Random 0/1-polytopes expand rapidly

TL;DR

This paper demonstrates that random 0/1-polytopes typically exhibit much stronger edge-expansion than previously conjectured, with expansion rates depending on the sampling probability.

Contribution

It establishes new probabilistic bounds showing that random 0/1-polytopes have significantly higher edge-expansion than the longstanding conjecture suggested.

Findings

Edge-expansion is (n) for p ter 1/2

Edge-expansion is n^{(\u00a0log log n)} for p ter 0

Improves previous bound of (1) for typical 0/1-polytopes

Abstract

A 0/1-polytope is the convex hull of a subset $V\subseteq \{0,1\}^n$. A celebrated conjecture of Mihail and Vazirani asserts that the graph of every 0/1-polytope has edge-expansion at least 1. In this paper, we show that typical 0/1-polytopes have significantly stronger expansion. Specifically, if $V$ is formed by sampling each vertex of $\{0,1\}^n$ independently with constant probability $p$, then with high probability the edge-expansion is $\Theta(n)$ for $p \in (1/2, 1)$, and $n^{\Theta(\log \log n)}$ for $p \in (0, 1/2)$. This improves the previously best known bound $\Omega(1)$ due to Ferber, Krivelevich, Sales and Samotij.