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Box-total dual integrality, box-integrality, and equimodular matrices

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Abstract

Box-totally dual integral (box-TDI) polyhedra are polyhedra described by systems which yield strong min-max relations. We characterize them in several ways, involving the notions of principal box-integer polyhedra and equimodular matrices. A polyhedron is box-integer if its intersection with any integer box \(\{\ell \le x \le u\}\) is integer. We define principally box-integer polyhedra to be the polyhedra P such that \( kP \) is box-integer whenever \( kP \) is integer. A rational \(r\times n\) matrix is equimodular if it has full row rank and its nonzero \(r\times r\) determinants all have the same absolute value. A face-defining matrix is a full row rank matrix describing the affine hull of a face of the polyhedron. Our main result is that the following statements are equivalent.

  • The polyhedron P is box-TDI.

  • The polyhedron P is principally box-integer.

  • Every face-defining matrix of P is equimodular.

  • Every face of P has an equimodular face-defining matrix.

  • Every face of P has a totally unimodular face-defining matrix.

  • For every face F of P, \(\mathrm{lin}(F)\) has a totally unimodular basis.

Along our proof, we show that a polyhedral cone is box-TDI if and only if it is box-integer, and that these properties are carried over to its polar. We illustrate these charaterizations by reviewing well known results about box-TDI polyhedra. We also provide several applications. The first one is a new perspective on the equivalence between two results about binary clutters. Secondly, we refute a conjecture of Ding, Zang, and Zhao about box-perfect graphs. Thirdly, we discuss connections with an abstract class of polyhedra having the Integer Carathéodory Property. Finally, we characterize the box-TDIness of the cone of conservative functions of a graph and provide a corresponding box-TDI system.

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Notes

  1. In the standard definition, the emptyset is a face. It is not the case in this paper in order to lighten the statements.

  2. When we write that a face F has a face-defining matrix M, we mean that M is face-defining for the face F, which is more restrictive than being a face-defining matrix of the polyhedron F.

  3. Between the submission and the publication of this paper, the question was answered in [4].

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Acknowledgements

We are grateful to András Sebő for his invaluable comments and suggestions. We also thank the referees for their very careful reading and useful suggestions.

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Authors and Affiliations

  1. Lycée Olympe de Gouges, rue de Montreuil à Claye, 93130, Noisy le Sec, France

    Patrick Chervet

  2. LIPN, CNRS, UMR 7030, Université Sorbonne Paris Nord, 93430, Villetaneuse, France

    Roland Grappe

  3. Université de Genève, rue du Lièvre 2-4, 1227, Geneva, Switzerland

    Louis-Hadrien Robert

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  1. Patrick Chervet
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Correspondence to Roland Grappe.

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Louis-Hadrien Robert: Supported by the NCCR SwissMAP, funded by the Swiss National Science Foundation.

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Chervet, P., Grappe, R. & Robert, LH. Box-total dual integrality, box-integrality, and equimodular matrices. Math. Program. 188, 319–349 (2021). https://doi.org/10.1007/s10107-020-01514-0

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