24 research outputs found

    Round and Bipartize for Vertex Cover Approximation

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    The vertex cover problem is a fundamental and widely studied combinatorial optimization problem. It is known that its standard linear programming relaxation is integral for bipartite graphs and half-integral for general graphs. As a consequence, the natural rounding algorithm based on this relaxation computes an optimal solution for bipartite graphs and a 2-approximation for general graphs. This raises the question of whether one can interpolate the rounding curve of the standard linear programming relaxation in a beyond the worst-case manner, depending on how close the graph is to being bipartite. In this paper, we consider a round-and-bipartize algorithm that exploits the knowledge of an induced bipartite subgraph to attain improved approximation ratios. Equivalently, we suppose that we work with a pair (?, S), consisting of a graph with an odd cycle transversal. If S is a stable set, we prove a tight approximation ratio of 1 + 1/?, where 2? -1 denotes the odd girth (i.e., length of the shortest odd cycle) of the contracted graph ?? : = ?/S and satisfies ? ? [2,?], with ? = ? corresponding to the bipartite case. If S is an arbitrary set, we prove a tight approximation ratio of (1+1/?) (1 - ?) + 2 ?, where ? ? [0,1] is a natural parameter measuring the quality of the set S. The technique used to prove tight improved approximation ratios relies on a structural analysis of the contracted graph ??, in combination with an understanding of the weight space where the fully half-integral solution is optimal. Tightness is shown by constructing classes of weight functions matching the obtained upper bounds. As a byproduct of the structural analysis, we also obtain improved tight bounds on the integrality gap and the fractional chromatic number of 3-colorable graphs. We also discuss algorithmic applications in order to find good odd cycle transversals, connecting to the MinUncut and Colouring problems. Finally, we show that our analysis is optimal in the following sense: the worst case bounds for ? and ?, which are ? = 2 and ? = 1 - 4/n, recover the integrality gap of 2 - 2/n of the standard linear programming relaxation, where n is the number of vertices of the graph

    Advances on Strictly Δ\Delta-Modular IPs

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    There has been significant work recently on integer programs (IPs) min{cx ⁣:Axb,xZn}\min\{c^\top x \colon Ax\leq b,\,x\in \mathbb{Z}^n\} with a constraint marix AA with bounded subdeterminants. This is motivated by a well-known conjecture claiming that, for any constant ΔZ>0\Delta\in \mathbb{Z}_{>0}, Δ\Delta-modular IPs are efficiently solvable, which are IPs where the constraint matrix AZm×nA\in \mathbb{Z}^{m\times n} has full column rank and all n×nn\times n minors of AA are within {Δ,,Δ}\{-\Delta, \dots, \Delta\}. Previous progress on this question, in particular for Δ=2\Delta=2, relies on algorithms that solve an important special case, namely strictly Δ\Delta-modular IPs, which further restrict the n×nn\times n minors of AA to be within {Δ,0,Δ}\{-\Delta, 0, \Delta\}. Even for Δ=2\Delta=2, such problems include well-known combinatorial optimization problems like the minimum odd/even cut problem. The conjecture remains open even for strictly Δ\Delta-modular IPs. Prior advances were restricted to prime Δ\Delta, which allows for employing strong number-theoretic results. In this work, we make first progress beyond the prime case by presenting techniques not relying on such strong number-theoretic prime results. In particular, our approach implies that there is a randomized algorithm to check feasibility of strictly Δ\Delta-modular IPs in strongly polynomial time if Δ4\Delta\leq4

    Extended formulations for a class of polyhedra with bimodular cographic constraint matrices

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    We are motivated by integer linear programs (ILPs) defined by constraint matrices with bounded determinants. Such matrices generalize the notion of totally-unimodular matrices. When the determinants are bounded by 22, the matrix is called bimodular. Artmann et al. give a polynomial-time algorithm for solving any ILP defined by a bimodular constraint matrix. Complementing this result, Conforti et al. give a compact extended formulation for a particular class of bimodular-constrained ILPs, namely those that model the stable set polytope of a graph with odd cycle packing number 11. We demonstrate that their compact extended formulation can be modified to hold for polyhedra such that (1) the constraint matrix is bimodular, (2) the row-matroid generated by the constraint matrix is cographic and (3) the right-hand side is a linear combination of the columns of the constraint matrix. This generalizes the important special case from Conforti et al. concerning 4-connected graphs with odd cycle transversal number at least four. Moreover, our results yield compact extended formulations for a new class of polyhedra

