Edge-Graph Diameter Bounds for Convex Polytopes with Few Facets

TL;DR

This paper proves the Hirsch conjecture for polytopes with a small difference between facets and dimension by establishing diameter bounds for 6-dimensional polytopes with 12 facets, using matroid polytopes and satisfiability reductions.

Contribution

It verifies the d-step conjecture for d=6 and extends the diameter bound to all polytopes with n-d ; it also introduces a reduction to satisfiability problems for matroid polytopes.

Findings

Diameter of 6-polytope with 12 facets is at most 6

Hirsch conjecture holds for n-d  6

Reduction to satisfiability problems for proof

Abstract

We show that the edge graph of a 6-dimensional polytope with 12 facets has diameter at most 6, thus verifying the d-step conjecture of Klee and Walkup in the case of d=6. This implies that for all pairs (d,n) with n-d \leq 6 the diameter of the edge graph of a d-polytope with n facets is bounded by 6, which proves the Hirsch conjecture for all n-d \leq 6. We show this result by showing this bound for a more general structure -- so-called matroid polytopes -- by reduction to a small number of satisfiability problems.