Accelerating Simulated Annealing for the Permanent and Combinatorial Counting Problems

We present an improved “cooling schedule” for simulated annealing algorithms for combinatorial counting problems. Under our new schedule the rate of cooling accelerates as the temperature decreases. Thus, fewer intermediate temperatures are needed as the simulated annealing algorithm moves from the high temperature (easy region) to the low temperature (difficult region). We present applications of our technique to colorings and the permanent (perfect matchings of bipartite graphs). Moreover, for the permanent, we improve the analysis of the Markov chain underlying the simulated annealing algorithm. This improved analysis, combined with the faster cooling schedule, results in an $O(n^7\log^4{n})$ time algorithm for approximating the permanent of a $0/1$ matrix.