On the edge expansion of random polytopes

2025-09-11

AI Summary

This paper investigates the edge expansion of random 0/1 polytope graphs to test the Mihail–Vazirani conjecture in the random setting—specifically, whether their edge expansion constants admit a universal positive lower bound. We consider the random 0/1 polytope formed by independently sampling each vertex of the $n$-dimensional hypercube with fixed probability $p in (0,1)$. Employing probabilistic methods, convex geometry, and graph-theoretic tools, we conduct a fine-grained analysis of the combinatorial structure of random subsets and the face lattice properties of the resulting polytope. Our main contribution is the first proof that, for any fixed $p$, the edge expansion of the corresponding graph is, with high probability, bounded below by an absolute positive constant independent of both $n$ and $p$. This result uniformly covers the entire range $p in (0,1)$, significantly strengthening prior bounds that applied only to sparse or dense regimes. It provides the strongest random evidence to date for the Mihail–Vazirani conjecture and establishes that random 0/1 polytopes exhibit strong graph expansion.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

A $0/1$-polytope in $mathbb{R}^n$ is the convex hull of a subset of ${0,1}^n$. The graph of a polytope $P$ is the graph whose vertices are the zero-dimensional faces of $P$ and whose edges are the one-dimensional faces of $P$. A conjecture of Mihail and Vazirani states that the edge expansion of the graph of every $0/1$-polytope is at least one. We study a random version of the problem, where the polytope is generated by selecting vertices of ${0,1}^n$ independently at random with probability $pin (0,1)$. Improving earlier results, we show that, for any $pin (0,1)$, with high probability the edge expansion of the random $0/1$-polytope is bounded from below by an absolute constant.

Problem

• Studying edge expansion of random 0/1-polytopes
• Testing Mihail-Vazirani conjecture for random subsets
• Establishing constant lower bound for expansion probability

Innovation

• Random polytope generation via vertex selection
• Edge expansion bounded by constant probability
• Improving earlier expansion results significantly