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

TL;DR

This paper establishes a sharp upper bound on the number of columns in totally unimodular matrices with column sums of one, using Seymour's decomposition theorem, and applies this to bound vertices of unimodular polytopes.

Contribution

It introduces a new sharp bound on columns of totally unimodular matrices and relates this to vertex bounds of unimodular polytopes, leveraging Seymour's decomposition theorem.

Findings

Bound on columns of totally unimodular matrices with column sum 1

Bound on vertices of unimodular polytopes

Application to bipartite graph edge polytopes

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.