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

Benjamin Nill

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.

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

Abstract

We prove a sharp upper bound on the number of distinct columns of a totally unimodular matrix with column sums 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.
Paper Structure (10 sections, 11 theorems, 13 equations)

This paper contains 10 sections, 11 theorems, 13 equations.

Table of Contents

  1. The column number bound on polytopal TU-matrices
  2. Applying the column number bound to unimodular polytopes
  3. Definition and relation to polytopal TU-matrices
  4. The maximal number of lattice points/vertices of unimodular polytopes
  5. The proof of the column number bound for polytopal TU-matrices
  6. Network matrices
  7. Transposes of network matrices
  8. On $k$-sums and $\Delta$-sums of TU-matrices
  9. Technical inequalities regarding the bound in \ref{['theorem:tu']}
  10. Proof of \ref{['theorem:tu']}

Key Result

Theorem 1.2

Let $M$ be a TU-matrix with $m$ rows and $n$ columns such that all columns are pairwise distinct. If there exists $f \in (\mathbb{Z}^{m})^*$ such that $f$ evaluates to a positive number on every column of $M$ (e.g., all column sums are positive), then $n$ is bounded from above by $\frac{(m+1)m}{2}$.

Theorems & Definitions (33)

  • Definition 1.1
  • Theorem 1.2: Heller's bound
  • Definition 1.3
  • Theorem 1.4
  • Example 1.5
  • Remark 1.6
  • Definition 2.1
  • Example 2.2
  • Remark 2.3
  • Proposition 2.4
  • ...and 23 more