximum weight \nof the coefficients is 1). If we choose $\\delta$ such that $n\\delta$ is\n$o(OPT)$ it will suffice. For the main theorems, choosing $\\delta \\leq\n\\frac{1}{polylog n}$ will work.\n\nAfter $t$ iterations, the $j$-th constraint is denoted by\n$C_j^t : < V_j , X_t > \\ \\leq 1 + \\beta_t$ that has many of variables\n{\\it fixed}. We shall denote the set of unfixed variables in iteration $t$ \nby ${\\cal U}(t) \\subset\n\\{1, 2 \\ldots n\\}$ and the corresponding vector by $X^{{\\cal U}}_t$ where the\nthe fixed variables are set to 0. The constraint vector $V^t_j =\nV_j \\cap {\\cal U}(t)$, where only the coefficients in $V^t_j$ corresponding to \n${\\cal U}(t)$ can be non-zero. Then\\\\\n$|< V^t_j , X_t - X_{t+1}>| = |< \\frac{V_j}{\\twonorm{ V_j }} , \nX^{{\\cal U}}_t - X_{t+1}^{{\\cal U}}>| \\cdot\n\\twonorm{ V_j }$ is the change in the value of $C_j$ in\niteration $t$ as the remaining\nvariables do not change. \nNote that while $X^{\\cal U}$ changes in every step, $V^t_j$ changes only when\nsome variable is absorbed. \n\nFor convenience of the analysis, we will club successive iterations \ninto {\\it phases}, where\nwithin a phase $p$, $\\beta_p$ remains unchanged. Equivalently, $\\beta_p \n- \\beta_{p-1}$\nreflects the cumulative effect of a number of random walk steps within the \nphase $p$ referred to as the {\\it accumulated error} or simply {\\it error}.  \nThe phase $p$ corresponds to the $L_2$ norm of any\nconstraint $\\twonorm{V^{p}_j}$ that is bounded by $\\sqrt{n\/2^p}$. \nIntuitively, with additional random walk steps, we are more \nlikely to violate the original constraints and $\\beta_p$ is a measure of\nthe violation. \nIn terms of the above notation, it is obvious that\n$ \\twonorm{V^{p+1}_j} \\leq \\twonorm{V^{p}_j}$. We will use the index $p$ (\nrespectively $t$) to indicate reference to phases (resp. iterations).\n\nWlog, we assume that all constraints have at least 2 \nvariables and at most $n-1$\nvariables. \\footnote{A constraint with $n$ variables has a trivial solution\nwhere any one variable can be set to 1 and a constraint with one\nvariable is redundant.} \nThe successive Brownian motion steps (defined by multidimensional normal\nGaussian) form a martingale sequence.\nThe following result forms the crux of our analysis - see \\cite{bansal:10,LM:12}\nfor more details and proof.\n\\begin{observ}\nIn iteration $t$, let $Y_t = < V_j , U_t \\cdot \\gamma >$, that is the\nmeasure of the change in $< V_j , x >$ in the step $t$. Then $Y_t$\nis a Gaussian random variable with mean 0 and variance $\\gamma\n\\cdot\\twonormsq{ V_j }$.\n\\end{observ}\nNote that the scaled\nrandom variable $Y'_t = \\frac{Y_t}{\\gamma \\twonorm{V^t_j}} $ is a\nGaussian with mean 0 and variance $\\leq 1$. \nWhen $Y'_i$ correspond to standard Gaussian, then\n\\begin{lemma}\nFor any $\\beta > 0$,\n $\\Pr [ | Y'_1 + Y'_2 \\ldots Y'_T | > \\beta ] \\leq 2 \\exp ( - \\beta^2 \/2T\n) $.\n\\label{chernoff-mart}\n\\end{lemma}\n\nThe behavior of random walks starting from an arbitrary initial position\nand subsequently absorbed at 0 can be\nobtained from the {\\it gambler's ruin problem} where the underlying martingale\nis the Brownian motion.\nThere exists a wealth of literature on Brownian motion \\cite{feller1,Ross:2006},\nbut the specific form in which we invoke them for analyzing our algorithm\nis stated below. A proof is presented in the appendix.\n\n\\begin{lemma}\nConsider a random walk starting from position $a$ in an\ninterval of length $a +b$ with absorbing barriers at both end-points.\nThen the expected number of steps for the walk to get absorbed (at any of the\nends) is $a \\cdot b$.\nMoreover, the probability of the random walk being absorbed at 0 is\n$\\frac{b}{a+b} - 1\/k$\nafter $k\\cdot a \\cdot b $ steps for any $k > 1$.\n\\label{boost-prob-absorb}\n\\end{lemma}\n{\\it Remark} By choosing $b > k \\cdot a$,\nthe probability\ncan be made arbitrarily close to 1 for $k \\gg 1$ - conversely,\nthe probability of non-absorption at 0 is $O(\\frac{1}{k})$ after $k^2 a^2$\nsteps.\n\n\\ignore{\nThe success of the rounding algorithm depends on the {\\it race} between the\nhitting of constraints $C^t_j$ and the unfixed \nvariables of $V_j^t$. In particular, we would like to\nhit the variable (hyper)-planes $x_i =1$ or $x_i = 0$ more often than the\nhyperplanes $C^t_j$. \n The sooner the ${(X_t )}_i$ hits the \nboundaries 0 or 1, the smaller is the displacement $\\twonorm{ X_t }$ in \n${\\mathbb R}^n$. \nLovette and Meka \\cite{LM:12} also make use of this observation\nimplicitly and in their case, the discrepancy hyperplanes are much further\ncompared to the variable hyperplanes. In our case, the situation is much \ntighter and much of our technical analysis revolves around that.  \nIf there are $r^2$ {\\it unfixed} \nvariables (a subspace with dimension $r^2$)\nthen the variance of the displacement in each step is proportional to \nthe $L_2$ norm of $V_j$ which is $r$.\nSince the\nsum of the $r^2$ variables is bounded by 1 in any feasible LP solution, at \nleast $r^2 \/2$ variables have LP values bounded by $ x'_i \\leq \n\\frac{2}{r^2}$ which is the starting point of the random walk, ${(X_0 )}_i$. \nThe 1-dimensional projection is also a brownian walk ${(X_t)}_i$\nwhere each step is scaled by \n$\\gamma$, and consequently, the $r^2 \/2$ variables are within $\\frac{2}\n{\\gamma \\cdot r^2}$ from 0. Using a simpler argument based on number\nof $\\gamma$-length step sizes, one needs to move \n$\\frac{\\beta}{r \\gamma}$ steps towards some constraint $C_j$ with \nslack $\\beta$\nversus $\\frac{2}{\\gamma \\cdot r^2}$ steps to hit 0s of\nvariables associated with $C_j$. \nClearly the \nlatter is closer if $\\beta > \\frac{2}{r}$. This {\\it bias} in favor of \nhitting the variables holds the key to the success and efficiency of \nour algorithm and we need to capitalize on this by choosing $T_p$\nappropriately, so that the many variables become fixed at 0 before\n$X_p$ hits the constraints (of the scaled polytope). This in turn slows\ndown the Brownian motion further, as the norm reduces and we keep repeating\nthis process. \n\n\n\\section{Appendix}\n{\\bf Proof of Lemma \\ref{boost-prob-absorb}}\n\\begin{proof}\nThe proof of the gambler's ruin problem actually uses a {\\it quadratic\nmartingale}\\footnote{The proof that it is a martingale can be found in\nstandard books on stochastic process}  $B^2 (t) - t$ where $B(t)$ is the \nBrownian motion random variable.\nUsing the optional stopping theorem on this, we obtain\n\\[ \\mathbb{E} [ B^2 (T) - T ] = \\mathbb{E}[ B^2 (0) - 0 ] = a^2 \\]\nSince $p = \\frac{b}{a+b}$ is the probability that it is at 0, we obtain\n$\\mathbb{E}[T] = a \\cdot b$.\n\nNow, consider the events leading to the failure of being absorbed at 0 -\nthese correspond to the absorption of the random walk at either ends or\nthe non-absorption at either ends. \n\\[ \\Pr [ Failure ] = \\Pr [ Failure \\cap Absorption ] + \\Pr [ Failure \\cap\nNon-absorption ]\\]\n\\[  = \\Pr [ Failure | Absorption ] \\cdot \\Pr [ Absorption] \n+ \\Pr[ Failure | Non-absorption ] \\cdot \\Pr [Non-absorption]  \\]\n\\[\n\\leq \\Pr [ Failure | Absorption ] + \\Pr [Non-absorption] \\] \nFrom the preceding argument ,the probability that the random walk is not \nabsorbed after $k a\\cdot b$ steps \nis less than $1\/k$ using Markov's inequality. \nFrom optional stopping criteria, the probability of\nabsorption at 0 is $\\frac{b}{a+b}$. The probability that the random walk is\nnot absorbed at 0 after $k a  b $ steps is bounded by \n$ 1 - \\frac{b}{a+b} \n+ 1\/k$. Consequently, the probability of being absorbed at 0 is \nat least $\\frac{b}{a+b} - 1\/k$. \n\\end{proof}\n\\subsection{A Lower Bound on error}\n\\begin{lemma}\nFor $A \\in {\\cal A}^{m\\times n}_{\\log n}$ such that $A\\cdot \\bar{x} \\leq e^m$,\nwe cannot simultaneously obtain $\\sum_i \\hat{x}_i = \\Omega ( n\/\\log n )$ and\n${|| A \\hat{x}||}_\\infty  \\leq o(\\frac{\\log\\log n}{\\log\\log\\log n})$\nfor $\\hat{x}_i \\in \\{ 0, 1 \\}$.\n\\label{lbnd}\n\\end{lemma}\n\n\\begin{proof}\n\nAs mentioned before, it is easy to see that $x_i = 1\/k$ is a solution \nfor $\\bar{x}$ with objective\nfunction value $n\/k$. After rounding $\\bar{x}$ we want to have at least \n$\\Omega ( n\/k )$ 1s in the rounded vector, say $\\bar{y}$.\nWe must have $A \\cdot \\bar{y} \\leq t\\cdot e$ for some rounding guarantee\n$t$. Let $b$ be a fixed vector with $n\/k$ 1s. \n\nLet $r$ be a row vector with $k$ 1s in random location - this corresponds to\na row of $A$. The probability that $< r , \\bar{y} > \\ \\ \\geq t$ is given by\n \\[  \\frac{{ (n\/k) \\choose t} \n\\cdot { (n-n\/k) \\choose (k-t) }}{{n \\choose k }} \\]\nUsing $ {(\\frac{n}{k})}^k \\leq {n \\choose k} \\leq {(\\frac{ne}{k})}^k $, the \nabove expression is\n\\begin{eqnarray}\n \\geq & \\frac{ {( \\frac{n}{tk})}^t \\cdot {(\\frac{n -n\/k}{k-t})}^{k-t}}{\n{(\\frac{ne}{k})}^k} \\\\\n = & \\frac{ \\frac{1}{t^t} \\cdot {( \\frac{k-1}{k-t})}^{k-t}} {\n e^k } \\\\ \n = & \\frac{ k^{k-t}}{ {( k-t )}^{k-t} \\cdot e^k \\cdot t^t }\\\\\n = & \\frac{1}{ {( 1 - t\/k )}^{k-t} \\cdot e^k \\cdot t^t } \\\\\n \\geq & \\frac{1}{{( 1 - t\/k )}^{k\/2} \\cdot e^k \\cdot t^t } \\text{ for } k \\gg t\\\\\n = & \\frac{1}{ e^{ k - t\/2} \\cdot t^t }\n\\end{eqnarray}\nSo the probability that for $m$ independently chosen rows, the probability that\n$< r , \\bar{y} > \\ \\ < t$ is \n\\begin{eqnarray}\n  p(m,k,t) &  \\leq & {\\left( 1 - \\frac{1}{ e^{ k - t\/2} \\cdot t^t } \\right)}^m \n\\leq  {\\left( 1 - \\frac{1}{ e^{ k } \\cdot t^t } \\right)}^m =\n{\\left( 1 - \\frac{1}{ e^{ k } \\cdot t^t } \\right)}^{e^k t^t \\frac{m}{e^k t^t}}\n\\\\\n & \\leq & {\\left( \\frac{1}{e} \\right)}^{\\frac{m}{e^k t^t}} \n\\end{eqnarray} \nSince there are no more than ${n \\choose\n(n\/\\log n)}$ choice of columns with $n\/\\log n$ 1s, the probability that all\nthe dot products are less than $t$ is less than ${n \\choose (n\/\\log n)} \\cdot\np(m,k,t)  \\leq p(m,k,t) \\cdot {(\\frac {n e}{n\/\\log n})}^{n\/\\log n} \n \\leq e^n \\cdot p(m,k,t)$. If $e^n \\cdot p(m,k,t) < 1$ then there must exist\nmatrices in ${\\cal A}^{m \\times n}_k$ that do not satisfy the error bound $t$.\nSo,\n\\[    1  > {\\left( \\frac{1}{e} \\right)}^{\\frac{m}{e^k t^t}} \\cdot e^{n} \n    \\Rightarrow {( e ) }^{\\frac{m}{e^k t^t}}  >  e^n \\] \n which is equivalent to the condition \n\\begin{equation}\n \\frac{m}{n}  > e^k t^t \\text{  or  }\n\\log(m\/n) > k + t\\log t \n\\label{lbndreq}\n\\end{equation}\n \n For $k = \\log n$ and $m = n\\cdot polylog(n) $, this holds for some $t \\geq \n\\frac{\\alpha \\log \\log n}{\\log\\log\\log n})$, i.e., for some constant \n$\\alpha > 0$, i.e., \nthe error cannot be\n$o (\\frac{\\log\\log n}{\\log\\log\\log n})$. \n\\end{proof}\n\n\\ignore{\n\\subsection{An alternate proof for the RT bound}\nOur algorithm also provides an alternate proof for the \n$O(\\frac{\\log m}{\\log\\log m})$ error bound in constraints incurred in \ncase of classical randomized rounding by Raghavan and \nThompson \\cite{RT:87}\n\nWe run our algorithm on the above problem(with scaling $S=1$).\nAs shown in the proof of Lemma \\ref{absorp_rate},the expected number of \nunfixed variables in $i^{th}$ constraint $=E(u_i^p) (=\\|V_i^p\\|_2 ^2) \\leq \\frac{n}{2^p}$ (when $B = S = 1$).The above holds with high probability when $p \\leq \\log(n)-\\log(\\log(m))(=p*)$.\\\\\nTo simplify our analysis we define variables indexed by the number of phases after p* phases are over.\nLet q indicate the number of phases after $p*$ phases are over.\nLet $w_i^q=u_i^{q+\\log (n)-\\log (\\log (m))}$ (that is the number of unfixed variables in $i^{th}$ constraint indexed by number of phases after p*)and let $\\overset{\\sim}{T_q}=T_{q+\\log (n)-\\log (\\log (m))}$ (that is the number of iterations as a function of number of phases after p*).\\\\\nNow $E(w_i^q)=\\frac{n}{2^{\\log (n)-\\log (\\log (m))+q}}=\\frac{\\log m}{2^q}$.\nTo get bound with sufficiently high probability,bound we need to run our algorithm for a sufficiently large number($>>\\overset{\\sim}{T_q}$) of steps so that,number of unabsorbed variables is bounded by $\\frac{\\log (m)}{2^q}$ with high probability.\n\n\n\nWe claim the following:-\n\\begin{lemma}\nIf r=$2^q \\cdot 2^{2^q +1}$, then after $\\overset{\\sim}{T_r}=\\frac{2^{r+\\log(n)-\\log(\\log(m))} \\cdot 2^{r+\\log(n)-\\log(\\log(m))}}{n^2 \\gamma^2}$ steps,the number of unfixed variables in $i^{th}$ equation is bounded by $\\frac{\\log(m)}{2^q}$ with probability $\\geq 1-\\frac{1}{m}$.\n\\end{lemma}\n\\begin{proof}\nBy definition $w_i^r$ is the number of unfixed variables in the $i^{th} constraint$ after $\\overset{\\sim}{T_r}$ iterations.\\\\\nWe run it for $\\overset{\\sim}{T_r}=\\frac{2^{\\log (n)-\\log (\\log (m))+r} \n2^{\\log (n)-\\log (\\log (m))+r}}{n^2 \\gamma^2}$ steps, where r is some function of q, so that $E(w_i^r)=\\frac{\\log (m)}{2^r}$\\\\\nThus,$Pr(w_i^r> \\frac{\\log (m)}{2^q})$\\\\\n$=Pr(w_i^r>E(w_i^r)(1+\\delta ))$, where $(1+\\delta)\\frac{\\log (m)}{2^r}=\\frac{\n\\log (m)}{2^q}$ \\\\\n$\\leq (\\frac{e^\\delta}{(1+\\delta)^{1+\\delta}})^{\\frac{\\log (m)}{2^r}}$\\\\\n$=e^{\\delta \\frac{\\log (m)}{2^r}-(1+\\delta )\\ln(1+\\delta )\\frac{\\log (m)}{2^r}}$,  where $(1+\\delta )\\frac{\\log (m)}{2^r}=\\frac{\\log (m)}{2^q}$\\\\\n$=e^{\\frac{\\log (m)}{2^q}-\\frac{\\log (m)}{2^r}-\\frac{\\ln(1+\\delta)\\log (m)}{2^q}}  \\text{  since } (1+\\delta )\\frac{\\log (m)}{2^r}=\\frac{\\log (m)}{2^q} \\text{   and  hence  }  \\delta \\frac{\\log (m)}{2^r}=\\frac{\\log (m)}{2^q}-\\frac{\\log \n(m)}{2^r}$\\\\\n$\\leq e^{\\frac{\\log (m)}{2^q}(1-ln(1+\\delta))}$\\\\\nTo bound the probability inverse polynomial in m,\nwe need to choose$\\frac{\\log (m)}{2^q}(ln(1+\\delta)-1) \\geq O(\\log (m))$\\\\\nor $\\ln(1+\\delta) \\geq 2^q+1$\\\\\nor $1+\\delta  \\geq 2^{2^q+1}$\\\\\nor $ 2^r \\geq 2^q 2^{2^q+1}$\\\\\nThus it suffices to run for $\\overset{~}{T_r}=\\frac{2^{r+\\log (n)-\\log \n(log (m))}.2^{r+\\log (n)-\\log (\\log (m))}}{n^2 \\gamma^2}$ iterations so as to have $w_i^r$ close to O($\\frac{\\log (m)}{2^q}$), with high probability, where r=$2^{2^q}.2^q$\\\\\n\\end{proof}\n\nTo prove a bound on the error, \nwe proceed similar to the analysis in Section \\ref{}. \nConsider a fixed constraint $C_j:V_j.\\bar{x} \\leq 1$.\n\\begin{lemma}For all constraints the error in $q^{th}$ phase after p* is \n$\\overset{\\sim}{\\delta_q} \\leq \\delta_{q+\\log(n)-\\log(\\log(m))}=2^{\\frac{q}{2}} \\cdot 2^{2^q+1}$\n\\begin{align*}\n\\text{Consider } &Pr(\\vert < \\gamma (\\sum_{i=\\overset{\\sim}{T_{r-1}}+1}^{\\overset{\\sim}{T_{r}}} U_i),V_j>\\vert \\geq \\beta_q)\\\\ \n&=Pr(\\vert < \\sum_{i=\\overset{\\sim}{T_{r-1}}+1}^{\\overset{\\sim}{T_{r}}} U_i, \\frac{V_j}{\\| V_j \\|}> \\vert \\geq \\frac{\\beta_q}{\\gamma \\| V_j \\|}).\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\nNow $<U_i,\\frac{V_j}{\\vert \\vert V_j \\vert \\vert}> \\sim \\mathcal{N}(0, \\sigma^2)$ where $\\sigma^2 \\leq 1$.\\\\\nThus the above is bounded by $e^{-\\frac{\\beta_q^2}{\\gamma^2 \\vert \\vert V_j \\vert \\vert ^2 2 (\\overset{\\sim}{T_r}-\\overset{\\sim}{T_{r-1}})}}$\nThus for the $q+\\log (n)-\\log(\\log (m))$ phase,we will need to choose $\\beta_q$ satisfying:-\n$\\frac{\\beta_q^2}{\\gamma^2 \\vert \\vert V_j \\vert \\vert ^2 2 (\\overset{\\sim}{T_j}-\\overset{\\sim}{T_{j-1}})} \\geq \\log (m)$\\\\\ni.e. if $\\beta_q \\geq \\gamma \\vert \\vert V_j \\vert \\vert \\sqrt{2.\\log (m).(\\overset{\\sim}{T_j}-\\overset{\\sim}{T_{j-1}})}$\\\\\nSince $\\vert \\vert V_j \\vert \\vert \\leq \\sqrt{\\frac{\\log n}{2^i}}$ and $\\overset{\\sim}{T_j}=\\frac{2^{q+\\log (n)-\\log (\\log (m))} 2^{2^q+1}.2^{q+\\log (n)-\n\\log (\\log (m)) 2^{2^q+1}}}{n^2 \\gamma^2}$,\\\\ \nit suffices to choose \\begin{align*}\n \\beta_q &\\geq 2^{q\/2}.2^{2^q+1}\n\\end{align*}\nFor $q \\leq \\log (\\log (\\log (m)))-2$, the error is bounded by $2^{\\frac{\\log\n(\\log (\\log (m)))-2}{2}}.2^{2^{\\log (\\log (\\log (m)))-2}}=\\sqrt{\\log \n(m).\\log (\\log (m))}$\\\\\nThis leaves the number of unfixed random variables bounded by O($\\frac{\\log \n(m)}{2^q}=\\frac{\\log (m)}{\\log (\\log (m))}$).Increasing $q$ \nbeyond this value will not give any advantage since the error will then exceed the error due obtained by setting variables to 1.\n\\end{proof}\n\n\\subsection{Proof of the  general approximation bound: Proof of Theorem \\ref{mainthm3} }\n\nIf we are interested in maintaining feasibility of constraints, we can\nensure it by sacrificing the value of the objective function according to\nsome trade-offs. The same algorithm works, where we use the known method \nof damping the probabilities of rounding (\\cite{RT:87,AS:95}).\n\nLet $X_1 , X_2 \\ldots X_u$ denote the unabsorbed random \nvariables in a constraint\nafter the Brownian walk phase and let $ U = \\sum_i X_i  = \\frac{B}{\\beta}$\nfor some $\\beta \\geq 1$. Let $\\hat{X_i}$ be the $\\{0, 1\\}$ random variables\nwhere $\\Pr [ \\hat{X_i} =1] = X_i$ and so $\\mathbb{E} [ \\sum_i  \\hat{X_i} ] = \n\\sum_i X_i = \\frac{B}{\\beta}$. To apply LLL, we need\n$\\Pr [ \\sum_i \\hat{X_i} > B ] < \\frac{1}{d}$.\n\nSince $U = \\sum_i \\hat{X_i}$, then from Chernoff bounds, we know that \n\\begin{equation}\n \\Pr [ U \\geq (1+ \\Delta)\\mathbb{E}[ U ] \\leq {\\left[ \\frac{e^\\Delta }{ { (1+ \n\\Delta )}^{ 1+ \\Delta }} \\right]}^{E[U]} \n\\label{chernoff}\n\\end{equation}\nUsing $\\mathbb{E} [ U ] = \\frac{B}{\\beta}$ and $\\beta = 1+ \\Delta $ we obtain\n$\\Pr [ U \\geq B ] \\leq {\\left[ \\frac{e^{\\beta -1}}{ {(\\beta )}^{\\beta}}\\right]\n}^{B\/\\beta} \\leq {\\left( \\frac{e}{\\beta} \\right)}^B \\leq \\frac{1}{\\alpha d}$\nfor some scale parameter $\\alpha \\geq 1$.\nIt follows that \n\\begin{equation}\n {( \\beta \/e )}^{B} > d\\alpha \n\\label{damp_req}\n\\end{equation}\nTherefore $\\beta = e {(d \\alpha)}^{1\/B}$.\nSince our matrices are 0-1, the\nminimum infeasibility happens at $B+1$, so we can substitute $B+1$ instead\nof $B$ in the previous calculations to obtain $\\beta = e {(d \\alpha)}^{1\/B+1}$.  \nUsing $S \\geq 1$ as the initial scaling for Brownian walk,\nwe obtain the following result using Lemma \\ref{err_bnd}\n\n\\begin{lemma}\nFor $B \\geq 1$,\nwe can round the fractional solution to a feasible integral solution\nwith objective function value $\\Omega ( OPT \/S )$ for S=max(${(\\frac{m}{OPT}})^\\frac{1}{B-1},d^\\frac{1}{B-1}$).\nIf $B \\in \\mathbb{Z}$,B can be replaced by B+1 in above exprssion and proof \n\\label{colbndapprox}\n\\end{lemma}\n\\begin{proof}\nIf we start from $\\overset{\\sim}{x}=\\frac{x'}{S}$, the solution produced by brownian walk satisfies the constraints with RHS  \n$\\frac{B}{\\beta}$(as shown in Lemma $\\ref{err_bnd}$).