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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.

Cite As Get BibTex

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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