Expansion of random $0/1$ polytopes

TL;DR

This paper proves that the edge expansion of the graph of a random 0/1 polytope in d-dimensional space is at least 1/(12d) with high probability, supporting a weaker version of a conjecture related to random walks.

Contribution

It establishes a probabilistic lower bound on the edge expansion of random 0/1 polytopes, advancing understanding of their combinatorial properties.

Findings

Edge expansion of random 0/1 polytope graph ≥ 1/(12d) with high probability

Supports a weaker form of Mihail and Vazirani's conjecture

Implications for rapid mixing of random walks on these graphs

Abstract

A conjecture of Mihail and Vazirani states that the edge expansion of the graph of every $0/1$ polytope is at least one. Any lower bound on the edge expansion gives an upper bound for the mixing time of a random walk on the graph of the polytope. Such random walks are important because they can be used to generate an element from a set of combinatorial objects uniformly at random. A weaker form of the conjecture of Mihail and Vazirani says that the edge expansion of the graph of a $0/1$ polytope in $\mathbb{R}^d$ is greater than 1 over some polynomial function of $d$. This weaker version of the conjecture would suffice for all applications. Our main result is that the edge expansion of the graph of a $\textit{random}$ $0/1$ polytope in $\mathbb{R}^d$ is at least $\frac{1}{12d}$ with high probability.