\\\\\nNow we define events for LLL similar to the proof of Lemma \\ref{bnd_col}.\\\\\n$\\forall 1 \\leq i \\leq m$ define $E_i\\equiv (A_i^T.x >B)$.\\\\\nand define $E_{m+1}\\equiv c^T.x <(1-\\epsilon).OPT$.\\\\\nAs shown above, $\\forall 1\\leq i \\leq m$, $Pr(E_i)\\leq (\\frac{e}{\\beta})^B$.\\\\\nAlso by chernoff's bound $Pr(E_{m+1})\\leq e^{{-\\epsilon^2}{OPT\/2}}$.\\\\\nWe apply LLL with weights $y_i=\\frac{1}{\\alpha d}$, for $1\\leq i \\leq m$\\\\\nand $y_{m+1}=\\frac{1}{2}$.\\\\,\nfor some appropriately defined $\\alpha \\geq 1$.\\\\\nNow we need $(\\frac{e}{\\beta})^B \\leq \\frac{1}{\\alpha.d}(1-\\frac{1}{\\alpha d})^d.\\frac{1}{2}$\\\\\nand $e^{\\frac{-\\epsilon^2 OPT}{2 \\beta}} \\leq \\frac{1}{2}.(1-\\frac{1}{\\alpha .d})^m \\leq \\frac{1}{2}.e^{-\\frac{m}{\\alpha.d}}$(From Equation (\\ref{eqnobj}))\\\\\nThe above is satisfied for $\\frac{OPT}{\\beta}\\geq \\frac{m}{\\alpha.d}$\\\\\nThus we need to choose $\\frac{OPT}{(\\alpha.d)^\\frac{1}{B}}\\geq \\frac{m}{\\alpha.d}$\\\\\nOR $\\beta=(\\alpha.d)^{1\/B} \\geq (\\frac{m}{OPT})^{\\frac{1}{B-1}}$.\\\\\nAlso $\\alpha \\geq 1$.For $B \\in \\mathbb{Z}$, it would have sufficed to use $B'=B+1$ as opposed to choosing B which proves the stated lemma.\n\\end{proof}\nTo prove Theorem \\ref{mainthm3}, we can observe that for $c_i \\geq \\frac{1}{p}$, we have $OPT \\geq \\frac{n}{kp}$.\nFor $\\frac{mk}{n}=O(\\log(m))$, we get the scaling as $max((\\log(m))^\\frac{2}{B},(p \\log(m))^\\frac{1}{B})$, which proves Theorem \\ref{mainthm3}\\\\\n\n\n\\color{black}\n\n\n\\section{Applications of our rounding results}\n\\label{sec6}\n\nIn this section we briefly sketch the applications of our rounding theorems.\nAlthough these do not significantly improve prior results,\nour parameterization could simplify further applications.\n\\ignore{For some of the applications, we need a weighted version of Theorem\n\\ref{mainthm2} that we state without proof - which follows from our\nearlier applications of LLL.\n\\begin{lemma}\nFor $A \\in {\\{0,1 \\}}^{m \\times n}$ with a maximum of\n$\\rho$ 1's in each column, we can round\nthe optimum solution $x^{*}$ of the linear program\n$\\max_{x} \\sum_i c_i x_i \\ \\ s.t. A x \\leq 1 \\ \\ , 0 \\leq\nx_i \\leq 1 $ and\n $1 = \\max_i c_i \\geq c_i \\geq 1\/\\alpha \\mbox{ for some } \\alpha \\geq e $\nto $\\hat{x} \\in {\\{ 0, 1 \\}}^n$ such that\n\\[\n{|| A \\hat{x}||}_\\infty \\leq \\left( \\frac{\\log \\rho + \\log \\alpha + \\log\\log n}{\\log\\log\n(\\rho \\alpha \\log n)}\\right) \\ \\ \\text{ and } \\sum_i c_i \\hat{x}_i \\geq\n(1 - \\epsilon ) OPT \\text{ for } OPT \\geq \\Omega ( m\/(\\alpha d ))\n\\label{bnd_col}\n\\]\n\\end{lemma}\n}\n\n\\subsection{Application to Switching circuits}\n\nConsider an $n$-input butterfly network (with $n\\log n$ total nodes) \nwith $( s_i , t_i ) \\ \\ i \\leq n\\log n$ source destination\npairs where each input\/output node has $\\log n$ sources and $\\log n$ \ndestinations. \nFor any arbitrary instance of routing, we can do a two phase\nrouting with a random intermediate destination, say, we choose a random\nintermediate destination $r_i$ for the source-destination pair $(s_i , t_i )$. \nWe want to route a maximum number of pairs subject to some edge capacity\nconstraints. \n\nIf $n = 2^L$, then the the expected\ncongestion $k$ of an edge for a random permutation is $\\log n \\cdot \n\\frac{ 2^{\\ell} \\times\n2^{L - \\ell} }{2^L} = \\log n$ and moreover it can be \nbounded by $c\\log n$ with high probability for some constant $c$. \nSo there exists a fractional solution with flow value $\\frac{1}{\\log n}$\nfor each of the $n\\log n$ paths with objective function value $n$. \n\nLet $A$ be an $m \\times t$ matrix where $m$ is the number of edges\nand $t = n\\log n$ is the number of paths. The edges are denoted \nby $e_1 , e_2 \\ldots e_m$\nand the paths are denoted by $\\Pi_1 , \\Pi_2 \\ldots \\Pi_t$. Then \n$A_{i,j} =1$ iff \nthe flow $\\Pi_j$ passes through edge $e_i$. The number of edges in a path\nis bounded by $L = \\log n$. The value of the flow through \npath $\\Pi_j$ (\ndenoted by $f_j$) is the amount of flow and let $\\bar{f}$ be the\nvector denoting all the flows. Then $A \\cdot f \\leq \\bar{c}$ where $\\bar{c} =\n( c_1 , c_2 \\ldots c_m )$ is the vector corresponding to the congestion\nin edges $( e_1 , e_2 \\ldots e_m )$.\n\nSince the congestion is bounded by $c \\log n $, $A \\in {\\cal A}^{m \\times \nn\\log n}_{c\\log n}$ and there exists a fractional solution \n$\\bar{f} = ( 1\/c\\log n , 1\/c\\log n \\ldots 1\/c\\log n$ with objective value\n$\\Omega (n)$. Using $\\rho = O(\\log n)$ in \nLemma \\ref{bnd_col}, we can round it to\na 0-1 solution that yields the following result. \n\\begin{theorem}\nIn an $n$ input BBN butterfly network having $2(n \\log n)$ edges, \nwe can route the $\\Omega (n)$ source-sink pairs\nwith congestion $O(\\log\\log n\/\\log\\log\\log n )$. From the capacity \nconstraint, it follows that no node contains more than $O(\\log\\log n)$\nsources or destination. \n\\end{theorem}\nNote that we can also handle weighted objective functions. \n\nThe above result matches the previous results of \\cite{MS:99,CMHMRSSV:98} \nwhich have the advantage of being online. Using $B = \n\\log\\log n\/\\log\\log\\log n$, in Lemma \\ref{colbndapprox} we can obtain an\noptimal bound. Note that, in this case we can match asymptotically \nthe optimal fractional flow of $n\\log\\log n\/\\log\\log\\log n)$ with \n$x_i = \\frac{\\log\\log n}{c \\log\\log\\log n\\cdot \\log n}$.\n\\begin{theorem}\nIn an $n$ input multi-butterfly network having $n\\log n$ edges, \nwe can route $\\Omega (\\frac{n\\log\\log n}{\\log\\log\\log n})$\nflows with congestion $O(\\log\\log n\/\\log\\log\\log n )$ \nwhere each node can be the source or sink of at most \n\\\\\n$O(\\log\\log n\/\\log\\log\\log n )$ flows.\n\\end{theorem}\nThe above result marginally extends a similar result of Maggs and Sitaraman\n\\cite{MS:99} where they can route $(n\/\\log^{1\/B} n)$ pairs in the online\ncase. It remains an open problem to find a fast online implementation of\nour rounding algorithms. \n\\subsection{Maximum independent set of rectangles}\n\nConsider a $\\sqrt{n} \\times \\sqrt{n}$ grid and a set of axis-parallel\nrectangles that are aligned with the grid points,i.e., the upper-left\nand the lower-bottom corners are incident on the grid\\footnote{We can\nassume that all the grid-points that are flush with the sides are\nin the interior by slightly enlarging the rectangle}. Each rectangle\ncontains $A$ grid points for some $A \\leq n$ but they can have any\naspect ratio. Trivially, the different types of such rectangles can be\nat most $A$ where $A = a \\cdot b$ for $a = 1, 2 \\ldots A$. Two such\nrectangles $r_1$ and $r_2$ are {\\it overlapping} if $r_i \\cap r_2$\ncontains one or more grid points. \n\nFrom the $A$ possible rectangles whose upper-right corners are anchored\nat a specific grid point $p$, the input consists of one such rectangle.\nTo avoid messy calculations, we assume that the grid is actually embedded on\na torus. Given a set $S$ of $n$ such rectangles, our goal is to select a\nlarge non-overlapping set of rectangles. Although this is a restricted\nversion of the general MISR problem it still captures many applications. \n\nFor any given grid point $p$, let $S(p)$ denote the set of rectangles\ncontaining $p$. This can be bounded by $\\sum_{a = 1}^{A} a \\times A\/a = \nO( A^2 )$. After solving the relevant packing linear program, the\n$n \\times n$ point-rectangle matrix ${\\cal A}$ contains 1 in the $(i,j)$\nposition if the $i$-th grid point is incident on the $j$-th rectangle.\nFrom our previous observation, no row contains more than $A^2$ 1's. \n\nBy setting $k = A^2$, we can apply our rounding results to obtain a wide\nrange of trade-offs for this problem. For example, if $A$ is polylog($n$), \nthen using $\\rho = A$ in Lemma \\ref{bnd_col}, \nwe can choose $\\Omega (OPT)$ rectangles with no more than \n$O(\\log\\log n\/ \\log\\log\\log n )$ overlapping rectangles on any clique, \nwhere $OPT$ is the fractional\noptimal solution of the corresponding packing problem. \n\nFurther, using a result of \\cite{ENO:04}, we\ncan obtain $\\Omega ( \\frac{OPT \\log\\log\\log^2 n}{\\log\\log^2 n })$ \nnon-overlapping rectangles as well as a $\\frac{\\log\\log^2 n}\n{\\log\\log\\log^2 n}$ approximate solution in the weighted case.\n\\subsection{$b$-matching in random hypergraphs}\n\nIn a hypergraph $H$ on $n$ vertices has hyper-edges defined by subsets of \nvertices.\nWe wish to choose the maximum set of hyper-edges such that no more than\n$b$ hyperedges are incident on any vertex. An integer linear program\ncan be written easily for this problem with $n$ constraints and $m$\nvariables corresponding to each of the $m$ edges. Let $A$ be an $n \\times m$\nmatrix where $A_{i,j} = 1$ if vertex $i$ is incident on the $j$-th hyperedge.\nLet $x_i = \\{ 0,1 \\}$ depending on if the $i$-th hyperedge is selected in the\n$b$-matching.\n\\[ \\textbf{Maximize} \\sum_i x_i \\ \\ s.t. \\ A x \\leq b \\cdot e^m \\ \\ x_i \\in\n\\{ 0, 1 \\} \\] \nThe relaxed LP can be solved and the solution may be rounded. The weighted\nversion can be formulated similarly by associating weights $w_i$ for $x_i$.\n\nIf the $m$ edges of $H$ are random \nsubsets of vertices, then we can represent the fractional \nsolution as an $n \\times\nm$ random matrix with each entry of the matrix set to 1 with probability\n$k\/n$. That is, each hyperedge has $k$ vertices and chooses $k$ vertices \nrandomly. So the expected number of edges incident on a vertex is \n$\\frac{mk}{n}$. There exists a feasible fractional solution using\n${x_i} \\ \\ 1 \\leq i \\leq m = \\frac{n}{mk}$ with objective value $OPT \\geq n\/k$.\n\nFor $b = \\Omega ( \\log\\log n\/\\log\\log\\log n )$, we can can obtain a \n$b$ matching of\nsize $\\Omega (OPT)$ for {\\it most} hypergraphs which matches the best\npossible size given by the fractional optimum. In general, we obtain \nan approximation $k^{1\/b}$ for $b \\geq 2$ using the result of Theorem\n\\ref{mainthm3} including the weighted version. \n\nIt is known that $k$-uniform $b$-matching problem cannot be approximated\nbetter than $\\frac{k}{b\\log k}$ for $b \\leq k\/\\log k$ unless \n$P = NP$ \\cite{OFS:11,HSS:06}. So our\nresult shows that the bound can be much better for many $k$-uniform\nhypergraphs, for example $k = \\log n$ and $b \\geq 2$. \n A closely related result for bounded column case was observed by\nSrinivasan \\cite{AS:96}.\n\n\\section{Concluding Remarks}\n\n\n\n\\subsection{A general approximation bound :Proof of theorem \\ref{mainthm3}}\n\nIf we are interested in maintaining feasibility of constraints, we can\nensure it by sacrificing the value of the objective function according to\nsome trade-offs. The same algorithm works, where we use the known method \nof damping the probabilities of rounding (\\cite{RT:87,AS:95}).\n\nLet $X_1 , X_2 \\ldots X_u$ denote the unabsorbed random \nvariables in a constraint\nafter the Brownian walk phase and let $ U = \\sum_i X_i  = \\frac{B}{\\beta}$\nfor some $\\beta \\geq 1$. Let $\\hat{X_i}$ be the $\\{0, 1\\}$ random variables\nwhere $\\Pr [ \\hat{X_i} =1] = X_i$ and so $\\mathbb{E} [ \\sum_i  \\hat{X_i} ] = \n\\sum_i X_i = \\frac{B}{\\beta}$. To apply LLL, we need\n$\\Pr [ \\sum_i \\hat{X_i} > B ] < \\frac{1}{d}$.\n\nSince $U = \\sum_i \\hat{X_i}$, then from Chernoff bounds, we know that \n\\begin{equation}\n \\Pr [ U \\geq (1+ \\Delta)\\mathbb{E}[ U ] \\leq {\\left[ \\frac{e^\\Delta }{ { (1+ \n\\Delta )}^{ 1+ \\Delta }} \\right]}^{E[U]} \n\\label{chernoff}\n\\end{equation}\nUsing $\\mathbb{E} [ U ] = \\frac{B}{\\beta}$ and $\\beta = 1+ \\Delta $ we obtain\n$\\Pr [ U \\geq B ] \\leq {\\left[ \\frac{e^{\\beta -1}}{ {(\\beta )}^{\\beta}}\\right]\n}^{B\/\\beta} \\leq {\\left( \\frac{e}{\\beta} \\right)}^B \\leq \\frac{1}{\\alpha d}$\nfor some scale parameter $\\alpha \\geq 1$.\nIt follows that \n\\begin{equation}\n {( \\beta \/e )}^{B} > d\\alpha \n\\label{damp_req}\n\\end{equation}\nTherefore $\\beta = e {(d \\alpha)}^{1\/B}$.\nSince our matrices are 0-1, the\nminimum infeasibility happens at $B+1$, so we can substitute $B+1$ instead\nof $B$ in the previous calculations to obtain $\\beta = e {(d \\alpha)}^{1\/B+1}$.  \nUsing $S \\geq 1$ as the initial scaling for Brownian walk,\nwe obtain the following result using Lemma \\ref{err_bnd}\n\\begin{lemma}\nFor $S \\geq c'\\cdot {(d\\alpha )}^{1\/(B+1)}$ , and $B \\geq 1$,\nwe can round the fractional solution to a feasible integral solution\nwith objective function value $\\Omega ( OPT \/S )$ for $OPT \\geq \\Omega \n( mS\/(\\alpha d)))$. \n\\label{colbndapprox}\n\\end{lemma}\n The last condition in the Lemma follows since Equation (\\ref{eqnobj}) is \nsatisfied. \n\nIt would be worthwhile to compare and contrast our results and techniques\nwith \\cite{AS:95,AS:96}. For the case of the number of 1's bounded\nby $\\rho$ in a 0-1 matrix\\footnote{Our techniques and proofs must be\nmodified to work for more general matrices having entries from $[0,1]$} $A$, an objective value $OPT\/ {\\rho}^{1\/B}$  \nwas obtained which is marginally superior to our result of\n$OPT\/{(\\rho \\log n)}^{1\/(B+1)}$.\nStill, we believe, that our techniques are \neasier to use and build upon for various applications compared to the FKG\ninequality based techniques. Following the Brownian motion, the number\nof non-zero coefficients in any constraint is at most $O(\\log n)$ but it\nis not clear how this can be exploited by FKG based analysis of \\cite{AS:95,\nBS:00} which is independent of the number of columns.\n\nFor weighted objective function with $c_i \\geq 1\/p$, we can choose\nthe value of the scaling factor $\\alpha$ more carefully so that $p =\n\\alpha^{1 - 1\/B'} \\cdot {(\\log n)}^{1 -2\/B'} $, and $mk\/n \\leq \\log m$,\nusing $S = {(\\alpha \\log^2 n )}^{1\/B'}$ gives an objective value\nof at least $ nB'\/(p Sk)  \\geq n\/(k \\alpha \\log n) $\nwhich satisfies condition \\ref{eqnobj}. Note that $S = {( p \\log n)}^{1\/(B'-1)}\n$ from the above parameter settings.\\\\\nBy subsituting back $B+1$ for $B'$ in the previous calculations,\n$S = {( p \\log n)}^{1\/B}$. So for $B = 1$, the approximation is\n$O(p\\log n)$.\nThis completes the proof of Theorem \\ref{mainthm2}. \\\\[0.1in]\n\n\\section{Introduction}\nMany combinatorial optimization problems can be modeled\nusing a weighted packing integer program.\n$ \\mbox{ Maximize } \\sum_{i=1}^{i=n} c_i \\cdot x_i$\nsubject to\n$ \\sum_{ i \\in S_j } x_i \\leq 1 \\ \\ 1 \\leq j \\leq m \\ \\ \\ \n x_i \\in \\{0, 1 \\}$.\\\\ \nWlog, we can assume that $\\max_i c_i = 1$.\nAlthough the above formulation appears somewhat restrictive\nhaving a fixed right hand side of each inequality, the\nmethods that we develop extend to more general parameters.\nOften the constraints are expressed as \n$ C_j: V_j \\cdot x \\leq 1$ \nwhere $V_j$ is a 0-1 incidence vector corresponding to set $S_j \\subset\n\\{ 1, 2\\ldots n \\}$. We will also use $x_i$ to denote the $i$-th coordinate\nof a vector $x$.\nSince this version is also NP-hard as it captures many intractable\nindependent set problems,\na common strategy is to solve the Linear Program (LP)\n corresponding to the relaxation $0 \\leq x_i \\leq 1$. Then\nuse the LP optimum $OPT$ to obtain a good approximation of the optimal\nintegral solution. \nSuppose the optimum is achieved by the\nvector $x' = [ x'_1 , x'_2 \\ldots x'_n ]$ \nIn the conventional {\\it randomized rounding} \\cite{RT:87}, we \nround each $x_i$ independently in the following manner.\\\\ \n$\\hat{x}_i =\n\\{ 1 \\text{ with probability $x'_i$ and } \n0 \\text{ otherwise} \\}$.\n\\\\\nSince $\\mathbb{E}[ \\hat{x}_i ] = x'_i$, it follows that $\\mathbb{E} [ \\sum_i c_i \\hat{x}_i ] \n=  \\sum_i c_i \\cdot x'_i $ and using Chernoff bounds, we can show that for\nall $j$, w.h.p. $ \\sum_{ i \\in S_j } \\hat{x}_i \n\\leq \\left(\\frac{\\log m}{\\log\\log m}\n\\right) $.\n\nThe primary motivation for improved randomized rounding\nis improved approximation of\ninteger linear packing problems. Historically, the original randomized\nrounding technique was proposed for obtaining better approximation of \nmulticommodity\nflows \\cite{RT:87}. \nSrinivasan \\cite{AS:95,AS:99} presents an extensive survey of many \nsophisticated variations of\nrandomized rounding techniques and applications to approximation algorithms.\nIt was established much later \\cite{CGKT:07}\nthat the Raghavan-Thompson bound cannot be improved as a consequence of the\nfollowing result.\n\\begin{lemma}[\\cite{CGKT:07}]\nThere exists a constant $\\delta > 0$ such that the integrality gap of the \nmulticommodity flow relaxation problem is $\\Omega ( 1\/c' \\cdot \nn^{\\frac{1}{3c' + 13}} )$ for any congestion $c'$, $1 \\leq c' \\leq \n(\\delta \\log n)\/\n\\log\\log n $ where $n$ is the number of vertices in the graph and the \nintegrality gap for $c'$ is with respect to congestion 1. \n\\end{lemma}\n\nTherefore no rounding algorithm can simultaneously achieve error\n$c' = o( \\log n\/\\log\\log n)$, \nand an objective function value of $\\Omega ( OPT )$ for all\npolynomial size input ($m \\leq n^{O(1)}$). \nNote that this does not preclude improvement of the multicommodity flow problem\napproximation by alternate formulation - for this, there are alternate \nhardness bounds given by \\cite{CGKT:07}. \n\nSubsequent to the work of Raghavan-Thompson, Srinivasan \\cite{AS:95,AS:96}\nBaveja and Srinivasan \\cite{BS:00}, Kolliopoulous and Stein \\cite{KS:98}\nobtained results that focussed on circumventing the above bottleneck\nby making use of some special properties of the constraint matrices like\ncolumn-restricted matrices. In \nparticular, the focus shifted to designing approximation algorithms that\nare feasible (unlike the basic independent rounding of RT) by bounding\ndependencies between inequalities and using {\\it Lovasz Local Lemma} (LLL)\nor even more sophisticated corelation inequalites like FKG \\cite{AS:96}.\nLeighton, Rao and Srinivasan \\cite{LRS:98} made very clever use of\nLLL to achieve nearly optimal results in graphs with {\\it short} flow paths.