Canonical Paths for MCMC: from Art to Science

Markov Chain Monte Carlo (MCMC) method is a widely used algorithm design scheme with many applications. To make efficient use of this method, the key step is to prove that the Markov chain is rapid mixing. Canonical paths is one of the two main tools to prove rapid mixing. However, there are much fewer success examples comparing to coupling, the other main tool. The main reason is that there is no systematic approach or general recipe to design canonical paths. Building up on a previous exploration by McQuillan, we develop a general theory to design canonical paths for MCMC: We reduce the task of designing canonical paths to solving a set of linear equations, which can be automatically done even by a machine. Making use of this general approach, we obtain fully polynomial-time randomized approximation schemes (FPRAS) for counting the number of $b$-matching with $b\leq 7$ and $b$-edge-cover with $b\leq 2$. They are natural generalizations of matchings and edge covers for graphs. No polynomial time approximation was previously known for these problems.