Binary scalar products

Kupavskii, Andrey; Weltge, Stefan

doi:10.48550/arxiv.2008.07153

Cited by 1 publication

(1 citation statement)

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“…Polytopes that have a 0/1-valued slack matrix are called 2-level polytopes. These form a rich class of polytopes including stable set polytopes of perfect graphs, Birkhoff and Hanner polytopes, and stable matching polytopes (see [1,2,19] for more examples), and they have been the topic of many recent investigations (see [15,18]). In the second part of the paper, we use the tools developed in the first part to shed some light on this intriguing class.…”

Section: Introductionmentioning

confidence: 99%

Slack matrices, $k$-products, and $2$-level polytopes

Aprile¹,

Conforti²,

Faenza³

et al. 2021

Preprint

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In this paper, we study algorithmic questions concerning products of matrices and their consequences for recognition algorithms for polyhedra.The 1-product of matriceswhose columns are the concatenation of each column of S1 with each column of S2. The k-product generalizes the 1-product, by taking as input two matrices S1, S2 together with k − 1 special rows of each of those matrices, and outputting a certain composition of S1, S2.Our first result is a polynomial time algorithm for the following problem: given a matrix S, is S a k-product of some matrices, up to permutation of rows and columns? Our algorithm is based on minimizing a symmetric submodular function that expresses mutual information from information theory.Our study is motivated by a close link between the 1-product of matrices and the Cartesian product of polytopes, and more generally between the k-product of matrices and the glued product of polytopes. These connections rely on the concept of slack matrix, which gives an algebraic representation of classes of affinely equivalent polytopes. The slack matrix recognition problem is the problem of determining whether a given matrix is a slack matrix. This is an intriguing problem whose complexity is unknown. Our algorithm reduces the problem to instances which cannot be expressed as k-products of smaller matrices.In the second part of the paper, we give a combinatorial interpretation of k-products for two well-known classes of polytopes: 2-level matroid base polytopes and stable set polytopes of perfect graphs. We also show that the slack matrix recognition problem is polynomial-time solvable for such polytopes. Those two classes are special cases of 2-level polytopes, for which we conjecture that the slack matrix recognition problem is polynomial-time solvable.

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Section: Introductionmentioning

confidence: 99%