\nThe notion of short flow paths was developed further in the context of \nthe {\\it unsplittable} \nmulticommodity flow problem with an explicit objective function in many\npapers including \\cite{KR:96,BS:00,KS:06,CCGK:07}. Intuitively, short\nflow paths reduce dependencies between conflicting routing\npaths and makes the algorithms more efficient in terms of assigning paths\nwithout exceeding the maximum allowed congestion (for edge-disjoint paths\nit is 1). \n\nIn this paper, we attempt to provide a uniform framework behind many prior\nuses of independent rounding by recasting the process as\na Brownian walk in high dimension and analyzing its convergence. For readers\nfamiliar with this technique, it may appear to be an overkill since the\nfinal outcome of Brownian motion and independent rounding are identical \n(we shall formalize this in the next section). However, we will demonstrate\nthat this framework offers a cleaner exposition and simpler explanations \nof many of the previous clever, yet {\\it ad-hoc} techniques. \nWe have presented\na detailed characterization of the time-dependent behavior of the random variables \nassociated with each inequality executing Brownian motion which was not studied before\nto the best of our knowledge.   \nIt could potentially offer new tools for analysis of more complicated rounding\nmethods where the random variables may not be independent. \n\nWe demonstrate two distinct applications of our new framework\nby addressing the rounding problem for the {\\it average} case of\nan input family that we will define more precisely. \n\\\\\n(i) A simple approximation algorithm for {\\it weighted} objective\nfunction. \n\\\\\n(ii) An improvement of the Raghavan-Thompson rounding error for\na restricted class of weighted objective function.\n\nAs opposed to\nthe one-shot rounding of \\cite{RT:87} , also referred as {\\it independent\nrounding}, our rounding is iterative and based on successive refinements\nof the LP solution that has a lower variance per step, to yield a sparser\nset of constraints. This reduces dependencies between inequalities\nand makes the use of techniques like Lovasz Local Lemma (LLL) more effective. \n\\subsection{Main results and some applications}\n\\begin{theorem}\nGiven an $m \\times n$ 0-1 matrix $A$, such that $A \\cdot x' \\leq b$ for \n$x'_i \\in [0,1]$,\nand an objective function $\\sum_i c_i x_i$, \nin randomized polynomial time $x'$ can be transformed into $y'$ which \nare $[0,1]$ valued random variables such that for all $(\\log n - \\log\\log m \n+ \\log b ) \\geq p \\geq 0$\\\\\n(i) All constraints have $ \\leq \\frac{nB}{2^p}$ non-zero variables from \n$y'$ and\n\\\\\n(ii) For all $i$, $\\sum_i A_i \\cdot y' \\leq O(\\sqrt{\\frac{2^p B \\log m}{n}})$ where \n$A_i$ is the $i$-th row of $A$.\n\\\\\n(iii) $\\mathbb{E} [ c_i y'_i ] = \\sum_i c_i x_i$\n\\label{sparsify}\n\\end{theorem}\nIn other words we have a sparse system of equivalent constraints with the objective\nfunctional value unchanged in the expection.\n\\\\\n\n{\\bf Remark} For $p = \\log n - \\log\\log m + \\log B$, there are $\\log m$ \nvariables in each inequality summing upto $O(1)$. This specific result has\nbeen observed before by using a clever application of independent rounding\nin the following manner. Round each $x'_i < \\frac{1}{\\log m}$ to \n$\\frac{1}{\\log m}$ with probability $x'_i \\log m$. Then $\\mathbb{E} [x_i] =\nx'_i$ and no inequality has more than $\\log m$ variables with high probability.\n\\ignore{\nOur transformation does not use any extra variables unlike the \nprevious methods that makes multiple copies of each variable \n(see for example \\cite{CC:09}). \nThey transform the initial matrix $A$ to an $m\\times n\\log n$ matrix $A'$ and \n$x'$ is transformed into $y'$ that has dimension $n\\log n$, using \nanother application of independent rounding (some papers present it as \na technique to do $\\frac{1}{\\log n}$ rounding) so\nthat the fractional solution remains feasible. Subsequently,\ncare has to be taken to ensure that the rounded solution chooses exactly\none copy of a variable which is an implicit case of dependent rounding. \nThe {\\it path decomposition} strategy for multicommodity flow \nin the original paper of Raghavan and Thompson \\cite{RT:87} is also \nalong similar lines.  While it works for specific instances, it will be\ntechnically challenging to formalize it as a {\\it blackbox} \ntechnique similar to the previous theorem.    \n}\n\nUsing the above sparsification, we obtain a number of interesting \nresults arguably in simpler ways than known previously.  \n\\ignore{\n\\begin{theorem}\nLet $A \\in {\\{0,1\\}}^{m \\times n}$, with no more than $\\rho$\n1's per column.\nSuppose $x' \\in {[0,1]}^n$ maximizes $\\sum_i c_i x_i $ and\nsatisfies $ A \\cdot x \\leq \\cdot e^m $ with $OPT = \\sum_i c_i x'_i$.\nThen $x'$ can be rounded to $\\hat{x} \\in {\\{0,1 }\\}^n$ in polynomial time \nsuch that\nthe following holds with high probability, \n\\begin{eqnarray}\n {|| A \\hat{x}||}_\\infty & \\leq & 1\n\\text{ and } \\\\\n\\sum_i c_i \\hat{x}_i & \\geq & \\Omega ( OPT\/(\\rho \\log m) )\n\\end{eqnarray}\n\\label{mainthm2}\n\\end{theorem}\n\\vspace{-0.3in}\nThe above result follows by applying a simple greedy procedure to the \nsparsified matrix.\n\nFor a more general RHS, $b \\geq 1$, we use LLL-based techniques to obtain the following\nresults.\n}\n\\begin{theorem}\nLet ${\\cal A}^{m \\times n}_k$ denote the family of $m \\times n$ \nmatrices where each row (independently) has $k$ ones in randomly \nchosen columns from $\\{ 1, 2, \\ldots n \\}$ and 0 elsewhere\\footnote{Alternately\nwe can set every entry to be 1 with probability $k\/n$ independently}. \nLet $A \\in \n{\\cal A}^{m \\times n}_k$, then\nfor any point $x' = ( x'_1 , x'_2 \\ldots x'_n ) , \\\n0 \\leq x'_i \\leq 1$ such that\n$ A \\cdot x' \\leq e^m $, where $e^m$ is a vector of $m$ 1's,\nand $OPT = \\sum_i c_i \\cdot x'_i$ where \n$1 = \\max_i \\{ c_i \\}$ and\n$\\forall i\\ c_i \\geq \\frac{1}{p}$. Then,\n$\\bar{x'}$ can be rounded to $\\hat{x} \\in {\\{0,1 }\\}^n$ such that\nthe following holds with probability $\\geq 1 - \\frac{1}{m}$\n\\begin{eqnarray}\n{|| A \\hat{x}||}_\\infty & \\leq & O\\left(\\frac{\\log(mkp\\log m\/n) + \\log\\log m }{\\log\\log (mkp\\log m\/n + \\log m)}\\right)\n\\text{ and } \\label{cond1}\\\\\n\\sum_i c_i \\hat{x}_i & \\geq & \\Omega ( OPT )\n\\end{eqnarray}\nMoreover such an $\\hat{x}$ can be computed in randomized polynomial time.\n\\label{mainthm0-1}\n\\end{theorem} \n{\\bf Remark} \n(i) The result relates the rounding error to the average number of ones\nin a column, i.e., $\\frac{mk}{n}$ and characterizes tradeoffs between the\nparameters $m, n, k$. For example, for $k = \\sqrt{n}$, we can bound the error\nto $O(\\log\\log n)$ for $m \\leq \\sqrt{n} \\cdot polylog(n)$ for\n$p  \\leq \\frac{1}{\\log^{O(1)} n}$.\nThis is a significant improvement over the $O(\\frac{\\log n}{\\log\\log n})$\nbound of the independent rounding. \n\\\\\n(ii) For $k = \\log n$, the bound is indeed tight.\nA proof is given in the appendix (\\cite{V:15}).\n\\\\\n(iii) For the unweighted case, the result holds for arbitrary distribution \nof the $k$ 1's in each row.\\footnote{ \nSet each variable to 1 with\nprobability $1\/k$, and for each violated constraint that has more than $q =\n\\log ((mk)\/n )$\nones, zero all its variables (or zero enough of its variables, chosen\narbitrarily, so that it has only $q$ ones). At most $n\/2$\nof the variables are heavy in the sense that they appear in\nmore than $2mk\/n$ constraints. For a light variable, even if it comes up 1, the\nprobability that it is a member of a violated constraint is small (say, below\n1\/2), and hence the light variables (using linearity of expectation) guarantee\na value of $\\Omega(n\/k)$.\\\\\nThis proof sketch was given by an anonymous reviewer of an earlier version.\n}\n\\begin{theorem}\nLet $A \\in {\\cal A}^{m \\times n}_k$,\nsuch that  $x' \\in {[0,1]}^n$ maximizes $\\sum_i c_i x_i $ and\nsatisfies $ A \\cdot x \\leq b \\cdot e^m $ with $OPT = \\sum_i c_i x'_i$.\nIf $1 = \\max_i c_i \\geq 1\/p ,\\ \\ p \\geq 1$ \nand $m \\leq \\frac{n\\log m}{k}$,\nthen $x'$ can be rounded to $\\hat{x} \\in {\\{0,1 }\\}^n$ in polynomial time \nsuch that the following holds with high probability for $b \\geq 1$\n\\begin{eqnarray}\n {|| A \\hat{x}||}_\\infty & \\leq & b\n\\text{ and } \\\\\n\\sum_i c_i \\hat{x}_i & \\geq & \\Omega ( OPT\/(max(p \\log m ,\\log^2(m))^{1\/b})) \n\\end{eqnarray}\n\\label{mainthm3}\n\\end{theorem}\n\\vspace{-0.3in}\n\\color{black}\nWe observe a trade-off between the\nweights and the approximation factor ${(p \\log m ))}^{1\/b}$. For $b = O(\n\\frac{\\log\\log m}{\\log\\log\\log m})$, we can obtain $O(1)$ approximation for \nfor $c_i \\geq \\frac{1}{\\log^C n}$ for any constant $C \\geq 1$. \n\nThe random matrices provides a natural framework for combinatorial\nresults related to random hypergraphs. We sketch one such application to\n$b$-matching of $k$-regular hypergraphs that yields \nan approximation $k^{1\/b}$ for $b \\geq 2$ using the result of Theorem\n\\ref{mainthm3} and\nextends a result of Srinivasan \\cite{AS:96}. This implies that \nfor $b = \\Omega ( \\log\\log n\/\\log\\log\\log n )$, we can can obtain a\n$b$ matching of\nsize $\\Omega (OPT)$ for {\\it most} hypergraphs which matches the best\npossible size given by the fractional optimum. \nIt is known that $k$-uniform $b$-matching problem cannot be approximated\nbetter than $\\frac{k}{b\\log k}$ for $b \\leq k\/\\log k$ unless\n$P = NP$ \\cite{OFS:11,HSS:06}. \n\n\\ignore{For lack of space, we have sketched details of the other applications\nin the Appendix, section \\ref{sec6}.}\n\\subsection{An overview and related work}\nOur algorithm can be best characterized as a randomized iterated rounding where we begin\nfrom a fractional feasible (specifically optimal) solution $x'$ and iteratively\nconverge to a {\\rm good} integral solution. Our algorithm\nhas two distinct stages - {\\it random walk} stage (more precisely, Brownian walk) and \nsubsequently in the second stage invokes the Moser-Tardos iterative scheme \nfor constructive Lovasz\nLocal Lemma (LLL). In the first stage we effectively \n{\\it slow down} the RT rounding \nprocess. \nOur approach is intuitive  - starting from $x'$, for each\nvariable (dimension), we will roughly increment \n(actually a normal Gaussian increment)\n$x_i$ by $\\pm \\gamma$ for\na suitably chosen $1 > \\gamma > 0$ where the sign (direction) is a\nrandom variable. In each iteration,\nthe values of $x_i$'s are modified and we continue this\nprocess for each $x_i$ until it is in the range $[0, \\delta ] \\cup\n[1- \\delta,1]$ for an appropriate $1 > \\delta > 0$ such that\n$\\delta > \\gamma$. \nAt this point we {\\it fix} the variable\nand we terminate when all inequalities have less than some predetermined \nvalue $u$ of {\\it unfixed} variables . \nThis stage has some similarities with the method \nof Lovette and Meka \\cite{LM:12} but our analysis requires completely \ndifferent techniques.\nThe crux of the method called {\\it partial coloring lemma}\nis a rounding strategy of an arbitrary $ x \\in {[-1 , +1]}^n$ vector within\nthe discrepancy polytope defined by the constraints starting with\n$x = (0,0, \\ldots 0 )$.\nTheir method can be mapped to\n$\\{ 0, 1\\}$ rounding as well, that was observed by Rothvoss\\cite{R:13}.\nCompared to the setting of the discrepacy rounding, we are dealing with\nsmaller error margins and the variable and polytope constraints are\nnot widely separated in terms of distances from the starting point.\nMoreover, one needs\nto also account for the deviation of the objective function which is\nnot required for Spencer's discrepancy result.\n\n\\ignore{\nAt a high level, our framework provides an alternate \ntechnique for limiting dependencies. In the remark following Theorem \n\\ref{mainthm0-1} we noted that the rounding error is related to the average\nnumber of 1's in a column which is related to the path-lengths in switching\ncircuits (c.f. section \\ref{sec6}). \nWe show that $\\Omega (\\frac{n \\log\\log n}{ \\log\\log\\log n})$ \nconnections can be supported in a \n{\\it multi-butterfly} network with congestion\nbounded by $O(\\log\\log n\/\\log\\log\\log n )$, extending a result of \n\\cite{MS:99,CMHMRSSV:98} where they achieve a similar congestion for $n$ flows. \n\nWe also present approximation algorithms for the Maximum Independent Set of\nRectangles (MISR) problem where the rectangles are aligned with grid points.\nThe rectangles can have arbitrary aspect ratios but their areas are bounded.\nThis version can be useful for many applications including map labelling\nwhere the bounding boxes are not arbitrarily large and do not intersect \ntoo many cliques. For the case that the rectangles contain at most\npolylog grid points, we present a\n$O(\\frac{\\log\\log^2 n}{\\log\\log\\log^2 n})$ approximation bound for the \nweighted version. Obtaining an approximation factor $o(\\frac{\\log n}\n{\\log\\log n})$ \nbeen an important open problem (\\cite{CC:09,CH:09}) and has \npartly motivated the recent work in \nquasi-polynomial time approximation algorithms \\cite{AW:13}.   \n\nFurther, the random matrices provides a natural framework for combinatorial\nresults related to random hypergraphs. We sketch one such application to\n$b$-matching of $k$-regular hypergraphs that yields \nan approximation $k^{1\/b}$ for $b \\geq 2$ using the result of Theorem\n\\ref{mainthm2} and\nextends a result of Srinivasan \\cite{AS:96}. This implies that \nfor $b = \\Omega ( \\log\\log n\/\\log\\log\\log n )$, we can can obtain a\n$b$ matching of\nsize $\\Omega (OPT)$ for {\\it most} hypergraphs which matches the best\npossible size given by the fractional optimum. \nIt is known that $k$-uniform $b$-matching problem cannot be approximated\nbetter than $\\frac{k}{b\\log k}$ for $b \\leq k\/\\log k$ unless\n$P = NP$ \\cite{OFS:11,HSS:06}. \n\nFor lack of space, we have sketched details of the above applications\nin the Appendix, section \\ref{sec6}.\n\\subsection{Prior related work in randomized rounding}\n\nSubsequent to the work of Raghavan-Thompson, Srinivasan \\cite{AS:95,AS:96}\nBaveja and Srinivasan \\cite{BS:00}, Kolliopoulous and Stein \\cite{KS:98}\nobtained results that are similar in spirit to this paper. A direct\ncomparison of these results is a complex exercise \nsince they had not addressed this specific framework of random matrices. \nLater, we\nwill be able to compare \nthem with respect to our result on column-restricted \n0-1 matrices that we use as an intermediate step\nto prove the main results. Although, this intermediate\nresult falls a little short of the best known, we feel that our techniques\nare simpler and more general that could open up a new paradigm for rounding \nmore general integer programs.\n\nThere exists a very extensive and rich literature on the use of randomized\nrounding for solving various generalizations \nof the edge-disjoint path problem starting\nwith the seminal paper of Raghavan and Thompson. Leighton and Rao \\cite{LR:88,\nLR:99} formalized many algorithmic an combinatorial \naspects of the multicommodity flow problem. \nThe problem of edge\ndisjoint paths in expander graphs was resolved by Bohman and Frieze\n\\cite{BF:01}. \n\n\n\nThe recent work on constructive discrepancy perhaps comes\nclosest to our technique of randomized iterative rounding. \nBansal's \\cite{bansal:10} seminal\npaper on a constructive proof of Spencer's discrepancy theorem is based\non rounding solutions of successive\na semi-definite program that captures the discrepancy constraints.\nThis method was further refined and simplified in the work of\nLovette and Meka\\cite{LM:12} who\nderived an alternate proof\nbased on a very elegant analysis of a multidimensional random walk.\nThe crux of the method called {\\it partial coloring lemma}\nis a rounding strategy of an arbitrary $ x \\in {[-1 , +1]}^n$ vector within\nthe discrepancy polytope defined by the constraints starting with\n$x = (0,0, \\ldots 0 )$. \nTheir method can be mapped to\n$\\{ 0, 1\\}$ rounding as well, that was observed by Rothvoss\\cite{R:13}.\nCompared to the setting of the discrepacy rounding, we are dealing with\nsmaller error margins and the variable and polytope constraints are \nnot widely separated in terms of distances from the starting point. \nMoreover, one needs\nto also account for the deviation of the objective function which is\nnot required for Spencer's discrepancy result.\n}\n\n\\subsection{A Lower Bound on error}\n\\begin{lemma}\nFor $A \\in {\\cal A}^{m\\times n}_{\\log n}$ such that $A\\cdot \\bar{x} \\leq e^m$,\nwe cannot simultaneously obtain $\\sum_i \\hat{x}_i = \\Omega ( n\/\\log n )$ and\n${|| A \\hat{x}||}_\\infty  \\leq o(\\frac{\\log\\log n}{\\log\\log\\log n})$\nfor $\\hat{x}_i \\in \\{ 0, 1 \\}$.\n\\label{lbnd}\n\\end{lemma}\n\n\\begin{proof}\n\nAs mentioned before, it is easy to see that $x_i = 1\/k$ is a solution \nfor $\\bar{x}$ with objective\nfunction value $n\/k$. After rounding $\\bar{x}$ we want to have at least \n$\\Omega ( n\/k )$ 1s in the rounded vector, say $\\bar{y}$.\nWe must have $A \\cdot \\bar{y} \\leq t\\cdot e$ for some rounding guarantee\n$t$. Let $b$ be a fixed vector with $n\/k$ 1s. \n\nLet $r$ be a row vector with $k$ 1s in random location - this corresponds to\na row of $A$. The probability that $< r , \\bar{y} > \\ \\ \\geq t$ is given by\n \\[  \\frac{{ (n\/k) \\choose t} \n\\cdot { (n-n\/k) \\choose (k-t) }}{{n \\choose k }} \\]\nUsing $ {(\\frac{n}{k})}^k \\leq {n \\choose k} \\leq {(\\frac{ne}{k})}^k $, the \nabove expression is\n\\begin{eqnarray}\n \\geq & \\frac{ {( \\frac{n}{tk})}^t \\cdot {(\\frac{n -n\/k}{k-t})}^{k-t}}{\n{(\\frac{ne}{k})}^k} \\\\\n = & \\frac{ \\frac{1}{t^t} \\cdot {( \\frac{k-1}{k-t})}^{k-t}} {\n e^k } \\\\ \n = & \\frac{ k^{k-t}}{ {( k-t )}^{k-t} \\cdot e^k \\cdot t^t }\\\\\n = & \\frac{1}{ {( 1 - t\/k )}^{k-t} \\cdot e^k \\cdot t^t } \\\\\n \\geq & \\frac{1}{{( 1 - t\/k )}^{k\/2} \\cdot e^k \\cdot t^t } \\text{ for } k \\gg t\\\\\n = & \\frac{1}{ e^{ k - t\/2} \\cdot t^t }\n\\end{eqnarray}\nSo the probability that for $m$ independently chosen rows, the probability that\n$< r , \\bar{y} > \\ \\ < t$ is \n\\begin{eqnarray}\n  p(m,k,t) &  \\leq & {\\left( 1 - \\frac{1}{ e^{ k - t\/2} \\cdot t^t } \\right)}^m \n\\leq  {\\left( 1 - \\frac{1}{ e^{ k } \\cdot t^t } \\right)}^m =\n{\\left( 1 - \\frac{1}{ e^{ k } \\cdot t^t } \\right)}^{e^k t^t \\frac{m}{e^k t^t}}\n\\\\\n & \\leq & {\\left( \\frac{1}{e} \\right)}^{\\frac{m}{e^k t^t}} \n\\end{eqnarray} \nSince there are no more than ${n \\choose\n(n\/\\log n)}$ choice of columns with $n\/\\log n$ 1s, the probability that all\nthe dot products are less than $t$ is less than ${n \\choose (n\/\\log n)} \\cdot\np(m,k,t)  \\leq p(m,k,t) \\cdot {(\\frac {n e}{n\/\\log n})}^{n\/\\log n} \n \\leq e^n \\cdot p(m,k,t)$. If $e^n \\cdot p(m,k,t) < 1$ then there must exist\nmatrices in ${\\cal A}^{m \\times n}_k$ that do not satisfy the error bound $t$.