The Diameters of Network-flow Polytopes satisfy the Hirsch Conjecture

TL;DR

This paper proves that the diameter of network-flow polytopes, including transportation polytopes, always satisfies the Hirsch conjecture, with a maximum diameter of m+n-1 for networks with n nodes and m arcs.

Contribution

It establishes the Hirsch conjecture for the diameters of all network-flow polytopes, a significant advance in combinatorial polytope theory.

Findings

Hirsch conjecture holds for network-flow polytope diameters

Maximum diameter is m+n-1 for networks with n nodes and m arcs

Proof includes classical transportation polytopes as a special case

Abstract

We solve a problem in the combinatorics of polyhedra motivated by the network simplex method. We show that the Hirsch conjecture holds for the diameter of the graphs of all network-flow polytopes, in particular the diameter of a network-flow polytope for a network with $n$ nodes and $m$ arcs is never more than $m+n-1$. A key step to prove this is to show the same result for classical transportation polytopes.