On the Expansion of Graphs of 0/1-Polytopes

TL;DR

This paper investigates the edge expansion of graphs of 0/1-polytopes, providing new bounds and proving the conjecture for several classes, which has implications for randomized algorithms and combinatorial optimization.

Contribution

The paper introduces novel techniques for bounding edge expansion and proves the conjecture for multiple classes of 0/1-polytopes, including all polytopes of dimension up to five.

Findings

All 0/1-polytopes of dimension at most five have edge expansion at least one.

Simple 0/1-polytopes, hypersimplices, stable set polytopes, and perfect matching polytopes have edge expansion at least one.

Abstract

The edge expansion of a graph is the minimum quotient of the number of edges in a cut and the size of the smaller one among the two node sets separated by the cut. Bounding the edge expansion from below is important for bounding the ``mixing time'' of a random walk on the graph from above. It has been conjectured by Mihail and Vazirani that the graph of every 0/1-polytope has edge expansion at least one. A proof of this (or even a weaker) conjecture would imply solutions of several long-standing open problems in the theory of randomized approximate counting. We present different techniques for bounding the edge expansion of a 0/1-polytope from below. By means of these tools we show that several classes of 0/1-polytopes indeed have graphs with edge expansion at least one. These classes include all 0/1-polytopes of dimension at most five, all simple 0/1-polytopes, all hypersimplices, all stable set polytopes, and all (perfect) matching polytopes.