\nSo,\n\\[    1  > {\\left( \\frac{1}{e} \\right)}^{\\frac{m}{e^k t^t}} \\cdot e^{n} \n    \\Rightarrow {( e ) }^{\\frac{m}{e^k t^t}}  >  e^n \\] \n which is equivalent to the condition \n\\begin{equation}\n \\frac{m}{n}  > e^k t^t \\text{  or  }\n\\log(m\/n) > k + t\\log t \n\\label{lbndreq}\n\\end{equation}\n \n For $k = \\log n$ and $m = n\\cdot polylog(n) $, this holds for some $t \\geq \n\\frac{\\alpha \\log \\log n}{\\log\\log\\log n})$, i.e., for some constant \n$\\alpha > 0$, i.e., \nthe error cannot be\n$o (\\frac{\\log\\log n}{\\log\\log\\log n})$. \n\\end{proof}\n\n\n\n\\subsection{A general approximation bound }\n\nIf we are interested in maintaining feasibility of constraints, we can\nensure it by sacrificing the value of the objective function according to\nsome trade-offs. The same algorithm works, where we use the known method \nof damping the probabilities of rounding (\\cite{RT:87,AS:95}).\n\nLet $X_1 , X_2 \\ldots X_u$ denote the unabsorbed random \nvariables in a constraint\nafter the Brownan motion phase and let $ U = \\sum_i X_i  = \\frac{B}{\\beta}$\nfor some $\\beta \\geq 1$. Let $\\hat{X_i}$ be the $\\{0, 1\\}$ random variables\nwhere $\\Pr [ \\hat{X_i} =1] = X_i$ and so $\\mathbb{E} [ \\sum_i [ \\hat{X_i} ] = \n\\sum_i X_i = \\frac{B}{\\beta}$. To apply LLL, we need\n$\\Pr [ \\sum_i \\hat{X_i} > B ] < \\frac{1}{d}$.\n\nSince $U = \\sum_i \\hat{X_i}$, then from Chernoff bounds, we know that \n\\begin{equation}\n \\Pr [ U \\geq (1+ \\Delta)\\mathbb{E}[ U ] \\leq {\\left[ \\frac{e^\\Delta }{ { (1+ \n\\Delta )}^{ 1+ \\Delta }} \\right]}^{E[U]} \n\\label{chernoff}\n\\end{equation}\nUsing $\\mathbb{E} [ U ] = \\frac{B}{\\beta}$ and $\\beta = 1+ \\Delta $ we obtain\n$\\Pr [ U \\geq B ] \\leq {\\left[ \\frac{e^{\\beta -1}}{ {(\\beta )}^{\\beta}}\\right]\n}^{B\/\\beta} \\leq {\\left( \\frac{e}{\\beta} \\right)}^B \\leq \\frac{1}{\\alpha d}$\nfor some scale parameter $\\alpha \\geq 1$.\nIt follows that \n\\begin{equation}\n {( \\beta \/e )}^{B} > d\\alpha \n\\label{damp_req}\n\\end{equation}\nTherefore $\\beta = e {(d \\alpha)}^{1\/B}$.\nUsing $S \\geq 1$ as the initial scaling for Brownian motion,\nwe can obtain the following result using Lemma \\ref{err_bnd}\n\\begin{lemma}\nFor $S \\geq c'\\cdot {(d\\alpha )}^{1\/B}$ , and $B \\geq 1$,\nwe can round the fractional solution to a feasible integral solution\nwith objective function value $\\Omega ( OPT \/S )$ for $OPT \\geq \\Omega \n( mS\/(\\alpha d)))$. \n\\label{colbndapprox}\n\\end{lemma}\n The last condition in the Lemma follows from Equation \\ref{eqnobj} is \nsatisfied. \nThis completes the proof of Theorem \\ref{mainthm2}.\n\nIt would be worthwhile to compare and contrast our results and technqiues\nwith \\cite{AS:95,AS:96}. For the case of the number of 1's bounded\nby $\\rho$ in a 0-1 matrix $A$, an objective value $OPT\/ {\\rho}^{1\/B}$  \nis obtained which is marginally superior to our result of\n$OPT\/{(\\rho \\log n)}^{1\/B}$. Still, we believe, that our technqiues are \neasier to use and build upon for various applications compared to the FKG\ninequality based technqiues. Following the Brownian motion, the number\nof non-zero coefficients in any constraint is at most $O(\\log n)$ but it\nis not clear how this can be capitalized by FKG based analysis.\n\n\n\n\\subsection{Random matrices}\n\\label{sec5}\n\nLet ${\\cal A}^{m\\times n}_k$ denote the family of \n$m \\times n$ 0-1 matrix with exactly $k$ 1s in each of\nthe $m$ rows chosen uniformly at random. \nClearly $x'_i = 1\/k$ is a feasible solution with objective\nfunction value $n\/k$. After rounding $x'$ to $\\hat{x}$, \nwe want to achieve an objective value\n$\\Omega ( n\/k )$ 1s in the rounded vector. \nIn addition, $A \\cdot x' \\leq t\\cdot e^m$ for error guarantee\n$t$. \n\nTo compute the dependency $d$ for $C_r$, we observe that \nanother constraint $C_i$ will contain a 1 in $j_1 (r)$ with \nprobability $\\frac{k}{n}$, \ni.e., if it had one of the $k$ randomly chosen 1's in that column, which\nis $\\frac{k}{n}$. Since all the rows were chosen independently, the expected\nnumber of rows among $m$ rows that have a 1 in column $j_1 (r)$ can\nbe bounded by $\\frac{m k}{n}$ and by $O(\\max\\{ \\frac{m k }{n} , \\log m \\} )$ \nwith probability greater than $1 - 1\/m$. Since this holds for all the positions,\n$j_i (r)$, by the union bound, and \nusing $\\max \\{ a , b \\} \\leq (a+b)$, we can claim the following\n\\begin{claim}\nThe total number of constraints that are \ncorrelated to $C_r$ can be\nbound by $O(\\frac{m k \\log m}{n} +  \\log^2 m )$ with high \nprobability.\n\\label{deg_bnd}\n\\end{claim}\nIn our context, for $k = \\log m$, $d = O( \\frac{m\\log^2 m}{n} + \n\\log m ^2)$. \nWe can verify this by \nexhaustively computing the dependence - if it exceeds this then our algorithm\nis deemed to have failed which is bounded by inverse polynomial probability.\nIf we choose $t$ such that $e \\cdot 2^{-t} \\cdot (d+1) \\leq 1$ or\nequivalently $t = \\Omega ( \\log (\\frac{m\\log^2 m}{n}+ \\log^2 m  ) ) $, \nthen we can apply\nthe previous theorem to obtain a rounding that satisfies an error bound of\n$O( \\log (\\frac{m\\log^2 m}{n}+ \\log^2 m ) )$. \\\\\nFor $m$ bounded by \n$n \\log^{O(1)} m$ this is\n$O(\\log\\log m)$ which is substantially better than the Raghavan-Thompson\nbound.  \\\\However for $m \\geq n^{1 + \\epsilon}$ for any constant $\\epsilon > 0$,\nit is no better.\n\\begin{proof} (Completing the Proof of Theorem \\ref{mainthm0-1})\nNow we formally define our bad events $E_i$ here.\\\\\n$\\forall 1\\leq i \\leq m$, define $E_i\\equiv(A_i^T x > \\delta )$.(where we choose error to be $\\delta$).\nDefine $E_{m+1}\\equiv(c^T.x <(1-\\epsilon)OPT)$.\nWe have $Pr(E_i)\\leq \\frac{e^{\\delta}}{{1+\\delta}^{1+\\delta}}$.\nand $Pr(E_{m+1})<e^{\\frac{-\\epsilon^2.OPT}{2}}$.\nWe choose $y_i=\\frac{1}{\\alpha.d}$,for some $\\alpha\\geq1$.\nand choose $y_{m+1}=\\frac{1}{2}$.\nTo apply LLL we need,\n$Pr(E_i)<\\frac{1}{\\alpha.d}(1-\\frac{1}{\\alpha.d})^d.(1-\\frac{1}{2})$\nand $Pr(E_{m+1})<\\frac{1}{2}(1-\\frac{1}{\\alpha.d})^m$.\nTo satisfy these equations we get error,$\\delta=\\frac{\\log(\\alpha.d)}{\\log(\\log(\\alpha.d))}$.\\\\\nAlso we get $e^{\\frac{-\\epsilon^2.OPT}{2}}<e^{\\frac{-m}{\\alpha.d}}$.\nThe second equation is satisfied for $OPT>\\frac{m}{\\alpha.d}$ or $\\alpha.d >\\frac{m}{OPT}$.\nCombining the 2 results we have error $\\delta$ in constraints as $max\\left\\{\\frac{\\log(d)}{\\log(\\log(d))},\\frac{\\log(\\frac{m}{OPT})}{\\log(\\log(\\frac{m}{OPT}))}\\right\\}$\\\\which is $max\\left\\{\\left( \\frac{\\log (\\frac{mk\\log(m)}{n}) +  \\log\\log m}{\\log\\log \n(\\frac{mk \\log(m)}{n} \\log m)}\\right),\\frac{\\log(\\frac{m}{OPT})}{\\log(\\log(\\frac{m}{OPT}))}\\right\\}$.\nWhen $c_i \\geq \\frac{1}{p}$ we have $OPT \\geq \\frac{n}{k.p}$.\nSubstituting the value of OPT in the equation and using the fact that $\\frac{mk}{n}=O(\\log(m))$, we prove Theorem \\ref{mainthm0-1}. \n\\ignore{\n\\color{red}\n\n Then, using \n$d \\leq \\frac{mk\\log m}{n}$ (Claim \\ref{deg_bnd}), $\\exp (- m\/(d\\alpha ) \\leq\n\\exp (- n\/(\\alpha k\\log n))$ so the condition \\ref{eqnobj} is \nsatisfied for LLL to be applicable. In general, Equation \\ref{eqnobj} can be\nsatisfied when the objective function value $OPT$ \nexceeds $\\frac{n}{\\alpha k\\log n}$ for choosing a suitable $\\alpha \\geq e$. \n\n\\subsection{Putting it together}\nFor a random $m \\times n$ matrix $A \\in {\\cal A}^{m \\times n}_k$ \nfor $k \\geq \\log n$ \nin $A^k$, we first apply the Brownian\nwalk based algorithm to sparsify it to $\\log m$ unfixed variables\nper constraint. \n\n\nFor weighted objective function, observe that Equation \\ref{eqnobj} \ncan be satisfied for value of the objective function at least $\\Omega \n(n\/(k\\log n)$. This can be ensured by the condition $c_i \\geq 1\/\\log n$.\nMore generally, for $c_i \\geq 1\/(\\alpha \\log n)$, \nwe can use a scale factor $\\alpha \\geq 1$ to obtain\nan error bound of $O(\\log (d \\alpha) \/\\log\\log (d \\alpha))$ with\nobjective value $\\Omega ( n\/(k\\alpha \\log n))$.\\\\ \nThe rest follow from LLL and the algorithm of Moser-Tardos.} The expected \nrunning time for the second phase is $O( \\frac{ m^2 }{n} polylog (n))$ in\nour case, due to application of Moser Tardos.\\\\ \nHere the error bound may fail for the additional reason that\nthe dependence may exceed $\\frac{m k polylog (n)}{n}$ after the Brownian \nmotion phase. This can happen with probability $\\frac{1}{m^{\\Omega (1) }}$. \nThis part is related the input distribution of ${\\cal A}^{m\\times n}_k$ and\nrepeating the algorithm may not work.\n\\end{proof}\n\n\n{\\it Remark} A direct application of LLL (without running the Brownian \nmotion) would increase the dependence to $O(\\frac{m k^2 \\log n}{n} +\\log m \n\\log n )$. In order to maintain the same asymptotic error bound, the number\nof rows in the matrix, $m$, could be significantly less. For example, if\n$m,k = n^{1\/2}$, then the difference would be $O(\\log n\/\\log\\log n)$\nversus $O(\\log\\log n\/\\log\\log\\log n )$.  \n\nThe proof of Theorem \\ref{mainthm3} is along similar lines after appropriate \nscaling and is given in the appendix.\n\n\\ignore{\nTo obtain the approximation factor of \nTheorem \\ref{mainthm2} we use the previous scaling technique with\n$d = \\alpha \\max\\{ \\frac{e mk u}{n} , u\\log m\\}$, where $u$ is the\nnumber of unabsorbed variables in any constraint and $\\alpha \\geq 1$ is an\nappropriate scaling factor. Let $B' = B+1$, then \n\\[ \\beta = e \\alpha^{1\/B'} \\cdot \\max\\{ {(\\frac{e mk u}{n})}^{1\/B'} ,\n {(u\\log m)}^{1\/B'} \\} = e \\cdot {(\\alpha u)}^{1\/B'} \n\\cdot \\max\\{ {(\\frac{e mk }{n})}^{1\/B'} ,\n {(\\log m)}^{1\/B'} \\} . \\]\n\n\nFor weighted objective function with $c_i \\geq 1\/p$, we can choose\nthe value of the scaling factor $\\alpha$ more carefully so that $p = \n\\alpha^{1 - 1\/B'} \\cdot {(\\log n)}^{1 -2\/B'} $, and $mk\/n \\leq \\log m$,\nusing $S = {(\\alpha \\log^2 n )}^{1\/B'}$ gives an objective value\nof at least $ nB'\/(p Sk)  \\geq n\/(k \\alpha \\log n) $ \nwhich satisfies condition \\ref{eqnobj}. Note that $S = {( p \\log n)}^{1\/(B'-1)} \n$ from the above parameter settings.\\\\  \nBy subsituting back $B+1$ for $B'$ in the previous calculations,\n$S = {( p \\log n)}^{1\/B}$. So for $B = 1$, the approximation is\n$O(p\\log n)$.\nThis completes the proof of Theorem \\ref{mainthm2}. \\\\[0.1in]\n}\n\\color{black}\n\n\\section{Brownian walk analysis}\n\nWe will divide the analysis into two components - \nFirst we will compute the rate at which the variables are absorbed at 0. \nSecond, we compute the increments of all variables during each iteration that \ncauses an inequality to be violated. \nThis causes the right hand side of any inequality to \nbehave as a martingale and we will refer to it as the {\\it error}. \n Note that the increase in {\\it error} is related to the number of \nunabsorbed variables as they execute random walk. Once all variables are\nabsorbed, then the error doesn't change. For subsequent applications, we\nwill derive the bounds for starting position scaled by $S \\geq 1$, i.e.,\nfrom $\\frac{x'}{S}$. Readers who are familiar with \n\\cite{LM:12} may note that our analysis focuses on variables associated with\neach constraint as opposed to the global number of variables.\n\\begin{lemma}\n\nLet $x' \\in \\mathbb{R}^n$ be a feasible solution to $A\\cdot x \\leq B\\cdot \\vec{1}^m$, \n$A \\in {\\{0,1\\}}^{m\\times n}$, and \n$\\overset{\\sim}{x}=\\frac{x'}{S}$, $S \\geq 1$ be chosen as the starting point \nfor brownian walk. Then,\n after $T_p=\\frac{2^p. 2^p}{n^2 \\gamma^2}$ steps, with probability \n$\\geq 1-\\frac{1}{m^{\\Omega(1)}}$, the number of unfixed variables in $i^{th}$ \nconstraint is $O(\\frac{n B}{S \\cdot 2^p}$) \nfor $2^p \\leq \\frac{n B}{S  (\\log (m)))}$\\footnote{For $m$ polynomial in $n$\nwe need not distinguish between $\\log m$ and $\\log n$}.\n\\footnote{Applying a union bound over all constraints, we satisfy the conditions of the lemma for each of the phases and constraints with high probability}\n\\label{absorp_rate}\n\\end{lemma}\n First, we can\nuse Lemma \\ref{boost-prob-absorb} to get a bound on the probability of \nabsorption.\n\\begin{claim}\nFor $b \\geq r a$,  after $O( \\frac{ r^2 a^2}{\\gamma^2} )$ Brownian motion \nsteps, the probability of non-absorption at 0 is $\\leq O(\\frac{1}{r})$. \n\\label{eq_var_bnd}\n\\end{claim}\nThe above claim is trivially true\nfor $r \\leq 1$.\nIn Lemma \\ref{boost-prob-absorb} , use $k = r $, $b = ra\/\\gamma$ \nthat gives absorption probability $\\leq O(1\/r)$ after $r\\cdot a\/\\gamma \\cdot\na r\/\\gamma = \\frac{ r^2 a^2 }{\\gamma^2 }$ steps.   \n\\begin{proof}(of Lemma \\ref{absorp_rate})\nChoosing $T_p=\\frac{2^p 2^p}{n^2 \\gamma^2}=\\frac{(\\overset{\\sim}{x_i})^2}{\\gamma^2}(\\frac{2^p}{n \\overset{\\sim}{x_i}})^2$,\\\\\nwe can apply Claim \\ref{eq_var_bnd} with $r=\\frac{2^p}{n \\overset{\\sim}{x_i}}$ to obtain that\nthe probability that $x_i$ is not absorbed at 0 is $\\leq \\frac{n \\overset{\\sim}{x_i}}{2^p}$.\nLet $V_i^p\\:x\\leq B$ be the $i^{th}$ constraint after $p$ phases.\n\nIf $u^p_i$\n(= $\\twonormsq{ V_i^p }$) denotes the number of unabsorbed variables in \nconstraint $i$ after $p$ phases then\n$\\mathbb{E} ( u^p_i ) \n\\leq \\sum_{j=1}^{n} \\frac{n \\overset{\\sim}{x_{j}}}{2^p} A_{i,j} \\leq \n\\frac{n B}{S \\cdot 2^p}$ since \n $\\sum_{j=1}^{n} A_{i,j}\\overset{\\sim }{x_{j}} \\leq \\frac{B}{S}$.\n \n Note that the variables are independently executing Brownian walks.\nFor $\\frac{n B}{S \\cdot 2^p} \\geq  \\log{(m)}$ or equivalently, \n$p \\leq \\log ({n})- \\log (\\log (m) )+\\log (B)-\\log (S)$,\n we can apply Chernoff bound to claim that $u_i^p$\nis $O(\\frac{n B}{S \\cdot 2^p})$ with probability $\\geq 1-\\frac{1}{m }$ \n\\end{proof}\n\\begin{cor}\nSince $\\mathbb{E} ( u^p_i ) =  \\mathbb{E} ( \\twonormsq{V_i^p} )$ for $V_i^p = {\\{ 0, 1 \\}}^n$,\nwe can bound $ \\twonormsq{V_i^p} $ by $\\frac{n B}{S \\cdot 2^p}$ with high\nprobability for $p \\leq \\log ({n})- \\log (\\log (m \\log(n)) )+\\log (B)-\\log (S)$.\n\\label{normbnd}\n\\end{cor}\n\\textbf{Remark}\n(i) For $A_{i,j} \\in [0,1]$, the above proof on $\\mathbb{E} ( u_i^p )$ \ndoesn't hold directly but the bound on $ \\twonormsq{V_i^p} $ is still valid.\n\\\\\n(ii) If we proceed upto $p* = \\log ({n})-\\log (\\log (m  ))+\n\\log (B)-\\log (S)-\\log (c)$, all constraints have\nat most $O(\\log m )$ \nunfixed coordinates.\n \n\\subsection{Error Bound}\n\nFrom the previous result, \nfor all $j , \\twonormsq{V_j^p} \\leq  O(\\frac{nB}{S2^p}$) \nwith high probability.\nTo bound the error consider a fixed constraint $C_j:V_j.x \\leq B$,\nwe denote the error accumulated in the $p^{th}$ phase by $\\delta_p$, so\nthat $\\sum_{q=1}^p \\delta_q = \\beta_p$.\n\n\\begin{lemma}\nFor all constraints $C_j$, the error \n $\\delta_p \\leq\n\\sqrt{\\frac{2 B \\log m } {S\\cdot n }} 2^{\\frac{p}{2}} .$\\\\ \nThus the total error $\\beta_p$ upto $\\log n - \\log (\\log m )+ \n\\log (B)- \\log (S)$ is bounded by $c' \\frac{B}{S}$ for some constant $c'$. \n\\label{err_bnd} \n\\end{lemma}\n\\begin{proof}\nConsider $\\Pr(< \\gamma (\\sum_{i=T_{p-1}+1}^{T_{p}}U_i ),V_j>\\vert \\geq \\delta_p )$\\\\ \n=$\\Pr(\\vert < \\sum_{i=T_{p-1}+1} ^{T_p} U_i, \\frac{V_j}{\\vert \\vert V_j \\vert \\vert}> \\vert \\geq \\frac{\\delta_p}{\\gamma \\vert \\vert V_j \\vert \\vert}).$\\\\\n\nNow $<U_i,\\frac{V_j}{\\vert \\vert V_j \\vert \\vert}> \n\\sim \\mathcal{N}(0, \\sigma^2)$ where $\\sigma^2 \\leq 1$. From Lemma \n\\ref{chernoff-mart}, it follows that\nthe above is bounded by $\\exp (-\\frac{(\\delta_p)^2}\n{\\gamma^2 \\vert \\vert V_j \\vert \\vert ^2 2 (T_p-T_{p-1})})$.\nIt will hold simultaneously for all the $m$ constraints in the $p^{th}$ \nphase, if $\\delta_p$ satisfies :-\n\n$\\frac{(\\delta_p)^2}{\\gamma^2 \\vert \\vert V_j \\vert \\vert ^2 \n2 (T_p-T_{p-1})} \\geq \\Omega( \\log m )$\\\\\ni.e. if $\\delta_p \\geq \\gamma \\vert \\vert V_j \\vert \\vert \\sqrt{2 \\cdot\n\\log  m .(T_p-T_{p-1})}$\\\\\nFrom Corollary \\ref{normbnd} $\\twonormsq{V^{p-1}_j } \\leq \\frac{nB}{S\\cdot 2^{p-1}}$ and $T_p=\\frac{2^p 2^p}{n^2 \\gamma^2}$,\\\\ \nit suffices to choose \n \\[ \\delta_p \\geq \\gamma \\sqrt{\\frac{nB}{S\\cdot 2^{p-1}}}\\cdot \n\\sqrt{2 \\log m }\\cdot \\sqrt{\\frac{2^p\\cdot 2^p}{n^2 \\gamma^2}}\n =2 \\sqrt{\\frac{ B \\log m  2^p}{S\\cdot n}}\n\\]\nThe above bound for $\\delta_p$ holds with high probability when $p \\leq \\log n -\n\\log (\\log m )+ \\log (B)- \\log (S)$ since only in this situation \nwe can bound effective value of $V_j$.\\\\\nThus the total error upto $\\log n - \\log (\\log m )+ \\log (B)- \\log (S)$ is \nbounded by \n\\begin{eqnarray}\n\\sum_{p=1}^{\\log n -\\log (\\log m )+\\log (B)-\\log (S)} 2\\sqrt{\\frac{ B\\log m }\n{S\\cdot n}} 2^{\\frac{p}{2}} &\n \\leq & 2\\sqrt{\\frac{ B\\log m }{S\\cdot n}} \\cdot O(2^{\\frac{\\log n - \n\\log (\\log m )+\\ log B - \\log (S)}{2}}) \\\\\n & = & 2\\sqrt{\\frac{ B \\log m }{S\\cdot n}} \\cdot O(\\sqrt{\\frac{nB}\n{S\\cdot \\log m }})=O(\\frac{B}{S})\n\\end{eqnarray}\nTherefore, the total error in every constraint is bounded by\n $<V_j,x^p>=<V_j,\\overset{\\sim}{x}+\\gamma\\sum_{i=1} ^{T_p} U_i>=\nO(\\frac{B}{S})$.\n\\end{proof}\nThus the solution obtained satisfies the constraints \n$A \\cdot x \\leq \\frac{B}{S}$.\n\n The above method can be extended to obtain an alternate proof the \n$O(\\log m\/\\log\\log m)$ error bound of Raghavan-Thompson that we\nhave ommitted from this version. \n\nBased on the above results, we summarize as follows.\n\\begin{cor}\nIn the first stage of Algorithm {\\bf Iterative Randomized Rounding}, we \nrun the algorithm for\n$T = \\frac{B^2}{S^2 \\log^2 m \\gamma^2}$ steps. Then with high \nprobability\\\\ \n(i) all constraints have $ \\leq \\log m$ unfixed variables and\n\\\\\n(ii) the total error in any constraint is bounded by $c'B\/S$ for some\nconstant $c'$. \n\\label{browniansteps}\n\\end{cor}\n\nUsing {\\it multiple copies} of a variable, results similar\nto this section have been used before that are {\\it ad hoc} to specific \napplications \\cite{CC:09,AS:96}.