Random 0/1-polytopes expand rapidly

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 $Θ(n)$ for $p \in (1/2, 1)$, and $n^{Θ(\log \log n)}$ for $p \in (0, 1/2)$. This improves the previously best known bound $Ω(1)$ due to Ferber, Krivelevich, Sales and Samotij.

• The paper establishes strong lower bounds on edge-expansion using probabilistic multicommodity flows and explicit path constructions.
• It demonstrates that for p > 1/2, the expansion grows linearly with n, while for smaller p, it achieves super-polynomial rates.
• These findings imply rapid mixing for random walks and enhanced performance for algorithms on high-dimensional random 0/1-polytopes.

Rapid Edge Expansion in Random 0/1-Polytopes

Introduction and Context

The study focuses on a fundamental property of 0/1-polytopes: the edge-expansion (Cheeger constant) of their graphs (1-skeletons). This concept is crucial for both the structural understanding of polytopal graphs and their algorithmic applications, such as analyzing the mixing of random walks on combinatorial structures and the efficiency of algorithms like the Simplex method. The Mihail–Vazirani conjecture posits that every 0/1-polytope has edge-expansion at least 1, a property verified for several specific families of polytopes but open in general.

This work addresses the behavior of edge-expansion in random 0/1-polytopes generated by independently sampling each vertex of $\{0,1\}^n$ with probability $p$. Previous results, specifically those of Ferber, Krivelevich, Sales, and Samotij, proved that edge-expansion is at least $\Omega(1)$ with high probability but could not confirm the conjecture's prediction for arbitrary (random) 0/1-polytopes (Ferber et al., 11 Sep 2025). The present paper demonstrates that, with high probability, random 0/1-polytopes not only meet the conjectured bound but, in most regimes, vastly exceed it: the edge-expansion is typically $\Theta(n)$ or even super-polynomial.

Main Results

The principal findings establish strong, high-probability lower bounds on the edge-expansion $h(G)$ of the graph $G$ of a random 0/1-polytope:

• For $p \in (1/2, 1)$, $h(G) = \Theta(n)$ with high probability.
• For $p \in (n^{-0.05}, 1/2)$, $h(G) = n^{\Theta(\log\log n + \log(1/p))}$ with high probability.

These statements significantly improve previous bounds, confirming the Mihail–Vazirani conjecture for this case and solving questions raised in (Ferber et al., 11 Sep 2025). The analysis also covers much sparser sampling via projection arguments and previous results, confirming super-polynomial expansion for $p$0 as low as $p$1.

Technical Approach

The core methodological advance is the construction of probabilistic multicommodity flows of low maximum congestion in the graphs of random 0/1-polytopes. The approach starts by leveraging the high symmetry of the Hamming-distance-$p$2 graph $p$3, constructing an explicit A-flow (all-pairs unit-demand multicommodity flow) and showing that, with appropriate path selection, every edge supports a bounded amount of flow. The key steps are:

• Flow Construction in the Hamming Graph: The paper demonstrates that $p$4 admits an A-flow with congestion at most $p$5, yielding edge-expansion $p$6.
• Embedding into Polytopal Graphs: It is shown that a substantial fraction of $p$7's structure is preserved in $p$8 for random $p$9, specifically via subgraphs $\Omega(1)$0. The authors replace edges in $\Omega(1)$1 by short paths ("pure paths") of length 7 in $\Omega(1)$2, carefully controlling congestion and preserving expansion up to constant factors.
• Management of Bottlenecks: The proofs account for rare vertices with small complements in $\Omega(1)$3 (the so-called $\Omega(1)$4-full vertices), which could serve as local bottlenecks. Such configurations are shown to be statistically negligible or manageable, depending on the regime for $\Omega(1)$5.
• Robust Concentration: The use of sharp concentration inequalities (Chernoff, McDiarmid's), combinatorial analysis, and path rerouting ensures that the main lower bounds are not destroyed by atypical randomness in $\Omega(1)$6.

Strong Numerical and Qualitative Results

The paper establishes that for $\Omega(1)$7 bounded away from zero and $\Omega(1)$8, the edge-expansion of a random 0/1-polytope's graph rises rapidly with $\Omega(1)$9, greatly surpassing the universal conjectured lower bound of 1. Specifically:

• For $\Theta(n)$0: With high probability, $\Theta(n)$1, implying linear expansion and exceptionally strong connectivity, which is a strict strengthening over the Mihail–Vazirani conjecture.
• For $\Theta(n)$2 polynomially small in $\Theta(n)$3 (but still $\Theta(n)$4): $\Theta(n)$5 is super-polynomial, with lower bounds matching $\Theta(n)$6.
• For $\Theta(n)$7 down to exponential thresholds: By building on e.g. (Babecki et al., 3 Jul 2025), the expansion becomes super-polynomial or the graph degenerates to a clique, recovering precise structural transitions.

Such numerical improvements are not only significant theoretically but provide concrete guarantees for the rapid mixing of Markov chains and the typical geometry of random 0/1-polytopes.

Implications and Future Directions

These results have several notable theoretical and practical ramifications:

• Algorithmic Random Walks: The demonstrated high edge-expansion guarantees rapid mixing for random walks on these graphs, supporting efficient randomized approximation schemes for counting and sampling combinatorial structures.
• Random Polytopal Geometry: The findings imply that high-dimensional random 0/1-polytopes almost never have narrow bottlenecks, but are instead "well-connected," confirming conjectures about their typical geometry.
• Broader Expansion Conjectures: The probabilistic techniques might generalize to more intricate families of polytopes or suggest new directions for the deterministic Mihail–Vazirani conjecture.
• Sparse Regimes: Extending sharp expansion bounds to regimes where $\Theta(n)$8 is extremely small (e.g., $\Theta(n)$9 or $h(G)$0) remains a technically rich direction, with links to extremal combinatorics and phase transitions.
• Path-Following Algorithms: Results implicate that path-following algorithms over random 0/1-polytopes operate in exceptionally well-connected graphs, potentially leading to concrete algorithmic performance improvements.

Conclusion

The work decisively advances the quantitative understanding of edge-expansion in random 0/1-polytopes, not only confirming prior conjectures in typical cases but showing strongly enhanced expansion almost everywhere in the random regime. The probabilistic constructions and flow-based arguments introduced are widely applicable and represent a significant technical contribution to polyhedral combinatorics, probabilistic geometry, and randomized algorithms (2604.09520).