The University of Chicago Faster Markov Chain Monte Carlo Algorithms for the Permanent and Binary Contingency Tables a Dissertation Submitted to the Faculty of the Division of the Physical Sciences in Candidacy for the Degree of Doctor of Philosophy

Random sampling and combinatorial counting are important building blocks in many practical applications. However, for some problems exact counting in deterministic polynomial-time is provably impossible (unless P = NP ), in which case the best hope is to devise suitable approximation algorithms. Markov chain Monte Carlo (MCMC) methods give efficient approximation algorithms for several important problems. This thesis presents the fastest known (approximation) algorithms for two such problems, the permanent and binary contingency tables. The permanent counts the number of (weighted) perfect matchings of a given bipartite graph, and it has applications in statistical physics, statistics, and several areas of computer science. Binary contingency tables count the number of bipartite graphs with prescribed degree sequences, and they are used in statistical tests in a number of fields including ecology, education and psychology. Our MCMC algorithms use the simulated annealing approach, a popular heuristic in combinatorial optimization. It utilizes intuition from the physical annealing process, where a substance is heated and slowly cooled so that it moves from a disordered configuration into a crystalline (i. e., ordered) state. In optimization, simulated annealing starts at an easy instance at high temperature, and slowly “cools” into the desired hard instance at low temperature. Simulated annealing is widely used, with much empirical success but little theoretical backing. This work presents a new cooling schedule for simulated annealing, with provable guarantees for several counting problems. We also improve results on the rate of convergence for the Markov chains which are at the core of the simulated annealing algorithm. As a consequence, we obtain a new algorithm, which for a graph with n vertices, approximates the permanent of a 0/1 matrix (within an arbitrarily close approximation) in time O∗(n7). We also present a new simulated annealing algorithm for binary contingency tables, which relies on an interesting combinatorial lemma.