\nHowever, it requires two stages of rounding - once choosing exactly one\ncopy followed by a phase of independent rounding. In comparison, our\ntechnique and analysis are more general. \n \n\n\\section{Applications using LLL}\n\n\\label{short}\n\nAt the end of brownian walk(that is after $\\frac{B^2}{S^2\\log^2m \\gamma^2}$ steps), we have atmost $\\log(m)$ unconverged variables \noccurring(with non zero coefficients) in each equation.\nNow we need to bound the error in independantly rounding the unconverged \nNow we need to bound the error in independantly rounding the unconverged \nvariables.\nAs per the notations setup in section \\ref{sec:algo}, $V_i^{p*}$ represents the \ncoefficient vector of $i^{th}$ constraint after p* phases.\n\\\\\n\nDefine $\\overset{\\wedge}{A}$ as a matrix having rows as $V_i^{p*}$.\n\nNow if the unconverged variables are rounded from $\\overset{\\sim}{x}$ to $\\overset{\\wedge}{x}$,\nthe change in the value of RHS is \n$A\\cdot(\\overset{\\wedge}{x}-\\overset{\\sim}{x})=\\overset{\\wedge}{A} \\cdot (\\overset{\\wedge}{x}-\\overset{\\sim}{x})$.\n(Since only unconverged variables change).\n\nHence to calculate additional error due to independant rounding we only need to consider the \"`sparsified\"' matrix $\\overset{\\wedge}{A}$.\n\\\\\nBefore we prove the main results, we consider the simpler \ncase of columns with bounded number of ones -\nSuppose the matrix $A^{m\\times n}$ has no more than\n$\\rho$ 1's in any column. Let $OPT$ be the optimal fractional \nobjective value for the weighted objective function $\\sum_i c_i \\cdot x_i$. \nConsider a fixed constraint $C_r$, that contains $\\log m$ 1's after the Brownian\nmotion and let $j_1 (r) , j_2 (r) \\ldots $ denote the (at most) $\\log n$\ncolumns that contain 1.\nWe say that a constraint $C_y$ is {\\it dependent } on $C_r$ if they\nshare at least one column where the value is 1.\nSo, the dependency of any single constraint can be bound by $\\rho\\log m$\n(Since there are $\\log m$ 1's per row in the sparsified matrix $\\overset{\\wedge}{A}$).\nIf we use the independent rounding to round the \nfractional solution $x'$, the probability that the value of a constraint\nexceeds $t > 1$ is bounded $\\frac{1}{2^t}$ from Chernoff\nbounds \\footnote{This\nis a slightly weaker version to keep the expression simple}. Let\n$E_i$ denote the event that $C_i$ exceeds $t$ when we use randomized\nrounding.  We are interested to know the\nprobability of the event $\\bigcap_{1 \\leq i \\leq m} \\bar{E_i}$ \nsince this implies the event that all the inequalities are less than $t$.\nThis is tailor-made for Lovasz Local Lemma (LLL).\nWe also want to guarantee a large\nvalue of the objective function. For example, setting all variables equal to\nzero would guarantee feasibility but also return an objective function value 0.\nThus we define an additional event $A_{m+1}$ corresponding to the objective\nfunction value less than $(1- \\epsilon ) \\cdot OPT$ for some suitable $1 >\n\\epsilon > 0$.\nSince $A_{m+1}$ is a function of all the variables, it has dependencies with\nall other $A_i \\ \\ i = 1 \\ldots m$, therefore we have to use the generalized\nversion of LLL in this case. \n\\begin{theorem}[Lovasz Local Lemma \\cite{EL:75}]\nLet $A_i , 1 \\leq i \\leq N$ be events such that $\\Pr [ A_i ] = p$ and each\nevent is dependent on at most $d$ other events. Then if $ep(d+1) < 1$, then\\\\\n$ \\Pr ( \\bar{A_1} \\cap \\bar{ A_2 } \\ldots \\bar{ A_N } ) > 0 $.\n\\\\\nAlternately, in a more general (asymmetric) case, where the dependencies are\ndescribed by a graph $(\\{ 1, 2 \\ldots N \\}, E)$\nwhere an edge between $i, j$ denotes dependency between\n$A_i , A_j$ and $y_i$ are real numbers such that $\\Pr ( A_i ) \\leq\ny_i \\cdot \\prod_{(i,j) \\in E} ( 1 - y_j )$ then\n$ \\Pr \\left( \\bigcap_{i=1}^{N} \\bar{ A_i } \\right) \\geq \\prod_{i=1}^{N}\n( 1 - y_j )$.\\\\\nMoreover, such an event can be computed in randomized polynomial time using\nan algorithm of Moser and Tardos \\cite{MT:10}.\n\\label{lll}\n\\end{theorem}\n\nIf we choose $t$ such that $e \\cdot 2^{-t} \\cdot (d+1) \\leq 1$ or\nequivalently $t = \\log d$, then we can apply\nthe previous theorem to obtain a rounding that satisfies an error bound of\n$O( \\log \\rho + \\log\\log m )$. \nThe error $t$ can be improved to $O(\\frac{\\log d}{\\log\\log d})$\nby using a tighter version of the Chernoff bound (Equation \\ref{chernoff}).\n\n\nWe define $A_{m+1}$\nas the event where the objective value is less than $(1- \\epsilon ) OPT$. From\nChernoff-Hoeffding bounds, we know that $\\Pr ( A_{m+1} ) \\leq \n\\exp ( - \\epsilon^2  OPT\/2 )$.\nWe define $y_i$ for $i = 1, 2 \\ldots m$ as before, corresponding\nto the probability of exceeding $t = \\log d\/\\log\\log d$.\n\\\\ By choosing\n$y_i = 1\/(\\alpha d) \\ i \\leq m$ and $y_{m+1} = \\frac{1}{2}$, \nfor some suitable scaling factor $\\alpha \\geq e$, we\nmust satisfy the following inequalities\n\\begin{eqnarray}\n\\Pr (A_i ) & \\leq & 1\/(\\alpha d) {( 1 - 1\/(\\alpha d) )}^{d} \\cdot \\frac{1}{2} \\ \\  i = 1, 2 \\ldots m \\\\\n\\Pr ( A_{m+1} ) & \\leq & \\frac{1}{2} \n{( 1 - 1\/(\\alpha d) )}^{m} \\leq\n\\frac{1}{2}\\cdot \\exp (- m\/(\\alpha d))\n\\label{eqnobj}\n\\end{eqnarray}\nThe first condition is easily satisfied when $\\Pr ( A_i ) = \\frac{1}{\\Omega \n(\\alpha d)}$.\\\\\nTo satisfy the second inequality, we can choose $\\alpha$\nso that  $OPT \\geq \\Omega ( \\frac{m}{\\alpha d})$. \nThis implies that $\\exp (- m\/(d\\alpha ) \\geq\n\\exp (- \\epsilon^2 OPT\/2))$,  so condition (\\ref{eqnobj}) is\nsatisfied for LLL to be applicable. \\\\\n\n\n\\begin{figure}\n\\fbox{\\parbox{6.0in}{\n\n{\\footnotesize \n\\begin{center}\nAlgorithm {\\bf LLL based Iterative Randomized Rounding }\n\\end{center}\nInput : $x'_i \\ \\ 1 \\leq i \\leq n, t$ (error parameter)\n\\\\\nOutput : $\\hat{x_i} \\in \\{0, 1\\}$ \\\\\n\nDo independent rounding on all the variables having values $x'_i$ to\n$\\hat{x_i}$.\n\\\\\nCompute the value of each constraint $C_i$ as $< V_i , \\hat{x} >$\n\n{\\bf While} any inequality exceeds $t$ {\\bf or} objective value is $< OPT\/2$\n\\begin{quote}\n\\begin{enumerate}\n\\item Pick an arbitrary constraint $C_j$ that exceeds $t$\nand perform independent rounding on all the variables in $V_j$.\n\\item Update the value of the constraints whose variables have changed\n\\end{enumerate}\n\\end{quote}\nReturn the rounded vector $\\hat{x}$.\n}}\n}\n\\caption{An iterative randomized rounding algorithm based on Moser-Tardos}\n\\label{algo2}\n\\end{figure}\n\n\n\nWe summarize our discussion as follows\n\\begin{lemma}\n\nFor $A \\in {\\{0,1 \\}}^{m \\times n}$ with a maximum of \n$\\rho$ 1's in each column, we can round \nthe optimum solution $x^{*}$ of the linear program\n$\\max_{x} \\sum_i c_i x_i \\ \\ s.t. A x \\leq 1 \\ \\ , 0 \\leq \nx_i \\leq 1 $ \nto $\\hat{x} \\in {\\{ 0, 1 \\}}^n$ such that \n\\[\n{|| A \\hat{x}||}_\\infty \\leq max(\\left( \\frac{\\log \\rho +  \\log\\log m}{\\log\\log \n(\\rho \\log m)}\\right),\\frac{\\log(\\frac{m}{OPT})}{\\log(\\log(\\frac{m}{OPT}))}) \\ \\ \\text{ and } \\sum_i c_i \\hat{x}_i \\geq \n(1 - \\epsilon ) OPT   \n\\label{bnd_col}\n\\]\n\\end{lemma}\n\\begin{proof} \nThe error bound follows from the previous discussion by setting $y_i = \\Omega (\n\\frac{1}{\\alpha d})$ and using the stronger form of Chernoff bound in\nEquation \\ref{chernoff}. \nNote that Equation \\ref{eqnobj} can be satisfied by ensuring\n$OPT \\geq \\frac{m}{\\alpha d}$ by choosing an appropriately large $\\alpha$.\\\\\nThe above is satisfied for $\\alpha \\geq \\frac{m}{OPT.d}$ i.e. if  $\\alpha.d=\\frac{m}{OPT}$.\\\\\nIn this case the discrepancy is $\\frac{\\log (\\frac{m}{OPT})}{\\log(\\log(\\frac{m}{OPT}))}$\\\\\nHence the result follows.\n\\end{proof}\n\nThe objective function still follows the martingale property since the\nvariables starting the random walk at $x'_i $ have probability\n$ x'_i $ of being absorbed at 1 which is identical to  the independent\nrounding that we use in the second phase\nfor Moser-Tardos algorithm.\nAlthough, the random walk is short-cut by a single step independent\nrounding, the distribution for absorption at 0\/1 remains unchanged. To\nsee this, consider the {\\it last} time a variable $x_i$ is rounded by the\nMoser-Tardos algorithm - the probability is $x'_i$ since it is done\nindependently every time.\n\n\\ignore{\nThe rest follow from LLL and the algorithm of Moser-Tardos. The expected\nrunning time for the second phase is $O( \\frac{ m^2 }{n} polylog (n))$ in\nour case.\\\\\nThat the error bound may fail if either the\nBrownian walk and the Moser-Tardos algorithm fail\nwith inverse polynomial probability because of the inherent randomization.\nThis probability can be boosted by repetition.\n\\end{proof}\nAs an example, consider $\\rho = \\log m$, $OPT \\geq \\frac{m}{\\rho \n\\log^{c+1} m }$ then for $\\alpha = \\log^{c} m$,\nthe error is $O(\\log\\log m\/\\log\\log\\log m )$.\n\\\\\n{\\it Remark} A direct application of LLL (without running the Brownian\nwalk) would increase the dependence to $O(\\rho \\cdot n)$ resulting in an\nerror bound of $O( \\frac{\\log \\rho + \\log m}{\\log\\log m })$ that is not \nbetter than the RT bound. \n}\n\\input{randmat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nA viable Machine Translation (MT) system must have a solution to anaphoric and elliptical ambiguities. It is not viable until it provides mechanism to handle the problem(s) of unidentified words such as names and abbreviations. Earlier, a discourse based approach is used to resolve anaphoric and elliptical ambiguities in Text-based Machine Translation (TBMT) \\cite{1}. To produce high quality translation, the source text is dissected into mono-sentential discourses where complex discourses require further dissection either directly into primitive discourses or first into compound and later into primitive discourses \\cite{2}. The resolution of ambiguities is performed during discourse processing stage \\cite{3}. However, less attention has been given to resolve unknown words in TBMT. This paper improves dissection and discourse processing procedure by providing an algorithm for the resolution of unidentified lexical units. Firstly, we discuss discourse analysis and the dissection of complex and compound discourses into primitive discourses. Secondly, we present our proposed algorithm and analyze its validity by applying it to real world text, i.e., newspaper fragments. Finally, we present conclusion to our work.    \n\n\\begin{table*}[!t]\n\\renewcommand{\\arraystretch}{1.3}\n\\caption{Primitive discourses and Generalized patterns (First complex discourse)}\n\\label{tab:1}\n\\centering\n\\begin{tabular}{|c|c|}\n\\hline\nPrimitive discourses &\tGeneralized patterns\\\\\n\\hline\nIPEC of ILO &\tA of B\\\\\n\\hline\nIPEC has initiated &\tA has B\\\\\n\\hline\nProject is new &\tA is B \\\\\n\\hline\n(Time\\textunderscore Bound\\textunderscore Program) of (Government\\textunderscore of\\textunderscore Pakistan) is supported &\tA of B is C\\\\\n\\hline\nElimination of WFCL\t& A of B\\\\\n\\hline\nElimination from Pakistan &\tA from C \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\begin{table*}[!t]\n\\renewcommand{\\arraystretch}{1.3}\n\\caption{Primitive discourses and Generalized patterns (Second complex discourse)}\n\\label{tab:2}\n\\centering\n\\begin{tabular}{|c|c|}\n\\hline\nPrimitive discourses &\tGeneralized patterns\\\\\n\\hline\nProject of IPEC &\tA of B\\\\\n\\hline\nIPEC will provide &\tA will B\\\\\n\\hline\nAssistance is technical &\tA is B\\\\\n\\hline\nProvided to the (Government\\textunderscore of\\textunderscore Pakistan) &\tA to the B\\\\\n\\hline\n(Convention\\textunderscore 182) of ILO &\tA of B\\\\\n\\hline\nImplement on children &\tA on B\\\\\n\\hline\nChildren are working &\tA are B\\\\\n\\hline\nWorking conditions are hazardous &\tA B are C\\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\section{Discourse analysis and discourse unit}\nThe term discourse is introduced by Zellig Harris in 1952. A discourse is a connected piece of text of more than one sentence spoken by one or more speakers \\cite{4}. Bellert defines discourse as a sequence of utterances $S_{1}, S_{2}, S_{3}....S_{n}$ such that the semantic interpretation of each utterance $S_{i}(2\\leq i \\leq n)$ is dependent on the interpretation of the utterances $S_{1}, S_{2}, S_{3}....S_{i-1}$ \\cite{5}. However, interpretation of $S_{i}$ may depend on any subset of the set (say $U$) of previous utterances where $U=S_{1}, S_{2}, S_{3}....S_{n}$ \\cite{2}. We think this definition needs further improvement in the context of cataphora resolution where interpretation of $S_{i}$ may sometimes depend on any subset of the set (say $\\overline{U}$) of the subsequent utterances. A discourse unit is an atomic utterance that has no reference beyond its limitations or boundaries and can be mono-sentential or poly-sentential.     \n\n\\section{Dissection and Discourse processing}  \nThe entire dissection procedure is divided into two phases, i.e., dissection phase and discourse processing phase. In dissection phase, the source text is converted into primitive discourses. These primitive discourses are used to get generalized predicates. In discourse processing phase, various ambiguities are resolved including anaphoric and elliptical ambiguities. Mathematically, the dissection procedure can be represented as\n\n\\begin{equation}\nT = (D_{1}, D_{2}, D_{3}......D_{i}...D_{l})l\\geq1 \n\\end{equation}\n\\begin{equation}\nT=\\sum_{i=1}^l D_{i}\n\\end{equation}\n\nwhere $T$= Source text and $D_{i}$= Poly or Mono sentential Discourse.\n\nThe resolution of the unidentified words is assumed to be processed during the discourse processing stage. The problem of unknown words resulted when we continued to apply the dissection concept to the real world text. For instance, consider the following newspaper fragment\n\n\\emph{[The International Labour Organization ILO's International Programme on the Elimination of Child Labour (IPEC) has initiated a new project to support the Government of Pakistan's Time Bound Programme on the elimination of the Worst Forms of Child Labour (WFCL) from Pakistan. The project will provide technical assistance to the Government of Pakistan to implement ILO Convention 182 on children working in hazardous working conditions.]}.\n\nThe two complex discourses are\n\n1- [The International Labour Organization ILO's International Programme on the Elimination of Child Labour (IPEC) has initiated a new project to support the Government of Pakistan's Time Bound Programme on the elimination of the Worst Forms of Child Labour (WFCL) from Pakistan].\n\n2- [The project will provide technical assistance to the Government of Pakistan to implement ILO Convention 182 on children working in hazardous working conditions].\n\nThe above discourses are dissected into the primitive discourses as given in Table \\ref{tab:1} and Table \\ref{tab:2}.\n\nIt is worth noting that one variable is used for the abbreviations as well as for the compound nouns. For example, Government of Pakistan is treated as a noun. Computer considers it noun by concatenating the text including some special characters (such as \\textunderscore) defined by the programmer. Government of Pakistan is therefore considered as Government\\textunderscore of\\textunderscore Pakistan. Recognizing the correct noun is a challenging task for the computer\/MT scientists. For instance, the Time Bound Program is considered noun from the previous discussion, but it might be treated otherwise. The decision must be based on the context. The concept of dissection is further explained in \\cite{1}.   \n\n\\section{Unidentified lexical units} \n\nCreation of a list or glossary that contains well-known nouns, i.e., abbreviation and names, could be the best solution. The computer should add the new abbreviations or names (as they come along) to the list but sometimes the repetitive uses of nouns create problems for MT system. A pseudo code is written to resolve this problem. The pseudo code could be useful in Question Answering (QA) system, Information Retrieval (IR) system and in MT system, respectively.\n\nThe proposed algorithm updates new names and abbreviations not present in the lexicon. Firstly, the existence of the noun is checked, i.e., whether the noun is correct or not. For instance, if a user enters Paksin Skidn Odind instead of Pakistan State Oil then the computer will subsequently reject it since these words not present in the lexicon. A problem could result if a user enters a word already in the lexicon, but not appropriate for the given abbreviation. For instance, the computer updates wrong abbreviation, if in case Pakistan Supreme Oil is entered instead of Pakistan State Oil. The best solution is that user is allowed to update inappropriate words for the given abbreviation, but these inappropriate words are less likely to be used by another user. Hence, the abbreviations that are less likely to be used are deleted automatically after a month. This could decrease the chances of errors. If these are not automatically deleted, then MT system could give invalid information, i.e., MT system may display Pakistan Supreme Oil instead of Pakistan State Oil.  The flow chart of the pseudo code is given in Fig \\ref{fig:1}. It identifies the existence of noun in a lexicon. It takes primitive discourse as a unit of analysis. If noun doesn't exist, the nature of the noun either abbreviation or name is initially identified. If the noun is abbreviation\/name, it is updated to the lexicon. Otherwise the user is requested to enter the required abbreviation\/name for the updating purposes.\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=5in]{fig}\n\\caption{Flow chart of the proposed algorithm}\n\\label{fig:1}\n\\end{figure*\n\n\\section{Evaluation} \n\nThe algorithm is applied to newspaper fragments and substantial numbers of unidentified lexical units are manually resolved. The evaluation of unknown words took place when we applied the dissection procedure to newspaper fragments . During our experiments, 124 names have been updated to the lexicon as compared to 57 abbreviations. Moreover, the number of unknown names appeared more than unknown abbreviations. However, updating new nouns depends on the lexicon. A poor lexicon results in considerable number of unidentified lexical units while a rich lexicon results in fewer unknown words.