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Rapid Mixing of the Down-Up Walk on Matchings of a Fixed Size

Authors Vishesh Jain , Clayton Mizgerd

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ACM Subject Classification
  • Mathematics of computing → Markov-chain Monte Carlo methods
Keywords
  • Down-up walk
  • Matchings
  • MCMC

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Abstract

Let G = (V,E) be a graph on n vertices and let m^*(G) denote the size of a maximum matching in G. We show that for any δ > 0 and for any 1 ≤ k ≤ (1-δ)m^*(G), the down-up walk on matchings of size k in G mixes in time polynomial in n. Previously, polynomial mixing was not known even for graphs with maximum degree Δ, and our result makes progress on a conjecture of Jain, Perkins, Sah, and Sawhney [STOC, 2022] that the down-up walk mixes in optimal time O_{Δ,δ}(nlog{n}). 
In contrast with recent works analyzing mixing of down-up walks in various settings using the spectral independence framework, we bound the spectral gap by constructing and analyzing a suitable multi-commodity flow. In fact, we present constructions demonstrating the limitations of the spectral independence approach in our setting.

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Vishesh Jain and Clayton Mizgerd. Rapid Mixing of the Down-Up Walk on Matchings of a Fixed Size. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 63:1-63:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.63

Author Details

Vishesh Jain
  • Department of Mathematics, Statistics, and Computer Science, University of Illinois Chicago, Chicago, IL, USA
Clayton Mizgerd
  • Department of Mathematics, Statistics, and Computer Science, University of Illinois Chicago, Chicago, IL, USA

Funding

  • Jain, Vishesh: NSF CAREER award DMS-2237646
  • Mizgerd, Clayton: NSF award ECCS-2217023

Acknowledgements

We thank Huy Tuan Pham for helpful discussions and anonymous referees for several useful suggestions.

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