On the Switch Markov Chain for Perfect Matchings

Author:

Dyer Martin1,Jerrum Mark2,Müller Haiko1

Affiliation:

1. University of Leeds, UK

2. Queen Mary University of London, UK

Abstract

We study a simple Markov chain, the switch chain, on the set of all perfect matchings in a bipartite graph. This Markov chain was proposed by Diaconis, Graham and Holmes as a possible approach to a sampling problem arising in Statistics. We ask: for which hereditary classes of graphs is the Markov chain ergodic and for which is it rapidly mixing? We provide a precise answer to the ergodicity question and close bounds on the mixing question. We show for the first time that the mixing time of the switch chain is polynomial in the case of monotone graphs, a class that includes examples of interest in the statistical setting.

Funder

EPSRC grants “Computational Counting”

“Randomized algorithms for computer networks”

Publisher

Association for Computing Machinery (ACM)

Subject

Artificial Intelligence,Hardware and Architecture,Information Systems,Control and Systems Engineering,Software

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1. Zero-freeness and approximation of real Boolean Holant problems;Theoretical Computer Science;2022-05

2. Rapid Mixing of the Switch Markov Chain for 2-Class Joint Degree Matrices;SIAM Journal on Discrete Mathematics;2022-01-06

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