\n\n\\section{Conclusions} \n\nThis paper explained the resolution of unidentified lexical units in TBMT. The resolution was considered as a part of discourse processing stage where apart from resolving other ambiguities, the resolution of unknown words was also considered. We presented an algorithm, which updates unknown nouns to the lexicon. The presented algorithm takes mono-sentential discourse as an input. The algorithm was manually applied to newspaper fragments and unidentified words were updated to the lexicon. In future this algorithm will be implemented as a part of our dissection model. Additionally, this could also be useful in QA and IR system.\n\\section*{Acknowledgement}\n\nThis research is supported by the MIC (Ministry of Information and Communication) South Korea, under ITRC (Information Technology Research Center) support program supervised by II TA  (Institute of Information Technology Advancement).\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nWhile wave functions of quantum systems may be complex, spectra of their\nenergy eigenvalues must be real, which is usually secured by restricting the\nunderlying Hamiltonian to be Hermitian \\cite{qm}. However, the condition of\nthe reality of the energy spectrum does not necessarily imply that it is\ngenerated by an Hermitian Hamiltonian. Indeed, it is well known that\nnon-Hermitian Hamiltonians obeying the parity-time ($\\mathcal{PT}$) symmetry\nmay also produce entirely real spectra~\\cite%\n{bender1,dorey,bender2,bender3,review,ptqm}. In terms of the single-particle\ncomplex potential,\n\\begin{equation}\nP(\\mathbf{r})\\equiv V(\\mathbf{r})+iW(\\mathbf{r}),  \\label{U}\n\\end{equation}%\nthe $\\mathcal{PT}$ symmetry requires its real and imaginary parts to be even\nand odd functions of coordinates \\cite{bender1}: $V(\\mathbf{r})=V(-\\mathbf{r}%\n),W(-\\mathbf{r})=-W(\\mathbf{r})$,~i.e.,~\n\\begin{equation}\nP(-\\mathbf{r})=P^{\\ast }(\\mathbf{r}),  \\label{minus}\n\\end{equation}%\nwhere the asterisk stands for the complex conjugate. Actually, Hamiltonians\nwhich keep $\\mathcal{PT}$ symmetry may be transformed into Hermitian ones\n\\cite{Mostafazadeh,Barash2,Barash}.\n\nIn the general case, the energy spectrum generated by the $\\mathcal{PT}$%\n-symmetric potential remains real (physically relevant) below a certain\ncritical value of the strength of the imaginary part of the underlying\npotential, $W(\\mathbf{r})$ in Eq. (\\ref{U}), which is a threshold of the $%\n\\mathcal{PT}$ symmetry breaking. Above the critical value, the system is\nmade unstable by emerging imaginary parts of energy eigenvalues. In some\nmodels, the breakup of the $\\mathcal{PT}$ symmetry may follow the onset of\nthe jamming anomaly, which means a transition from increase to decrease of\nthe power flux between the spatially separated gain and loss spots with the\ngrowth of the gain-loss coefficient \\cite{jamming1}. The fragility of the $%\n\\mathcal{PT}$ symmetry essentially limits the use of this property in\napplications, where new effects, such as unidirectional transmissivity \\cite%\n{uni}, enhanced absorption of light \\cite{Longhi}, lasing in microrings \\cite%\n{exp5}, acoustic sensors \\cite{sensors}, as well as the operation of $%\n\\mathcal{PT}$-symmetric metamaterials \\cite{exp4} and microcavities \\cite%\n{exp6} strengthen with the increase of the gain-loss coefficient.\n\nThus far, the $\\mathcal{PT}$ symmetry was not experimentally realized in\nquantum systems, and, moreover, it was argued that, strictly speaking, $%\n\\mathcal{PT}$-symmetric systems do not exist in the framework of the quantum\nfield theory \\cite{Szameit}. On the other hand, a possibility to implement\nthe concept of the $\\mathcal{PT}$ symmetry in terms of classical physics was\npredicted for optical media with symmetrically placed gain and loss elements\n\\cite{theo1}-\\cite{Kominis}, which is based on the similarity between the\nSchr\\\"{o}dinger equation in quantum mechanics and the paraxial-propagation\nequation for optical waveguides. Experimentally, this possibility was\nimplemented in several waveguiding settings \\cite{exp1}-\\cite{exp7}, as well\nas in other photonic media, including exciton-polariton condensates \\cite%\n{exci2,exci3}, and in optomechanical systems \\cite{om}. In these contexts,\nbreaking of the $\\mathcal{PT}$ symmetry was observed. Emulation of the $%\n\\mathcal{PT}$ symmetry was also demonstrated in acoustics \\cite{acoustics}\nand electronic circuits \\cite{electronics}, and predicted in atomic\nBose-Einstein condensates \\cite{Cartarius}, magnetism \\cite{magnetism}, and\nchains of coupled pendula \\cite{Peli}.\n\nThe $\\mathcal{PT}$ symmetry, being a linear feature, is often combined with\nintrinsic nonlinearity of settings in which it is realized. Most typically,\nit is the Kerr nonlinearity of underlying optical media, which gives rise to\nnonlinear Schr\\\"{o}dinger equations (NLSEs) with the cubic term and complex\npotentials, subject to constraint Eq. (\\ref{minus}). Such equations may\ngenerate $\\mathcal{PT}$-symmetric solitons, which were considered in many\ntheoretical works \\cite{soliton}, \\cite{Konotop}-\\cite{Alexeeva} (see also\nreviews \\cite{review1,review2}), and experimentally demonstrated too \\cite%\n{exp7}. Although these works were chiefly dealing with one-dimensional (1D)\nmodels, stable $\\mathcal{PT}$-symmetric solitons were also predicted in some\n2D settings \\cite{Yang}, \\cite{2D-1}-\\cite{2D-3}. A characteristic feature\nof $\\mathcal{PT}$-symmetric solitons is that, although existing in\ndissipative systems, they appear in continuous families, similar to their\ncounterparts in conservative models \\cite{families}, while usual dissipative\nsolitons exist as isolated solutions (\\textit{attractors}, if they are\nstable) \\cite{diss2,diss3}. The realization of the $\\mathcal{PT}$ symmetry\nin 2D geometry may provide essential extension of the above-mentioned\napplications, such as the unidirectional transmission, enhanced absorption,\nand lasing for broad optical beams.\n\nSolitons are also vulnerable to destabilization via the $\\mathcal{PT}$%\n-symmetry breaking at the critical value of the gain-loss coefficient \\cite%\n{breaking}. Nevertheless, it was found that, in some settings, the solitons'\n$\\mathcal{PT}$ symmetry can be made \\emph{unbreakable}, extending to\narbitrarily large values of the strength of the model's imaginary potential\n\\cite{unbreakable}-\\cite{China}, see also a brief review of the\nunbreakability concept in \\cite{book}. The particular property of these\nmodels is that self-trapping of solitons is provided not by the\nself-focusing sign of the nonlinearity, but by the defocusing sign, with the\ncoefficient in front of the cubic term growing fast enough from the center\nto periphery. In the absence of gain and loss, this scheme of stable\nself-trapping was elaborated for 1D, 2D, and 3D bright solitons \\cite%\n{Barcelona1}-\\cite{Barcelona5}. It is essential to stress that such models\nare \\textit{nonlinearizable}, which means that decaying tails of solitons\nare determined by the full nonlinear equation. In other words, the models\nhave no linear spectrum, the spectrum of eigenstates being represented by\nnonlinear self-trapped modes (solitons). Accordingly, the models elaborated\nin Refs. \\cite{unbreakable}-\\cite{China} realize the $\\mathcal{PT}$ symmetry\nin a sense different from that defined in the usual systems---not in terms\nof the linear spectrum, which does not exist in this case, but in the form\nof stable families of complex-valued solitons with real propagation\nconstants (eigenvalues), which exist in the presence of spatially odd\nimaginary potentials.\n\nThe present work introduces 2D models which maintain stable solitons,\nincluding (nearly) unbreakable ones, in the presence of the spatially\ngrowing self-defocusing nonlinearity and antisymmetric imaginary potentials,\n$iW\\left( x,y\\right) $ in Eq. (\\ref{U}). One model, with\n\\begin{equation}\nW(x,y)=\\gamma _{0}x\\exp \\left( -\\beta r^{2}\\right) ,~r^{2}=x^{2}+y^{2},\n\\label{x}\n\\end{equation}%\nwhere $\\gamma _{0}>0$ and $\\beta \\geq 0$ are constants, features the\nunbreakable or nearly unbreakable 2D $\\mathcal{PT}$ symmetry, represented by\nseveral species of families of stable solitons: single- and double-peak\nones, as well as 2D solitons with embedded integer vorticity (topological\ncharge), $m=1,2,3$. The second model, with\n\\begin{equation}\nW\\left( x,y\\right) =\\gamma _{0}xy\\exp \\left( -\\beta r^{2}\\right) ,\n\\label{xy}\n\\end{equation}%\nis not, strictly speaking, a $\\mathcal{PT}$-symmetric one, but it is equally\nrelevant for the realization in optics, and it shares basic manifestations\nof the $\\mathcal{PT}$ symmetry, maintaining families of single- and\nmulti-peak solitons [featuring up to five peaks, in accordance with the\nstructure of $W(x,y)$] and solitary vortices, also with $m=1,2,3$. The\nlatter result is a contribution to the general topic of constructing models\nmore general than the $\\mathcal{PT}$-symmetric ones with similar\nproperties(including the case of the partial $\\mathcal{PT}$ symmetry \\cite%\n{partial}), which has been addressed in various settings \\cite%\n{2D-0,2D-02,2D-03,zezyu,families,Kominis1,Kominis2,Nixon,Kominis3}, see also\nreview \\cite{review1}.\n\nIn both models, universal analytical forms are obtained for tails of\nsolitons, and full exact solutions are produced for particular species of\nsingle-peak solitons, with $\\beta =0$ in Eqs. (\\ref{x}) and (\\ref{xy}). In\nthe former case, the existence of the exact solitons at arbitrarily large\nvalues of $\\gamma _{0}$ in Eq. (\\ref{x}) explicitly demonstrates the\nunbreakability of the $\\mathcal{PT}$ symmetry. In the latter case two\ndifferent families of exact solutions are found, which, however, exist only\nfor $\\gamma _{0}\\leq 2$ in Eq. (\\ref{xy}) with $\\beta =0$. In addition, an\nanisotropic version of the latter model gives rise to particular exact\nsolutions for vortex solitons with topological charge $m=1$. Generic soliton\nfamilies with $m=0,1,2,3$, which include the exact single-peak solutions as\nparticular ones, are constructed in a numerical form in both models, and\ntheir stability is investigated numerically---both through computation of\neigenvalues for small perturbations and by means of direct simulations.\n\n\n\\section*{Results}\n\n\\subsection*{\\textbf{The models and analytical solutions for solitons}}\n\n\\subsubsection*{\\textbf{The underlying equations}}\n\nThe 1D NLSE for the amplitude of the electromagnetic field, $u(x,z)$, with\nthe local strength of the self-defocusing nonlinearity, $\\Sigma (x)$,\ngrowing from $x=0$ towards $x=\\pm \\infty $ faster than $|x|$ (this condition\nis necessary for self-trapping imposed by the self-repulsion \\cite%\n{Barcelona1}), which is capable to maintain bright solitons with unbreakable$%\n\\mathcal{\\ PT}$ symmetry, is \\cite{unbreakable}%\n\\begin{equation}\ni\\frac{\\partial u}{\\partial z}+\\frac{1}{2}\\frac{\\partial ^{2}u}{\\partial\nx^{2}}-\\Sigma (x)|u|^{2}u=iW(x)u.  \\label{qq}\n\\end{equation}%\nHere $z$ and $x$ are scaled propagation coordinate and transverse\ncoordinate, in terms of the planar optical waveguide. In work \\cite%\n{unbreakable}, the analysis was presented for a steep 1D modulation profile,%\n\\begin{equation}\n\\Sigma (x)=\\left( 1+\\sigma x^{2}\\right) \\exp \\left( x^{2}\\right) ,\n\\label{Gauss}\n\\end{equation}%\nwith $\\sigma \\geq 0$, where coefficients equal to $1$ may be fixed to these\nvalues by means of rescaling. The choice of this profile allows one to\nobtain a particular exact solution for solitons \\cite{Barcelona1}. Of\ncourse, in a real physical medium the local strength of the nonlinearity,\ndefined as per Eq. (\\ref{Gauss}), cannot grow to infinitely large values at $%\n|x|\\rightarrow \\infty $. However, in reality it is sufficient that it grows\naccording to Eq. (\\ref{Gauss}) to finite values, that correspond to $|x|$\nwhich is essentially larger than the width of the soliton created by this\nprofile. The growth of $\\Sigma (x)$ may be safely aborted at still larger $%\n|x|$ \\cite{Barcelona1}.\n\nFurther, the spatially-odd imaginary potential, which accounts for the $%\n\\mathcal{PT}$-symmetric gain-loss profile (cf. Eq. (\\ref{U})), was\nintroduced in Ref. \\cite{unbreakable} as%\n\\begin{equation}\nW(x)=\\gamma _{0}x\\exp \\left( -\\beta x^{2}\\right) ,  \\label{gamma}\n\\end{equation}%\nwith $\\gamma _{0}>0$ and $\\beta \\geq 0$. In the case of the spatially\nuniform self-focusing cubic nonlinearity, the 1D imaginary potential in the\nform given by Eq. (\\ref{gamma}) was introduced in Ref. \\cite{extra-China}.\n\nHere, we aim to introduce a 2D extension of the model, as the NLSE for the\npropagation of the electromagnetic field with amplitude $u\\left(\nx,y,z\\right) $ in the bulk waveguide with transverse coordinates $\\left(\nx,y\\right) $:\n\n\\begin{equation}\ni\\frac{\\partial u}{\\partial z}+\\frac{1}{2}\\left( \\frac{\\partial ^{2}u}{%\n\\partial x^{2}}+\\frac{\\partial ^{2}u}{\\partial y^{2}}\\right) -\\Sigma\n(r)|u|^{2}u=iW\\left( x,y\\right) u,  \\label{NLS}\n\\end{equation}%\nwhere $r\\equiv \\sqrt{x^{2}+y^{2}}$ is the radial coordinate, and the\nnonlinearity-modulation profile is chosen similar to its 1D counterpart (\\ref%\n{Gauss}):\n\\begin{equation}\n\\Sigma (r)=\\left( 1+\\sigma r^{2}\\right) \\exp \\left( r^{2}\\right)\n\\label{sigma}\n\\end{equation}%\nwith $\\sigma \\geq 0$. Further, we consider two different versions of the 2D\nimaginary potential. First, it is a $\\mathcal{PT}$-symmetric one given by\nEq. (\\ref{x}). The other imaginary potential, defined as per Eq. (\\ref{xy}),\nis not $\\mathcal{PT}$-symmetric, because the $\\mathcal{P}$ transformation, $%\n\\left( x,y\\right) \\rightarrow \\left( -x,-y\\right) $, does not reverse the\nsign of $W\\left( x,y\\right) $, in this case. However, in terms of the\nimplementation in optics the gain-loss distribution corresponding to Eq. (%\n\\ref{xy}) is as relevant as that defined by Eq. (\\ref{gamma}), and, as\nmentioned above, properties of solitons in models which are akin to $%\n\\mathcal{PT}$-symmetric ones is a subject of considerable interest.\n\nStationary states with a real propagation constant, $k$, are looked for as\nsolutions to Eq. (\\ref{NLS}) in the form of\n\\begin{equation}\nu\\left( x,y\\right) =\\exp \\left( ikz\\right) U\\left( x,y\\right) ,  \\label{uU}\n\\end{equation}%\nwith complex function $U\\left( x,y\\right) $ satisfying the following\nequation:%\n\\begin{equation}\nkU=\\frac{1}{2}\\left( \\frac{\\partial ^{2}U}{\\partial x^{2}}+\\frac{\\partial\n^{2}U}{\\partial y^{2}}\\right) -\\Sigma (r)|U|^{2}U-iW\\left( x,y\\right) U.\n\\label{UU}\n\\end{equation}\n\n\\subsubsection*{\\textbf{Asymptotic solutions}}\n\nAs mentioned above, Eqs. (\\ref{NLS}) and (\\ref{UU}) are \\textit{%\nnonlinearizable}, i.e., they cannot be characterized by a linear spectrum.\nIndeed, straightforward analysis of Eq. (\\ref{UU}) demonstrates that it may\nproduce localized solutions (solitons), with tails decaying at $r\\rightarrow\n\\infty $ according to an asymptotic expression which is determined by the\nfull nonlinear equation, rather than by its linearization. For the $\\mathcal{%\nPT}$-symmetric imaginary potential (\\ref{x}) with $\\beta =0$, it is%\n\\begin{equation}\nU_{\\mathrm{asympt}}\\left( x,y\\right) =\\frac{1}{\\sqrt{2\\sigma }}\\exp \\left( -%\n\\frac{1}{2}r^{2}-i\\gamma _{0}x\\right) ,  \\label{asympt1}\n\\end{equation}%\nprovided that $\\sigma \\neq 0$. In the case case of $\\sigma =0$, this\nasymptotic solution is replaced by%\n\\begin{equation}\nU_{\\mathrm{asympt}}\\left( x,y\\right) =\\frac{r}{\\sqrt{2}}\\exp \\left( -\\frac{1%\n}{2}r^{2}-i\\gamma _{0}x\\right) .  \\label{asympt2}\n\\end{equation}\nNote that asymptotic solutions given by Eqs. (\\ref{asympt1}) and (\\ref%\n{asympt2}) exist at \\emph{arbitrarily large} $\\gamma _{0}$, suggesting the\n\\textit{unbreakability} of the $\\mathcal{PT}$ symmetry in this case, as\ncorroborated by exact solution (\\ref{exact1}) produced below.\n\nThe imaginary potential defined by Eq. (\\ref{xy}) with $\\beta =0$ produces\nthe following result:%\n\\begin{equation}\nU_{\\mathrm{asympt}}\\left( x,y\\right) =\\sqrt{\\frac{1-\\left( \\gamma\n_{0}\/2\\right) ^{2}}{2\\sigma }}\\exp \\left( -\\frac{1}{2}r^{2}-\\frac{1}{2}%\ni\\gamma _{0}xy\\right) ,  \\label{asympt3}\n\\end{equation}%\nfor $\\sigma \\neq 0$, and if $\\sigma =0$, the result is%\n\\begin{equation}\nU_{\\mathrm{asympt}}\\left( x,y\\right) =\\sqrt{\\frac{1-\\left( \\gamma\n_{0}\/2\\right) ^{2}}{2}}r\\exp \\left( -\\frac{1}{2}r^{2}-\\frac{1}{2}i\\gamma\n_{0}xy\\right) .  \\label{asympt4}\n\\end{equation}%\nOn the contrary to the the above asymptotic solutions, given by Eqs. (\\ref%\n{asympt1}) and (\\ref{asympt2}), which are available for arbitrarily large $%\n\\gamma _{0}$, their counterparts produced by Eqs. (\\ref{asympt3}) and (\\ref%\n{asympt4}) exist only at $\\gamma _{0}<2$, i.e., if the gain-loss coefficient\nis not too large.\n\nIt is relevant to stress the \\textit{universal character} of all asymptotic\napproximations given by Eqs. (\\ref{asympt1}) - (\\ref{asympt4}): they depend\nsolely on coefficients $\\sigma $ and $\\gamma _{0}$ of the underlying model,\nand, unlike the commonly known asymptotic forms of solitons in usual\nsystems, do not depend on the propagation constant, $k$. The single\nexception is presented by exact solution Eq. (\\ref{exact0}) given below,\nwhose asymptotic form (actually coinciding with the exact soliton solution,\nin that case) explicitly depends on $k$, but this happens solely for\nspecially chosen parameters given by Eq. (\\ref{special}). In the generic\ncase, a dependence on $k$ appears in the next-order correction to the shape\nof the asymptotic tail. In particular, the correction to the tails given by\nEqs. (\\ref{asympt1}) and (\\ref{asympt2}) are%\n\\begin{equation}\n\\delta U_{\\mathrm{asympt}}\\left( x,y\\right) =-\\left( k\/r^{2}\\right) U_{%\n\\mathrm{asympt}}\\left( x,y\\right) .  \\label{delta}\n\\end{equation}%\nFurthermore, for more complex solutions, such as multi-peak solitons and\nsolitary vortices, as well as for higher-order radial states of the\nsingle-peak solitons, which are produced below in the numerical form, the\nasymptotic form at large $r$ is exactly the same as given by Eqs. (\\ref%\n{asympt1})-(\\ref{asympt4}).