    Round and bipartize for vertex cover approximation

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    The vertex cover problem is a fundamental and widely studied combinatorial optimization problem. It is known that its standard linear programming relaxation is integral for bipartite graphs and half-integral for general graphs. As a consequence, the natural rounding algorithm based on this relaxation computes an optimal solution for bipartite graphs and a 2-approximation for general graphs. This raises the question of whether one can interpolate the rounding curve of the standard linear programming relaxation in a beyond the worst-case manner, depending on how close the graph is to being bipartite. In this paper, we consider a round-and-bipartize algorithm that exploits the knowledge of an induced bipartite subgraph to attain improved approximation ratios. Equivalently, we suppose that we work with a pair (G, S), consisting of a graph with an odd cycle transversal. If S is a stable set, we prove a tight approximation ratio of 1 + 1/ρ, where 2ρ − 1 denotes the odd girth (i.e., length of the shortest odd cycle) of the contracted graph G̃:= G/S and satisfies ρ ∈ [2, ∞], with ρ = ∞ corresponding to the bipartite case. If S is an arbitrary set, we prove a tight approximation ratio of (1 + 1/ρ) (1 − α) + 2α, where α ∈ [0, 1] is a natural parameter measuring the quality of the set S. The technique used to prove tight improved approximation ratios relies on a structural analysis of the contracted graph G̃, in combination with an understanding of the weight space where the fully half-integral solution is optimal. Tightness is shown by constructing classes of weight functions matching the obtained upper bounds. As a byproduct of the structural analysis, we also obtain improved tight bounds on the integrality gap and the fractional chromatic number of 3-colorable graphs. We also discuss algorithmic applications in order to find good odd cycle transversals, connecting to the MinUncut and Colouring problems. Finally, we show that our analysis is optimal in the following sense: the worst case bounds for ρ and α, which are ρ = 2 and α = 1 − 4/n, recover the integrality gap of 2 − 2/n of the standard linear programming relaxation, where n is the number of vertices of the graph

    Sparse graphs with bounded induced cycle packing number have logarithmic treewidth

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    A graph is OkO_k-free if it does not contain kk pairwise vertex-disjoint and non-adjacent cycles. We show that Maximum Independent Set and 3-Coloring in OkO_k-free graphs can be solved in quasi-polynomial time. As a main technical result, we establish that "sparse" (here, not containing large complete bipartite graphs as subgraphs) OkO_k-free graphs have treewidth (even, feedback vertex set number) at most logarithmic in the number of vertices. This is proven sharp as there is an infinite family of O2O_2-free graphs without K3,3K_{3,3}-subgraph and whose treewidth is (at least) logarithmic. Other consequences include that most of the central NP-complete problems (such as Maximum Independent Set, Minimum Vertex Cover, Minimum Dominating Set, Minimum Coloring) can be solved in polynomial time in sparse OkO_k-free graphs, and that deciding the OkO_k-freeness of sparse graphs is polynomial time solvable.Comment: 28 pages, 6 figures. v3: improved complexity result

    Congruency-Constrained TU Problems Beyond the Bimodular Case

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    A long-standing open question in Integer Programming is whether integer programs with constraint matrices with bounded subdeterminants are efficiently solvable. An important special case thereof are congruency-constrained integer programs $\min\{c^\top x\colon\ Tx\leq b,\ \gamma^\top x\equiv r\pmod{m},\ x\in\mathbb{Z}^n\}withatotallyunimodularconstraintmatrix with a totally unimodular constraint matrix T.Suchproblemshavebeenshowntobepolynomialtimesolvablefor. Such problems have been shown to be polynomial-time solvable for m=2,whichledtoanefficientalgorithmforintegerprogramswithbimodularconstraintmatrices,i.e.,fullrankmatriceswhose, which led to an efficient algorithm for integer programs with bimodular constraint matrices, i.e., full-rank matrices whose n\times nsubdeterminantsareboundedbytwoinabsolutevalue.Whereastheseadvancesheavilyreliedonexistingresultsonwellknowncombinatorialproblemswithparityconstraints,newapproachesareneededbeyondthebimodularcase,i.e.,for subdeterminants are bounded by two in absolute value. Whereas these advances heavily relied on existing results on well-known combinatorial problems with parity constraints, new approaches are needed beyond the bimodular case, i.e., for m>2.Wemakefirstprogressinthisdirectionthroughseveralnewtechniques.Inparticular,weshowhowtoefficientlydecidefeasibilityofcongruencyconstrainedintegerprogramswithatotallyunimodularconstraintmatrixfor. We make first progress in this direction through several new techniques. In particular, we show how to efficiently decide feasibility of congruency-constrained integer programs with a totally unimodular constraint matrix for m=3.Furthermore,forgeneral. Furthermore, for general m$, our techniques also allow for identifying flat directions of infeasible problems, and deducing bounds on the proximity between solutions of the problem and its relaxation

    Reconstructing networks

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    Complex networks datasets often come with the problem of missing information: interactions data that have not been measured or discovered, may be affected by errors, or are simply hidden because of privacy issues. This Element provides an overview of the ideas, methods and techniques to deal with this problem and that together define the field of network reconstruction. Given the extent of the subject, we shall focus on the inference methods rooted in statistical physics and information theory. The discussion will be organized according to the different scales of the reconstruction task, that is, whether the goal is to reconstruct the macroscopic structure of the network, to infer its mesoscale properties, or to predict the individual microscopic connections.Comment: 107 pages, 25 figure

    Reconstructing networks

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    Complex networks datasets often come with the problem of missing information: interactions data that have not been measured or discovered, may be affected by errors, or are simply hidden because of privacy issues. This Element provides an overview of the ideas, methods and techniques to deal with this problem and that together define the field of network reconstruction. Given the extent of the subject, the authors focus on the inference methods rooted in statistical physics and information theory. The discussion is organized according to the different scales of the reconstruction task, that is, whether the goal is to reconstruct the macroscopic structure of the network, to infer its mesoscale properties, or to predict the individual microscopic connections
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