Coupling vs. conductance for the Jerrum-Sinclair chain

Abstract

We address the following question: is the Causal Coupling method as strong as the Conductance method in showing rapid mixing of Markov Chains? A causal coupling is a coupling which uses only past and present information, but not information about the future. We answer the above question in the negative by showing that there exists a bipartite graph G such that any causal coupling argument on the Jerrum-Sinclair Markov chain for sampling almost uniformly from the set of perfect and near perfect matchings of G must necessarily take time exponential in the number of vertices in G. In contrast, the above Markov chain on G has been shown to mix in polynomial time using conductance arguments.

Relational proofs for quantum programs

“…In the deterministic setting, relational Hoare logic can be shown to be relatively complete for terminating programs provided it includes sufficiently many one-sided rules. Relative completeness fails for probabilistic relational Hoare logic; a further potential complication is that the coupling method -upon which probabilistic relational Hoare logic builds -is itself not complete for proving convergence Markov chains [Anil Kumar and Ramesh 2001].…”

Relational proofs for quantum programs

“…In the deterministic setting, relational Hoare logic can be shown to be relatively complete for terminating programs provided it includes sufficiently many one-sided rules. Relative completeness fails for probabilistic relational Hoare logic; a further potential complication is that the coupling method -upon which probabilistic relational Hoare logic builds -is itself not complete for proving convergence Markov chains [Anil Kumar and Ramesh 2001].…”

Relational proofs for quantum programs

“…In the deterministic setting, relational Hoare logic can be shown to be relatively complete for terminating programs provided it includes sufficiently many one-sided rules. Relative completeness fails for probabilistic relational Hoare logic; a further potential complication is that the coupling method -upon which probabilistic relational Hoare logic builds -is itself not complete for proving convergence Markov chains [Anil Kumar and Ramesh 2001].…”

Relational proofs for quantum programs

“…In the deterministic setting, relational Hoare logic can be shown to be relatively complete for terminating programs provided it includes sufficiently many one-sided rules. Relative completeness fails for probabilistic relational Hoare logic; a further potential complication is that the coupling method -upon which probabilistic relational Hoare logic builds -is itself not complete for proving convergence Markov chains [Anil Kumar and Ramesh 2001].…”