Integer programs with bounded subdeterminants and two nonzeros per row

Fiorini, Samuel; Joret, Gwenaël; Weltge, Stefan; Yuditsky, Yelena

doi:10.1109/focs52979.2021.00011

Cited by 10 publications

(4 citation statements)

References 28 publications

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“…As far as we can tell, MIS could even be tractable in graphs with bounded iocp. This would constitute a surprising and formidable generalization of Conjecture 1.1 and of the same result for bounded ocp [FJWY21].…”

Section: Introductionmentioning

confidence: 56%

“…It was believed that Artmann et al 's result could even be lifted to graphs with bounded odd cycle packing number. Conforti et al [CFH + 20] proved it on graphs further assumed to have bounded genus, and Fiorini et al [FJWY21] con rmed that conjecture for graphs G satisfying ocp(G) = O(1), in general. A polynomial-time approximation scheme (PTAS), due to Bock et al [BFMR14], was known for MIS in the (much) more general case of n-vertex graphs G such that ocp(G) = o(n/ log n).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Sparse graphs with bounded induced cycle packing number have logarithmic treewidth

Bonamy¹,

Bonnet²,

Déprés³

et al. 2022

Preprint

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A graph is O k -free if it does not contain k pairwise vertex-disjoint and non-adjacent cycles. We prove that "sparse" (here, not containing large complete bipartite graphs as subgraphs) O k -free graphs have treewidth (even, feedback vertex set number) at most logarithmic in the number of vertices, which is sharp already for k = 2. As a consequence, most of the central NP-complete problems (such as M) can be solved in polynomial time in these graphs, and in particular deciding the O k -freeness of sparse graphs is polytime.

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Section: Introductionmentioning

confidence: 56%

Section: Introductionmentioning

confidence: 99%

Sparse graphs with bounded induced cycle packing number have logarithmic treewidth

Bonamy¹,

Bonnet²,

Déprés³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Recently, significant effort has been made to understand the computational complexity of integer programming problems defined by ∆-modular matrices. Three key results in this area are given in [4,15,22]. This task motivated the study of polyhedral geometry depending on the parameter ∆(A); see [5,9,10,20] for an incomplete collection of results concerning the distance of optimal integral solutions of an integer linear program and optimal vertex solutions of the corresponding relaxation, the lattice width of lattice-free polyhedra, and the diameter of polyhedra.…”

Section: Known Results On the Integer Carathéodory Rank Lead To Two I...mentioning

confidence: 99%

New Bounds for the Integer Carathéodory Rank

Aliev,

Henk,

Hogan

et al. 2024

SIAM J. Optim.

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Given a rational pointed n-dimensional cone C, we study the integer Carathéodory rank CR(C) and its asymptotic form CR a (C), where we consider "most" integer vectors in the cone. The main result significantly improves the previously known upper bound for CR a (C). We also study bounds on CR(C) in terms of ∆, the maximal absolute n × n minor of the matrix given in an integral polyhedral representation of C. If ∆ ∈ {1, 2}, we show CR(C) = n, and prove upper bounds for simplicial cones, improving the best known upper bound on CR(C) for ∆ ≤ n.

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“…First progress on Conjecture 1 was made by Artmann, Weismantel, and Zenklusen [AWZ17], who showed that it holds for ∆ = 2 (the bimodular case). Fiorini, Joret, Weltge, and Yuditsky [FJWY22] show that the conjecture is true for an arbitrary constant ∆ under the extra condition that the constraint matrix has at most two non-zero entries per row or column. Through a non-trivial extension of the techniques in [AWZ17], it was shown by Nägele, Santiago, and Zenklusen [NSZ22] that there is a randomized algorithm to check feasibility of an IP with a strictly 3-modular constraint matrix in polynomial time.…”

Section: Introductionmentioning

confidence: 95%

Advances on Strictly $Δ$-Modular IPs

Nägele¹,

Nöbel²,

Santiago³

et al. 2023

Preprint

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There has been significant work recently on integer programs (IPs) min{c x : Ax ≤ b, x ∈ Z n } with a constraint marix A with bounded subdeterminants. This is motivated by a well-known conjecture claiming that, for any constant ∆ ∈ Z >0 , ∆-modular IPs are efficiently solvable, which are IPs where the constraint matrix A ∈ Z m×n has full column rank and all n × n minors of A are within {−∆, . . . , ∆}. Previous progress on this question, in particular for ∆ = 2, relies on algorithms that solve an important special case, namely strictly ∆-modular IPs, which further restrict the n × n minors of A to be within {−∆, 0, ∆}. Even for ∆ = 2, such problems include well-known combinatorial optimization problems like the minimum odd/even cut problem. The conjecture remains open even for strictly ∆-modular IPs. Prior advances were restricted to prime ∆, 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 ∆-modular IPs in strongly polynomial time if ∆ ≤ 4.

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Integer programs with bounded subdeterminants and two nonzeros per row

Cited by 10 publications

References 28 publications

Sparse graphs with bounded induced cycle packing number have logarithmic treewidth

Sparse graphs with bounded induced cycle packing number have logarithmic treewidth

New Bounds for the Integer Carathéodory Rank

Advances on Strictly $Δ$-Modular IPs

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