Markovian coupling vs. conductance for the Jerrum-Sinclair chain

We show that no Markovian coupling argument can prove rapid mixing of the Jerrum-Sinclair Markov chain for sampling almost uniformly from the set of perfect and near perfect matchings of a given graph. In particular, we show that there exists a bipartite graph G such that any Markovian coupling argument on the Jerrum-Sinclair Markov chain for G must necessarily take time exponential in the number of vertices in G. This holds even when the coupling argument is time-variant, i.e., the transition probabilities used by the coupling process depend upon the history of the process. In contrast, the above Markov chain on G has been shown to mix in polynomial time using conductance arguments.