\n\n\\subsubsection*{\\textbf{Exact solutions for single-peak solitons}}\n\nPrecisely at the above-mentioned critical value $\\gamma _{0}=2$, the\nasymptotic solutions (\\ref{asympt3}) and (\\ref{asympt4}) vanish. However, in\nthe special case,\n\\begin{equation}\n\\sigma =0,\\gamma _{0}=2,\\beta =0,  \\label{special}\n\\end{equation}%\nthe vanishing asymptotic solution Eq. (\\ref{asympt4}) is replaced by a\ndifferent one, which, as can be easily checked, is an \\emph{exact solution}\nto Eq. (\\ref{UU}) (not just an asymptotic approximation valid at large $r$),%\n\\begin{equation}\n\\left( U_{\\mathrm{exact}}^{\\left( xy\\right) }\\right) _{\\gamma _{0}=2}=\\sqrt{%\n-\\left( 1+k\\right) }\\exp \\left( -\\frac{1}{2}r^{2}-\\frac{1}{2}i\\gamma\n_{0}xy\\right) .  \\label{exact0}\n\\end{equation}%\nIt exists, as the continuous family, at all values of $k<-1$.\n\nFurther, Eq. (\\ref{UU}) which includes the $\\mathcal{PT}$-symmetric\nimaginary potential Eq. (\\ref{x}), with $\\beta =0$, gives rise to an exact\nsolution at a special value $k_{0}^{(x)}$ of the propagation constant:%\n\\begin{equation}\nU_{\\mathrm{exact}}^{(x)}=\\frac{1}{\\sqrt{2\\sigma }}\\exp \\left( -\\frac{1}{2}%\nr^{2}-i\\gamma _{0}x\\right) ,  \\label{exact1}\n\\end{equation}%\n\\begin{equation}\n~k_{0}^{(x)}=-\\left( 1+\\frac{\\gamma _{0}^{2}}{2}+\\frac{1}{2\\sigma }\\right) ,\n\\label{exact1parameters}\n\\end{equation}%\nwhich exists at all values of coefficients $\\gamma _{0}$ and $\\sigma $,\nexcept for $\\sigma =0$. In other words, at $k=k_{0}^{(x)}$ the asymptotic\napproximation Eq. (\\ref{asympt1}) is tantamount to the exact solution. This\nsolution features the \\emph{unbreakable} $\\mathcal{PT}$ symmetry, as it\npersists at arbitrarily large values of the gain-loss coefficient, $\\gamma\n_{0}$. Moreover, although Eq. (\\ref{exact1}) yields the exact solution at\nthe single value of the propagation constant, given by Eq. (\\ref%\n{exact1parameters}), which is embedded in a generic family of numerically\nfound fundamental solitons, as demonstrated below in Figs. \\ref{fig1}-\\ref%\n{fig3}, the entire family asymptotically shrinks to the exact solution in\nthe limit of large $\\gamma _{0}$. Indeed, it is easy to find that, for $%\n\\gamma _{0}^{2}\\gg 1$ and a relatively small deviation of the propagation\nconstant from the special value (\\ref{exact1parameters}), $\\left\\vert \\delta\nk\\right\\vert \\equiv \\left\\vert k-k_{0}^{(x)}\\right\\vert \\ll \\gamma _{0}^{2}$%\n, the fundamental soliton is%\n\\begin{equation}\nU_{\\mathrm{approx}}^{(x)}\\approx \\frac{1}{\\sqrt{2\\sigma }}\\exp \\left[ -\\frac{%\n1}{2}\\left( r^{2}+\\frac{\\delta k}{\\gamma _{0}^{2}}x^{2}\\right) -i\\left(\n\\gamma _{0}-\\frac{\\delta k}{\\gamma _{0}}\\right) x\\right] ,  \\label{approx}\n\\end{equation}%\nfeaturing weak anisotropy of the shape, $\\left\\vert U_{\\mathrm{approx}%\n}^{(x)}\\left( x,y\\right) \\right\\vert $.\n\nNext, Eq. (\\ref{UU}) with the imaginary potential taken as per Eq. (\\ref{xy}%\n) with $\\beta =0$, and with $\\sigma \\neq 0$ in the nonlinearity-modulation\nprofile (\\ref{sigma}), gives rise to the following exact solution, at the\nrespective single value of $k$:%\n\\begin{equation}\n\\left( U_{\\mathrm{exact}}^{\\left( xy\\right) }\\right) _{\\gamma _{0}<2}=\\sqrt{%\n\\frac{1-\\left( \\gamma _{0}\/2\\right) ^{2}}{2\\sigma }}\\exp \\left( -\\frac{1}{2}%\nr^{2}-\\frac{1}{2}i\\gamma _{0}xy\\right) ,  \\label{exact2}\n\\end{equation}%\n\\begin{equation}\nk_{0}^{(xy)}=-\\left[ 1+\\frac{1}{2\\sigma }\\left( 1-\\left( \\frac{\\gamma _{0}}{2%\n}\\right) ^{2}\\right) \\right] .  \\label{exact2parameters}\n\\end{equation}%\nIn this case too, the asymptotic approximation Eq. (\\ref{asympt3}) becomes\nidentical to the exact solution at $k=k_{0}^{(xy)}$, both existing at $%\n\\gamma _{0}<2$, on the contrary to exact solution (\\ref{exact1}), which\nexists at all values of $\\gamma _{0}$.\n\nThus, the models considered here do not have the linear spectrum. Instead of\nit, they are characterized by spectra (families) of self-trapped nonlinear\nsolutions (solitons). The radical change of the concept of the system's\nspectrum implies a respective change in the concept of the $\\mathcal{PT}$\nsymmetry, which now applies not to the set of eigenvalues of the linearized\nsystem, but directly to the existence of families of nonlinear states.\nLastly, it is worthy to note that all the asymptotic and exact solutions\nproduced above, including the first correction (\\ref{delta}) to the\nasymptotic tails, feature isotropic shapes of $\\left\\vert U(x,y\\right\\vert $%\n, although the imaginary potentials Eqs. (\\ref{x}) and (\\ref{xy}) are\nobviously anisotropic.\n\n\\subsubsection*{\\textbf{Exact solutions for elliptic vortices in an\nanisotropic model}}\n\nIn addition to 2D fundamental solitons, similar to the exact ones presented\nhere, we also address below, by means of numerical methods, solitons with\nembedded vorticities, $m=1,2,3...$ . A challenging issue is to seek for\nexact solutions for vortex solitons. Such solutions can be found in the case\nof imaginary potential Eq. (\\ref{xy}) with $\\beta =0$, but for a more\ngeneral \\textit{anisotropic} version of the nonlinearity-modulation profile\nin Eq. (\\ref{NLS}) with $\\sigma =0$, namely,%\n\\begin{equation}\n\\Sigma \\left( x,y\\right) =\\exp \\left( x^{2}+gy^{2}\\right) ,  \\label{g}\n\\end{equation}%\nwhere positive $g\\neq 1$ accounts for the ellipticity of the modulation\nprofile. Then, an exact solution for elliptically deformed vortex solitons\nwith $m=1$ is given by the following ansatz [cf. Eq. (\\ref{exact2})]:\n\\begin{equation}\nU\\left( x,y\\right) =U_{0}\\left( x+iby\\right) \\exp \\left( -\\frac{1}{2}\\left(\nx^{2}+gy^{2}\\right) -iaxy\\right) ,  \\label{vortex}\n\\end{equation}%\nwhere real $b\\neq 1$ accounts for the ellipticity of the soliton's phase\nfield, and $a$ is another real constant. The substitution of this ansatz and\nexpressions Eq. (\\ref{g}) and Eq. (\\ref{xy}) (with $\\beta =0$) in the\naccordingly modified equation (\\ref{UU}) leads to the following relations\nbetween parameters of the ansatz:%\n\\begin{gather}\n\\left( 1+g\\right) a=-\\gamma _{0},  \\notag \\\\\n(g-1)b-\\left( 1+b^{2}\\right) a=0,  \\label{conditions} \\\\\nb^{2}\\left( 1-a^{2}\\right) +a^{2}=g^{2},  \\notag\n\\end{gather}%\nsupplemented by expressions for the propagation constant and soliton's\namplitude:%\n\\begin{equation}\nk=-\\left( 3\/2+g\/2+ab\\right) ,~U_{0}^{2}=\\left( 1-a^{2}\\right) \/2.  \\label{kU}\n\\end{equation}%\nThe system of three equations (\\ref{conditions}) for two free parameters $a$\nand $b$ demonstrates that the exact vortex solution is a nongeneric one, as\nit may exist only if an additional constraint, which can be derived by\neliminating $a$ and $b$ in Eq. (\\ref{kU}), is imposed on parameters $g$ and $%\n\\gamma _{0}$:%\n\\begin{equation}\n\\left( g^{2}-1\\right) ^{2}\\left[ g^{2}\\left( g+1\\right) ^{2}-\\gamma _{0}^{2}%\n\\right] \\left[ \\left( g+1\\right) ^{2}-\\gamma _{0}^{2}\\right] =\\gamma _{0}^{2}%\n\\left[ \\left( g^{2}+1\\right) \\left( g+1\\right) ^{2}-2\\gamma _{0}^{2}\\right]\n^{2}.\n\\end{equation}\n\nIn the isotropic model, with $g=1$, Eq. (\\ref{conditions}) has no nontrivial\nsolutions. However, they can be found for $g\\neq 1$. A particular example is%\n\\begin{gather}\nb=1\/\\sqrt{2}\\approx 0.707\\,1,a=-\\left( 3-\\sqrt{5}\\right) \/\\left( 4\\sqrt{2}%\n\\right) \\approx -0.1351, \\\\\nU_{0}=\\sqrt{3\\left( 3+\\sqrt{5}\\right) }\/\\left( 4\\sqrt{2}\\right) \\approx\n0.7006,\n\\end{gather}%\nwhich is a valid solution at $g=\\left( 3\\sqrt{5}-1\\right) \/8\\approx\n\\allowbreak 0.713\\,5\\ $and $\\gamma _{0}=\\left( 3+\\sqrt{5}\\right) \/\\left( 16%\n\\sqrt{2}\\right) \\approx \\allowbreak 0.231\\,4$. This value of $g$ corresponds\nto eccentricity $e\\equiv \\sqrt{1-g}=\\sqrt{(9-3\\sqrt{5})\/8}\\approx\n\\allowbreak 0.535\\,2$ of the elliptic profile in Eq. (\\ref{g}).\n\nNumerical results are reported below for the isotropic model, while the\nanisotropic one should be a subject for separate consideration.\n\n\\subsection*{\\textbf{Numerical results for zero-vorticity solitons}}\n\n\\subsubsection*{The\\textbf{\\ }$\\mathcal{PT}$\\textbf{-symmetric imaginary\npotential (\\protect\\ref{x}): single- and double-peak solitons}}\n\nThe isolated exact solution of the model with the $\\mathcal{PT}$-symmetric%\n\\textbf{\\ }gain-loss distribution, given by Eqs. (\\ref{exact1}) and (\\ref%\n{exact1parameters}), can be embedded in a continuous family of solitons,\nproduced by a numerical solution of Eq. (\\ref{UU}), with $\\Sigma (r)$ and $%\n\\gamma \\left( x\\right) $ taken as per Eqs. (\\ref{sigma}) and (\\ref{x}). The\nappropriate numerical algorithm is the Newton conjugate gradient method \\cite%\n{Yang-book}, which is briefly outlined in section Method below. The\nstability of the stationary states was identified by numerical computation\nof eigenvalues of small perturbations, using linearized equations (\\ref%\n{eigen}) for perturbations around the stationary solitons. Finally, the\nstability predictions, produced by the eigenvalues, were verified by\nsimulations of the perturbed evolution of the solitons (some technical\ndetails are reported elsewhere \\cite{book}).\n\nIt is relevant to stress that the convergence of the algorithm which\nproduces stationary states depends on appropriate choice of the initial\nguess. While stationary modes were not found in \\textquotedblleft holes\"\nappearing in stability charts which are displayed below in Figs. \\ref{fig2}, %\n\\ref{fig3}, \\ref{fig7}, \\ref{fig10}-\\ref{fig12} and \\ref{fig16}, \\ref{fig17}%\n, it is plausible that stationary solutions exist in the holes too, being,\nhowever, especially sensitive to the choice of the input. On the other hand,\nthe intricate alternation of stability and instability spots, which is also\nobserved in the charts, is a true peculiarity of the present model.\nMoreover, genuine structure of the stability charts may be fractal, but\nanalysis of this possibility is beyond the scope of the present work.\n\n\\begin{figure}[tbp]\n\\centering\\includegraphics[width=0.75\\textwidth]{Fig1_PT.pdf}\n\\caption{Typical examples of stable solitons produced by the model with the $%\n\\mathcal{PT}$-symmetric imaginary potential defined by Eq. (\\protect\\ref{x}%\n). (a) A fundamental single-peak soliton for $\\protect\\gamma _{0}=1.2$ in\nEq. (\\protect\\ref{x}) and propagation constant $k=-3.2$ in Eq. (\\protect\\ref%\n{uU}). (b) A higher-order radial state of the single-peak soliton for $%\n\\protect\\gamma _{0}=0.2$ and $k=-4$. (c) A double-peak soliton for $\\protect%\n\\gamma _{0}=1.4$ and $k=-4$. In all the cases, $\\protect\\sigma =1$ and $%\n\\protect\\beta =0$ are fixed in Eqs. (\\protect\\ref{sigma}) and (\\protect\\ref%\n{x}).}\n\\label{fig1}\n\\end{figure}\n\nGeneric examples of numerically found \\emph{stable} solitons with single-\nand double-peak shapes are displayed in Fig. \\ref{fig1}. Note that the\ndouble-peak modes have their two maxima separated in the direction of $x$,\nin accordance with the anisotropic shape of the imaginary potential in Eq. (%\n\\ref{x}). As concerns single-peak modes, two different varieties of stable\nones were found: fundamental solitons, with the shape similar to that of the\nexact solution given by Eqs. (\\ref{exact1}) and (\\ref{exact1parameters})\n[see Fig. \\ref{fig1}(a)], and higher-order states with a radial ring\nsurrounding the central peak, see Fig. \\ref{fig1}(b). It is worthy to note\nthat, unlike many other models, where higher-order radial states are\nunstable \\cite{Atai1}-\\cite{Atai7}, they are stable in the present case.\nNote also that shapes of both species of the single-peak solitons,\nfundamental and higher-order ones, seem isotropic in terms of $\\left\\vert\nU\\left( x,y\\right) \\right\\vert $, similar to exact solution (\\ref{exact1}).\nThe isotropy is obviously broken by double-peak modes, see Fig. \\ref{fig1}%\n(c).\n\nResults of the stability analysis, based on the computation of perturbation\neigenvalues, are summarized in the stability map in the plane of $\\left(\nk,\\gamma _{0}\\right) $ [the soliton's propagation constant and strength of\nthe gain-loss term in Eq. (\\ref{x})], for $\\beta =0$ and $\\beta =0.2$ in\nFigs. \\ref{fig2} and \\ref{fig3}, respectively. Several noteworthy features\nare revealed by these plots. First, it is worthy to note significant\nstability areas for both the double-peak and higher-order single-peak $%\n\\mathcal{PT}$-symmetric solitons in Figs. \\ref{fig2} and \\ref{fig3}.\nFurther, bistability is observed at many points, in the form of coexisting\nstable fundamental and double-peak solitons, or fundamental and higher-order\nsingle-peak ones. As concerns the possibility of maintaining the unbreakable\n$\\mathcal{PT}$ symmetry, Fig. \\ref{fig2} demonstrates shrinkage of the\nexistence and stability regions of the modes with the increase of $\\gamma\n_{0}$ at $\\beta =0$ to the exact soliton solution given by Eqs. (\\ref{exact1}%\n) and (\\ref{exact1parameters}), in agreement with the trend represented by\napproximate solution (\\ref{approx}). Eventually, the exact solution loses\nits stability at $\\gamma _{0}\\geq 2$. On the other hand, the introduction of\na relatively weak confinement of the gain-loss term, with $\\beta =0.2$ in\nEq. (\\ref{x}), demonstrates that the $\\mathcal{PT}$ symmetry remains\nunbreakable in \\ref{fig3}, where both the existence and stability regions\nextend in the direction of large values of $-k$ and $\\gamma _{0}$, without\nfeaturing any boundary.\n\nAs concerns unstable solitons, they typically blow up in the course of the\nevolution, see an example below in Fig. \\ref{fig18}. Although it shows the\nblowup of a vortex soliton, the instability development of zero-vorticity\nones is quite similar.\n\n\\begin{figure}[tbp]\n\\centering\\includegraphics[width=0.6\\textwidth]{Fig2_PT.jpg}\n\\caption{The stability map for the $\\mathcal{PT}$-symmetric solitons\nmaintained by imaginary potential (\\protect\\ref{x}), in the case of $\\protect%\n\\sigma =1$ and $\\protect\\beta =0$ in Eqs. (\\protect\\ref{sigma}) and (\\protect\n\\ref{x}). Stable fundamental single-peak solitons are marked by green dots.\nAll unstable solitons are marked by red crosses, irrespective of their\nstructure. Exact soliton solutions, given by Eqs. (\\protect\\ref{exact1}) and\n(\\protect\\ref{exact1parameters}), are indicated by green stars (except for\none at $\\protect\\gamma _{0}=2$, which is designated by the red cross, as the\nexact solutions are unstable at $\\protect\\gamma _{0}\\geq 2$). Green numbers $%\n\\geq 2$ in this figure and below denote stable solitons with the same number\nof peaks. Further, green numbers $1$ label stable single-peak solitons with\nthe higher-order radial structure, as in Fig. \\protect\\ref{fig1}(b). Green\nnumbers $1$ or $2$, placed close to green dots, imply bistability, i.e.,\ncoexistence of stable fundamental single-peak solitons and stable\nhigher-order or double-peak ones. Red crosses placed on top of green dots\nimply coexistence of fundamental single-peak solitons with some unstable\nmode. Soliton solutions were not found in white areas.}\n\\label{fig2}\n\\end{figure}\n\nThe stability charts, drawn in Figs. \\ref{fig2} and \\ref{fig3} for $\\sigma\n=1 $ in Eq. (\\ref{sigma}), are quite similar to their counterparts produced\nat other values of $\\sigma $, including $\\sigma =0$, when the exact solution\ngiven by Eqs. (\\ref{exact1}) and (\\ref{exact1parameters}) does not exist,\nwhile the asymptotic form of the solitons' tails is given by Eq. (\\ref%\n{asympt2}).\n\n\\begin{figure}[tbp]\n\\centering\\includegraphics[width=0.6\\textwidth]{Fig3_PT.pdf}\n\\caption{The same as in Fig. \\protect\\ref{fig2}, but for $\\protect\\beta =0.2$\nin Eq. ( \\protect\\ref{x}), i.e., with the gain-loss term subject to weak\nspatial confinement. In this case, there are no exact solitons solutions,\nwhile the asymptotic solution for the tails is given by Eq. (\\protect\\ref%\n{asympt1}) with $\\protect\\gamma _{0}=0$ (the confinement eliminates $\\protect%\n\\gamma _{0}$ from the asymptotic solution).}\n\\label{fig3}\n\\end{figure}\n\n\\subsubsection*{\\textbf{The\\ imaginary potential (\\protect\\ref{xy}): single-\nand multi-peak solitons}}\n\nA drastic difference revealed by the stability analysis of the model based\non Eqs. (\\ref{NLS}), (\\ref{sigma}) and (\\ref{xy}) is that the respective\nexact solutions, given by Eq. (\\ref{exact0}) for the special case (\\ref%\n{special}), and by Eqs.\\ (\\ref{exact2}) and (\\ref{exact2parameters}) for $%\n\\sigma >0$, $\\beta =0$ and arbitrary $\\gamma _{0}$, are completely unstable,\non the contrary to the stability of the exact solutions in the case of the $%\n\\mathcal{PT}$-symmetric imaginary potential Eq. (\\ref{x}) (at $\\gamma _{0}<2$%\n). Furthermore, all numerical solutions found in the full 2D model with $%\n\\beta =0$ in Eq. (\\ref{xy}) are unstable too. The stabilization in this\nmodel is provided by $\\beta >0$, i.e., by imposing the spatial confinement\non the gain-loss term in Eq. (\\ref{xy}). For fixed $\\sigma $, there is a\nminimum value $\\beta _{\\min }$ of $\\beta $ which secures the stabilization.\nFor instance, we have concluded that the solitons may be stable in the model\nwith $\\sigma =1$ in Eq. (\\ref{sigma}) at $\\beta \\geq \\beta _{\\min }\\approx\n0.2$ in Eq. (\\ref{xy}), still being completely unstable, e.g., at $\\beta\n=0.1 $.\n\nAs mentioned above, the steep growth of $\\Sigma \\left( r\\right) $ in Eq. (%\n\\ref{sigma}) cannot extend to infinity, it being sufficient to maintain the\nadopted profile of $\\Sigma (r)$ on a scale which is essentially larger than\na characteristic size of solitons supported by this profile. The same\npertains to the linear growth of the imaginary potential at large $|x|$ in\nEq. (\\ref{x}): in reality, it should not continue at distances much larger\nthan the size of the stable solitons considered in the previous section.\nHowever, the presence of $\\beta _{\\min }$ implies that the corresponding\n\\textquotedblleft tacit\" confinement of $\\gamma \\left( x,y\\right) $ in Eq. (%\n\\ref{xy}) is not sufficient to produce stable 2D solitons. At $\\beta >\\beta\n_{\\min }$, the numerical solution generates stable fundamental single-peak\nsolitons and their higher-order radial counterparts with isotropic shapes of\n$\\left\\vert U\\left( x,y\\right) \\right\\vert $, as shown in Fig. \\ref{fig4}%\n(a,b). Further, stable multi-peak solitons are found too. Due to the 2D\nstructure of the imaginary potential (\\ref{xy}), they feature a four- or\nfive-peak structure, built along both the $x$ and $y$ axes, as shown in Fig. %\n\\ref{fig4}(c,d), instead of the uniaxial double-peak modes supported by the\nquasi-1D imaginary potential (\\ref{x}), cf. \\ref{fig1}(c).