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arXiv · 1501.07725

On the switch Markov chain for perfect matchings

Martin Dyer·Mark Jerrum·Haiko Müller
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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 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.

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Martin Dyer, Mark Jerrum, Haiko Müller. 2017-01-26. On the switch Markov chain for perfect matchings. https://arxiv.org/abs/1501.07725

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