2026-07-23T18:19:11Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:21056 2024-09-16T06:02:39Z ddc:004 open_access

Rapid Mixing of the Down-Up Walk on Matchings of a Fixed Size Jain, Vishesh Mizgerd, Clayton Down-up walk Matchings MCMC 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Vishesh Jain and Clayton Mizgerd 2024 Is Part Of LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.APPROX/RANDOM.2024.63 urn:nbn:de:0030-drops-210563 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.63 eng https://creativecommons.org/licenses/by/4.0/legalcode