\n\n\\begin{figure}[tbp]\n\\centering\\includegraphics[width=0.75\\textwidth]{Fig4_PT.pdf}\n\\caption{Examples of stable single- and multi-peak $\\mathcal{PT}$-symmetric\nsolitons, found in the model based on Eqs. (\\protect\\ref{sigma}) and (%\n\\protect\\ref{xy}), with $\\protect\\sigma =1$ and (a) $\\protect\\beta =0.5$, $%\n\\protect\\gamma _{0}=1$, $k=-1$; (b) $\\protect\\beta =0.5$, $\\protect\\gamma %\n_{0}=0.2$, $k=-4$; (c) $\\protect\\beta =0.2$, $\\protect\\gamma _{0}=1.4$, $%\nk=-2.8$; (d) $\\protect\\beta =0.5$, $\\protect\\gamma _{0}=0.4$, $k=-1.8$. }\n\\label{fig4}\n\\end{figure}\n\nA typical stability chart for the 2D solitons generated by the model with $%\n\\beta >\\beta _{\\min }$ is displayed in Fig. \\ref{fig5}. It features\nbistability between the fundamental single-peak solitons and the\nhigher-order ones, or four- and five-peak complexes, in a relatively small\nregion of the $\\left( k,\\gamma _{0}\\right) $ plane, at sufficiently small\nvalues of $\\gamma _{0}$. Figure \\ref{fig5} clearly shows that no solitons\nwere found at $\\gamma _{0}\\geq 2$, this restriction coinciding with that for\nthe exact solution given by Eqs. (\\ref{exact2}) and (\\ref{exact2parameters}%\n). Thus, unlike the $\\mathcal{PT}$-symmetric imaginary potential (\\ref{x}),\nthe model based on potential (\\ref{xy}) does not produce unbreakable soliton\nfamilies.\n\n\\begin{figure}[tbp]\n\\centering\\includegraphics[width=0.6\\textwidth]{Fig5_PT.jpg}\n\\caption{The stability chart, defined as in Figs. \\protect\\ref{fig2} and\n\\protect\\ref{fig3}, but for the model including imaginary potential (\\protect\n\\ref{xy}), with $\\protect\\sigma =1$ and $\\protect\\beta =0.5$ in Eqs. (%\n\\protect\\ref{sigma}) and (\\protect\\ref{xy}). As indicated by the upper\ndashed red curve, no solitons were found at $\\protect\\gamma _{0}\\geq 2$,\nwhere the exact solution given by Eqs. (\\protect\\ref{exact2}) does not exist\neither.}\n\\label{fig5}\n\\end{figure}\n\n\\subsection{\\textbf{Vortex solitons}}\n\nSoliton solutions of Eq. (\\ref{UU}) with embedded vorticity were found\nnumerically by means of the above-mentioned Newton conjugate gradient\nmethod, initialized by the ansatz with integer vorticity $m\\geq 1$ added to\nthe previously found 2D stationary solutions of Eq. (\\ref{UU}):\n\n\\begin{equation}\nU\\left( x,y\\right) \\rightarrow U\\left( x,y\\right) r^{m}\\exp (im\\theta\n)\\equiv U\\left( x,y\\right) \\left( x+iy\\right) ^{m},  \\label{AngularPush}\n\\end{equation}%\nwhere $\\left( r,\\theta \\right) $ are the polar coordinates. The stability of\nresulting vortex solitons was again analyzed through the computation of\neigenvalues for modes of small perturbations around the vortex states, see\nEqs. (\\ref{eigen}), and then verified by direct simulations.\n\n\\subsubsection*{\\textbf{Vortex solitons in the case of the }$\\mathcal{PT}$%\n\\textbf{-symmetric imaginary potential}}\n\nIn the framework of the model with imaginary potential (\\ref{x}), stable\nvortex solitons were found in the case of $\\beta =0$ (no gain-loss\nconfinement) with $m=1$, while vortices with $m\\geq 2$ do not exist or are\nunstable. An example of stable vortices is shown in Fig. \\ref{fig6}, and the\nrespective stability charts for different values of $\\sigma $ in Eq. (\\ref%\n{sigma}) are presented in Fig. \\ref{fig7}. The strongly anisotropic shape of\nthe vortex is a consequence of the anisotropy of the underlying imaginary\npotential (\\ref{x}).\n\n\\begin{figure}[tbp]\n\\centering\\includegraphics[width=0.75\\textwidth]{Fig6_PT.pdf}\n\\caption{Three-dimensional (a) and top-view (b) shapes of $\\left\\vert\nU\\left( x,y\\right) \\right\\vert $ for a typical stable vortex soliton with $%\nm=1$, supported by the $\\mathcal{PT}$-symmetric imaginary potential (\\protect\n\\ref{x}) with $\\protect\\gamma _{0}=0.6$, $\\protect\\beta =0$, and $\\protect%\n\\sigma =0$ in Eq. (\\protect\\ref{sigma}), the propagation constant being $%\nk=-3 $. Panel (c) displays the phase structure of the vortex.}\n\\label{fig6}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\centering\n\\centering\\includegraphics[width=0.75\\textwidth]{Fig7_PT.pdf}\n\\caption{Stability charts for vortex solitons with topological charge $m=1$\nin the model including the $\\mathcal{PT}$-symmetric imaginary potential (%\n\\protect\\ref{x}) with $\\protect\\beta =0$, and $\\protect\\sigma =0$ or $1$ in\nEq. (\\protect\\ref{sigma}), in panels (a) and (b) panels, respectively. Green\ncircles and red crosses denote stable and unstable vortex solitons,\nrespectively. The same notation is used below in other stability charts for\nvortex solitons.}\n\\label{fig7}\n\\end{figure}\n\nThe introduction of the confinement of the gain and loss in Eq. (\\ref{x})\n(in particular, setting $\\beta =0.5$) makes it possible to construct stable\nvortex solitons with higher vorticities, corresponding to $m>1$ in Eq. (\\ref%\n{AngularPush}). An example of a stable vortex with $m=3$ is shown in Fig. %\n\\ref{fig8}.\n\n\\begin{figure}[tbp]\n\\centering\\includegraphics[width=0.75\\textwidth]{Fig8_PT.pdf}\n\\caption{The same as in Fig. \\protect\\ref{fig6}, but for stable vortex\nsoliton with $m=3$ and parameters $\\protect\\gamma _{0}=0.8$, $\\protect\\beta %\n=0.5$, $\\protect\\sigma =0$, $k=-4$.}\n\\label{fig8}\n\\end{figure}\n\nIn most cases, stable vortices generated by input (\\ref{AngularPush}) from\ndouble-peak stationary solutions have the same shape as those originating\nfrom their single-peak counterparts. However, in few cases the application\nof the lowest vorticity, with $m=1$ in Eq. (\\ref{AngularPush}), to the\ndouble-peak input leads to the creation of stable vortex solitons with a\ncomplex shape, see an example in Fig. \\ref{fig9}.\n\n\\begin{figure}[tbp]\n\\centering\\includegraphics[width=0.75\\textwidth]{Fig9_PT.pdf}\n\\caption{The same as in Fig. \\protect\\ref{fig6}, but for a case when the\nstable vortex soliton with $m=1$ and a complex shape is created, the\nparameters in Eqs. (\\protect\\ref{x}) and (\\protect\\ref{sigma}) being $%\n\\protect\\gamma _{0}=0.4$, $\\protect\\beta =0$, and $\\protect\\sigma =1$. The\npropagation constant is $k=-3.6$.}\n\\label{fig9}\n\\end{figure}\n\nStability charts for the vortex solitons with $m=1,2$, and $3$, supported by\nthe $\\mathcal{PT}$-symmetric imaginary potential which is subject to the\nspatial confinement, with $\\beta =0.5$ in Eq. (\\ref{x}), are shown in Figs. %\n\\ref{fig10} - \\ref{fig12}. While the stability area shrinks with the\nincrease of $m$, a few stable isolated modes were found even for $m=4$ (not\nshown here). The comparison of Figs. \\ref{fig10} and \\ref{fig7} shows that\nthe introduction of the spatial confinement of the gain-loss profile helps\nto expand the stability area for $m=1$ towards larger values of $\\gamma _{0}$%\n, thus upholding the trend to observe the unbreakable $\\mathcal{PT}$\nsymmetry in this 2D model. In direct simulations, the evolution of unstable\nvortex modes leads towards the blowup, via their fusion into a single peak,\nsimilar to what is displayed below in Fig. \\ref{fig18}.\n\n\\begin{figure}[tbp]\n\\centering\\includegraphics[width=0.75\\textwidth]{Fig10_PT.pdf}\n\\caption{Stability charts for solitons with vorticity $m=1$ in the case of\nthe $\\mathcal{PT}$-symmetric imaginary potential (\\protect\\ref{x}) with $%\n\\protect\\beta =0.5$, and $\\protect\\sigma =0$ or $1$ in Eq. (\\protect\\ref%\n{sigma}), in panels (a) and (b), respectively.}\n\\label{fig10}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\centering\n\\centering\\includegraphics[width=0.75\\textwidth]{Fig11_PT.pdf}\n\\caption{The same as in Fig.(\\protect\\ref{fig10}) (stability charts) but for\nvortex solitons with $m=2$.}\n\\label{fig11}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\centering\n\\centering\\includegraphics[width=0.75\\textwidth]{Fig12_PT.pdf}\n\\caption{The same as in Fig. (\\protect\\ref{fig10}), but for vorticity $m=3$.}\n\\label{fig12}\n\\end{figure}\n\n\\subsubsection*{\\textbf{Vortex solitons in the model with imaginary\npotential (\\protect\\ref{xy})}}\n\nStarting from input Eq. (\\ref{AngularPush}), stable vortices can be\nconstructed in the model with the gain-loss profile Eq. (\\ref{xy}) only if\nit is subject to the spatial confinement (recall the same is reported above\nfor zero-vorticity solitons). Examples of stable solitons with vorticities $%\nm=1,2$ and $3$ found in this model are shown in Figs. \\ref{fig13} - \\ref%\n{fig15}. Note that higher-order states with $m\\geq 2$ are actually compound\nstates built of $m$ unitary vortices, whose pivots do not merge into a\nsingle one, remaining separated, although with a small distance between\nthem, as can be seen for $m=2$ in Fig. \\ref{fig14}. The pivots form arrays\nalong axes $x$ or $y$, the particular direction being randomly chosen by the\ninitial conditions. Nevertheless, the overall shapes of the unitary and\nhigher-order vortices are nearly isotropic, due to the structure of the\ngain-loss term in Eq. (\\ref{xy}) (cf. strongly anisotropic shapes of\nvortices in Figs. \\ref{fig6}, \\ref{fig8}, and \\ref{fig9}, supported by the\nimaginary potential (\\ref{x})).\n\n\\begin{figure}[tbp]\n\\centering\\includegraphics[width=0.75\\textwidth]{Fig13_PT.pdf}\n\\caption{The same as in Fig. \\protect\\ref{fig6}, but for the stable vortex\nsoliton with $m=1$ in the case of imaginary potential Eq. (\\protect\\ref{xy}%\n), with $\\protect\\gamma _{0}=0.4$, $\\protect\\beta =0.5$, $\\protect\\sigma =0$%\n, and propagation constant $k=-3.4$.}\n\\label{fig13}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\centering\\includegraphics[width=0.75\\textwidth]{Fig14_PT.pdf}\n\\caption{The same as in Fig. \\protect\\ref{fig13}, but for stable vortex\nsolitons with $m=2$ and $k=-3.6$.}\n\\label{fig14}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\centering\\includegraphics[width=0.75\\textwidth]{Fig15_PT.pdf}\n\\caption{The same as in Fig. \\protect\\ref{fig13}, but for stable vortex\nsolitons with $m=3$ and parameters $\\protect\\gamma _{0}=0.2$, $\\protect\\beta %\n=0.5$, $\\protect\\sigma =0$, $k=-2.2$.}\n\\label{fig15}\n\\end{figure}\n\nStability charts obtained in this model for the solitons with embedded\nvorticities $m=1$ and $2$ are shown in Figs. \\ref{fig16} and \\ref{fig17}.\nOnly few examples of stable vortices with $m=3$, not shown here, have been\nfound in this case (for instance, at $\\sigma =0$, $\\gamma _{0}=0.4$, $k=-1.2$%\n).\n\n\\begin{figure}[tbp]\n\\centering\\includegraphics[width=0.75\\textwidth]{Fig16_PT.pdf}\n\\caption{Stability charts for vortex solitons with $m=1$ in the model\nincluding imaginary potential (\\protect\\ref{xy}), with $\\protect\\beta =0.5$\nand $\\protect\\sigma =0$ in (a) or $\\protect\\sigma =1$ in (b).}\n\\label{fig16}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\centering\\includegraphics[width=0.75\\textwidth]{Fig17_PT.pdf}\n\\caption{The same as in Fig. (\\protect\\ref{fig16}), but for vorticity $m=2$.}\n\\label{fig17}\n\\end{figure}\n\nFinally, a generic example of the evolution of an unstable vortex soliton is\nshown in Fig. \\ref{fig18}. The strong difference between vertical scales in\ndifferent panels of the figure clearly suggests that the instability leads\nto the blowup of the unstable mode, in the course of which the original\nvortex tends to fuse into a single peak. In fact, all unstable solitons\nconsidered in this work tend to develop the blowup in direct simulations.\n\n\\begin{figure}[tbp]\n\\centering\\includegraphics[width=0.75\\textwidth]{Fig18_PT.pdf}\n\\caption{The blowup of an unstable vortex soliton with $m=2$ and $\\protect%\n\\gamma _{0}=1.2$, $\\protect\\beta =0.5$, $\\protect\\sigma =1$, $k=-2.4$, in\nthe model with imaginary potential (\\protect\\ref{xy}). Panels display the\nfield at $z=60$ (a), $z=200$ (b) and $z=300$ (c). Note the difference in\nvertical scales between them.}\n\\label{fig18}\n\\end{figure}\n\n\\section*{Discussion}\n\nThe objective of this work is to elaborate 2D models with the spatially\nmodulated self-defocusing nonlinearity and gain-loss distributions\n[imaginary potentials, $iW\\left( x,y\\right) $] which give rise to families\nof stable single-peak, multi-peak, and vortical solitons, including ones\nwhich may persist and remain stable (\\textquotedblleft unbreakable\") at\narbitrarily large values of strengths $\\gamma _{0}$ of the imaginary\npotential. The unbreakability is possible in the case of the $\\mathcal{PT}$%\n-symmetric imaginary potential, which is given by Eq. (\\ref{x}). An asset of\nthe models, which can be implemented in bulk nonlinear optical waveguides\nwith embedded gain and loss elements, is that they produce universal\nasymptotic solutions for solitons' tails, along with full exact solutions\nfor selected species of 2D fundamental and vortex solitons (the latter one\nis available in the elliptically deformed version of the model). In\nparticular, in the limit of large $\\gamma _{0}$, the unbreakable family of\nfundamental solitons tends to shrink towards the exact solution. Generic\nfamilies of zero-vorticity solitons, including single- and multi-peak ones\nand higher-order radial states of single-peak solitons, as well as families\nof self-trapped modes with embedded vorticity $m=1,2$, and $3$, are\nconstructed in the numerical form, and their stability is identified by\nmeans of the numerical computation of eigenvalues for small perturbations,\nand verified by direct simulations. In the case of the $\\mathcal{PT}$%\n-symmetric imaginary potential (\\ref{x}) the solitons are stable in vast\nparameter regions, and feature a trend towards maintaining the unbreakable $%\n\\mathcal{PT}$ symmetry. Under the action of the imaginary potential (\\ref{xy}%\n), families of stable fundamental and vortex solitons exist too, provided\nthat the imaginary potential is subject to spatial confinement.\n\nA relevant extension of the analysis may be to address the elliptically\ndeformed model, which is considered in the present work in a brief form. A\nchallenging problem is the possibility of the fractal structure of the\nstability patterns in the models' parameter planes.\n\n\\section*{Methods}\n\n\\subsection*{\\textbf{The Newton conjugate gradient method for 2D robust }$PT$%\n\\textbf{-symmetry model}}\n\nSolutions of the stationary equation (\\ref{UU}) were constructed by means of\nthe Newton conjugate gradient method, which is presented in detail in book\n\\cite{Yang-book}. In terms of this method, the stationary-solution operator $%\n\\mathbf{L}_{0}$ is defined by Eq. (\\ref{UU}), while the respective\nlinearization operator $\\mathbf{L}_{1}$ is defined as\n\n\\begin{equation}\n\\mathbf{L}_{1}=%\n\\begin{bmatrix}\nA & B \\\\\nC & D%\n\\end{bmatrix}%\n,  \\label{L1_linearization_operator}\n\\end{equation}%\nwith matrix elements\n\\begin{eqnarray}\nA &=&-k+\\frac{1}{2}\\nabla ^{2}-\\Sigma (r)(\\left[ 3(\\mathrm{Re}U)^{2}+(%\n\\mathrm{Im}U)^{2}\\right] , \\\\\nB &=&-2\\Sigma (r)\\mathrm{Re}U\\cdot \\mathrm{Im}U+W(x,y), \\\\\nC &=&-2\\Sigma (r)\\mathrm{Re}U\\cdot \\mathrm{Im}U-W(x,y), \\\\\nD &=&-k+\\frac{1}{2}\\nabla ^{2}-\\Sigma (r)\\left[ 3(\\mathrm{Im}U)^{2}+(\\mathrm{%\nRe}U)^{2}\\right] ,\n\\end{eqnarray}%\nwhere the nonlinearity coefficient, $\\Sigma (r)$, and imaginary potential, $%\nW\\left( x,y\\right) $ are defined, respectively, by Eq. (\\ref{sigma}) and\nEqs. (\\ref{x}) or (\\ref{xy}).\n\n\\subsection*{\\textbf{Simulations of the evolution of the wave fields}}\n\nDirect simulations of the evolution equation (\\ref{NLS}), written as\n\n\\begin{equation}\ni\\frac{\\partial U}{\\partial z}=-\\frac{1}{2}\\left( \\frac{\\partial ^{2}U}{%\n\\partial x^{2}}+\\frac{\\partial ^{2}U}{\\partial y^{2}}\\right) +\\left[\nk+\\Sigma (r)|U|^{2}+i\\gamma \\left( x,y\\right) \\right] U,  \\label{EqOnU}\n\\end{equation}%\ncf. Eq. (\\ref{UU}), have been performed by means of the commonly known\nsplit-step method. Marching forward in $z$ at each step was split in two\nparts, according to the following equations:\n\n\\begin{eqnarray}\n\\mathbf{I} &\\mathbf{:~}&i\\frac{\\partial U}{\\partial z}=\\left[ k+\\Sigma\n(r)|U|^{2}+i\\gamma \\left( x,y\\right) \\right] U, \\\\\n\\mathbf{II} &\\mathbf{:~}&i\\frac{\\partial U}{\\partial z}=-\\frac{1}{2}\\left(\n\\frac{\\partial ^{2}U}{\\partial x^{2}}+\\frac{\\partial ^{2}U}{\\partial y^{2}}%\n\\right) .\n\\end{eqnarray}\n\nThe solutions were numerically constructed in the 2D spatial domain, $%\n\\left\\vert x,y\\right\\vert \\leq 9$, which was covered by a discrete grid of\nsize $N_{x}\\times N_{y}=512\\times 512$. The direct simulations were carried\nout with step $\\Delta z=10^{-5}$. This small step was selected to provide\nsufficient accuracy of the numerical solutions obtained in the presence of\nthe \\textquotedblleft exotic\" nonlinearity-modulation and gain-loss profiles\n(\\ref{sigma}) and (\\ref{x}) or (\\ref{xy}).\n\n\\subsection*{\\textbf{The stability analysis}}\n\nThe stability of the stationary states against small perturbations were\nbased, as usual, on the general expression for a perturbed solution,%\n\\begin{equation}\nu\\left( x,y,z\\right) =e^{ikz}\\left\\{ U\\left( x,y\\right) +\\varepsilon \\left[\ne^{\\Gamma z}v\\left( x,y\\right) +e^{\\Gamma ^{\\ast }z}w^{\\ast }\\left(\nx,y\\right) \\right] \\right\\} ,  \\label{pert}\n\\end{equation}%\nwhere $\\varepsilon $ is an infinitesimal perturbation amplitude, with\neigenmodes $\\left\\{ v\\left( x,y\\right) ,w\\left( x,y\\right) \\right\\} $ and\n(complex) eigenvalue $\\Gamma $, which should be found from the numerical\nsolution of the respective linearized equations,%\n\\begin{equation}\n\\begin{array}{c}\n\\left( -k+i\\Gamma \\right) v+\\frac{1}{2}\\left( \\frac{\\partial ^{2}}{\\partial\nx^{2}}+\\frac{\\partial ^{2}}{\\partial y^{2}}\\right) v-2\\Sigma\n(r)|U|^{2}v-\\Sigma U^{2}w=i\\gamma \\left( x,y\\right) v, \\\\\n\\left( -k-i\\Gamma \\right) w+\\frac{1}{2}\\left( \\frac{\\partial ^{2}}{\\partial\nx^{2}}+\\frac{\\partial ^{2}}{\\partial y^{2}}\\right) w-2\\Sigma\n(r)|U|^{2}w-\\Sigma U^{2}v=-i\\gamma \\left( x,y\\right) w,%\n\\end{array}\n\\label{eigen}\n\\end{equation}%\nsubject to zero boundary conditions at $\\left\\vert x,y\\right\\vert\n\\rightarrow \\infty $ (in fact, at borders of the solution domain). These\nequations were solved by means of the known spectral collocation method \\cite%\n{Yang-book}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}