Broder's Chain Is Not Rapidly Mixing

Abstract

We prove that Broder's Markov chain for approximate sampling near-perfect and perfect matchings is not rapidly mixing for Hamiltonian, regular, threshold and planar bipartite graphs, filling a gap in the literature. In the second part we experimentally compare Broder's chain with the Markov chain by Jerrum, Sinclair and Vigoda from 2004. For the first time, we provide a systematic experimental investigation of mixing time bounds for these Markov chains. We observe that the exact total mixing time is in many cases significantly lower than known upper bounds using canonical path or multicommodity flow methods, even if the structure of an underlying state graph is known. In contrast we observe comparatively tighter upper bounds using spectral gaps.