research-article

A strongly polynomial algorithm for bimodular integer linear programming

Authors:

Stephan Artmann,

Robert Weismantel,

Rico ZenklusenAuthors Info & Claims

STOC 2017: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing

Pages 1206 - 1219

https://doi.org/10.1145/3055399.3055473

Published: 19 June 2017 Publication History

Abstract

We present a strongly polynomial algorithm to solve integer programs of the form max{c^T x: Ax≤ b, xεℤⁿ }, for Aεℤ^mXn with rank(A)=n, bε≤^m, cε≤ⁿ, and where all determinants of (nXn)-sub-matrices of A are bounded by 2 in absolute value. In particular, this implies that integer programs max{c^T x : Q x≤ b, xεℤ_≥0ⁿ}, where Qε ℤ^mXn has the property that all subdeterminants are bounded by 2 in absolute value, can be solved in strongly polynomial time. We thus obtain an extension of the well-known result that integer programs with constraint matrices that are totally unimodular are solvable in strongly polynomial time.

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Cited By

Fiorini SJoret GWeltge SYuditsky Y(2025)Integer programs with bounded subdeterminants and two nonzeros per rowJournal of the ACM10.1145/369598572:1(1-50)Online publication date: 24-Jan-2025
https://dl.acm.org/doi/10.1145/3695985
Zhang YXia YZhan Y(2024)Total Unimodularity and Strongly Polynomial Solvability of Constrained Minimum Input Selections for Structural Controllability: An LP-Based MethodIEEE Transactions on Automatic Control10.1109/TAC.2023.326624269:1(387-394)Online publication date: Jan-2024
https://doi.org/10.1109/TAC.2023.3266242
Paat JWalsh ZXu L(2024)Extended formulations for the integer hull of strictly Δ-modular cographic polyhedral conesOperations Research Letters10.1016/j.orl.2024.107235(107235)Online publication date: Dec-2024
https://doi.org/10.1016/j.orl.2024.107235
Show More Cited By

Index Terms

A strongly polynomial algorithm for bimodular integer linear programming
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
      1. Combinatorial optimization
2. Theory of computation
  1. Design and analysis of algorithms
    1. Mathematical optimization
      1. Discrete optimization
      2. Mixed discrete-continuous optimization
        Integer programming
        Submodular optimization and polymatroids

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cover image ACM Conferences

STOC 2017: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing

June 2017

1268 pages

ISBN:9781450345286

DOI:10.1145/3055399

General Chairs:
Hamed Hatami
McGill University
,
Pierre McKenzie
Université de Montréal
,
Program Chair:
Valerie King
University of Victoria

Copyright © 2017 ACM.

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SIGACT: ACM Special Interest Group on Algorithms and Computation Theory

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New York, NY, United States

Publication History

Published: 19 June 2017

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Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung

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STOC '17

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STOC '17: Symposium on Theory of Computing

June 19 - 23, 2017

Montreal, Canada

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Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

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STOC '25

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57th Annual ACM Symposium on Theory of Computing (STOC 2025)

June 23 - 27, 2025

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Cited By

Fiorini SJoret GWeltge SYuditsky Y(2025)Integer programs with bounded subdeterminants and two nonzeros per rowJournal of the ACM10.1145/369598572:1(1-50)Online publication date: 24-Jan-2025
https://dl.acm.org/doi/10.1145/3695985
Zhang YXia YZhan Y(2024)Total Unimodularity and Strongly Polynomial Solvability of Constrained Minimum Input Selections for Structural Controllability: An LP-Based MethodIEEE Transactions on Automatic Control10.1109/TAC.2023.326624269:1(387-394)Online publication date: Jan-2024
https://doi.org/10.1109/TAC.2023.3266242
Paat JWalsh ZXu L(2024)Extended formulations for the integer hull of strictly Δ-modular cographic polyhedral conesOperations Research Letters10.1016/j.orl.2024.107235(107235)Online publication date: Dec-2024
https://doi.org/10.1016/j.orl.2024.107235
Gribanov DShumilov IMalyshev DZolotykh N(2024)Faster algorithms for sparse ILP and hypergraph multi-packing/multi-cover problemsJournal of Global Optimization10.1007/s10898-024-01379-z89:4(1033-1067)Online publication date: 1-Aug-2024
https://dl.acm.org/doi/10.1007/s10898-024-01379-z
Gribanov DShumilov IMalyshev DPardalos P(2024)On -modular integer linear problems in the canonical form and equivalent problemsJournal of Global Optimization10.1007/s10898-022-01165-988:3(591-651)Online publication date: 1-Mar-2024
https://dl.acm.org/doi/10.1007/s10898-022-01165-9
Nägele MNöbel CSantiago RZenklusen R(2024)Advances on strictly $$\Delta $$-modular IPsMathematical Programming10.1007/s10107-024-02148-2Online publication date: 30-Oct-2024
https://doi.org/10.1007/s10107-024-02148-2
Ferrarini LFiorini SKober SYuditsky Y(2024)Total Matching and SubdeterminantsCombinatorial Optimization10.1007/978-3-031-60924-4_15(192-204)Online publication date: 22-May-2024
https://doi.org/10.1007/978-3-031-60924-4_15
Celaya MKuhlmann SWeismantel R(2024)On Matrices over a Polynomial Ring with Restricted SubdeterminantsInteger Programming and Combinatorial Optimization10.1007/978-3-031-59835-7_4(43-56)Online publication date: 22-May-2024
https://doi.org/10.1007/978-3-031-59835-7_4
Liu SXu C(2024)On the Congruency-Constrained Matroid BaseInteger Programming and Combinatorial Optimization10.1007/978-3-031-59835-7_21(280-293)Online publication date: 3-Jul-2024
https://dl.acm.org/doi/10.1007/978-3-031-59835-7_21
Papalamprou KPitsoulis LKotnyek B(2023)A mathematical programming approach for recognizing binet matricesOptimization Letters10.1007/s11590-023-02066-w18:6(1511-1532)Online publication date: 6-Oct-2023
https://doi.org/10.1007/s11590-023-02066-w
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