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On a generalization of the Chvátal–Gomory closure

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Abstract

Many practical integer programming problems involve variables with one or two-sided bounds. Dunkel and Schulz (A refined Gomory–Chvátal closure for polytopes in the unit cube, http://www.optimization-online.org/DB_FILE/2012/03/3404.pdf, 2012) considered a strengthened version of Chvátal–Gomory (CG) inequalities that use 0–1 bounds on variables, and showed that the set of points in a rational polytope that satisfy all these strengthened inequalities is a polytope. Recently, we generalized this result by considering strengthened CG inequalities that use all variable bounds. In this paper, we generalize further by considering not just variable bounds, but general linear constraints on variables. We show that all points in a rational polyhedron that satisfy such strengthened CG inequalities form a rational polyhedron. We also extend this polyhedrality result to mixed-integer sets defined by linear constraints.

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Acknowledgements

We would like to thank two anonymous referees for their valuable feedback on this paper, and we also thank two anonymous referees for the IPCO version [25] of this paper.

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Authors and Affiliations

  1. IBM T.J. Watson Research Center, Yorktown Heights, NY, 10598, USA

    Sanjeeb Dash

  2. School of ORIE, Cornell University, Ithaca, NY, 14850, USA

    Oktay Günlük

  3. Discrete Math Group, Institute for Basic Science (IBS), Daejeon, 34126, Republic of Korea

    Dabeen Lee

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  1. Sanjeeb Dash
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Correspondence to Dabeen Lee.

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An extended abstract of this paper was published in the proceedings of the 21st International Conference on Integer Programming and Combinatorial Optimization (IPCO 2020) [25]. This research was supported, in part, by the Institute for Basic Science (IBS-R029-C1, IBS-R029-Y2).

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Dash, S., Günlük, O. & Lee, D. On a generalization of the Chvátal–Gomory closure. Math. Program. 192, 149–175 (2022). https://doi.org/10.1007/s10107-021-01697-0

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