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arXiv · 2405.13431

Unimodular polytopes and column number bounds on polytopal totally unimodular matrices via Seymour's decomposition theorem

Benjamin Nill
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Abstract

We prove a sharp upper bound on the number of distinct columns of a totally unimodular matrix with column sums $1$ improving upon Heller's classical bound. The proof uses Seymour's decomposition theorem. Such matrices are closely related to unimodular polytopes: lattice polytopes where the vertices of every full-dimensional subsimplex form an affine lattice basis. This is an interesting subclass of 0/1-polytopes and contains for instance edge polytopes of bipartite graphs. Our main result on totally unimodular matrices implies a sharp upper bound on the number of vertices of unimodular polytopes.

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Benjamin Nill. 2024-05-22. Unimodular polytopes and column number bounds on polytopal totally unimodular matrices via Seymour's decomposition theorem. https://doi.org/10.1137/24m1690904

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