Markov chains and rapid mixing

Abstract

This monograph is primarily concerned with positive results about counting and generating structures described by natural p-relations. With the exception of certain relations of unary type, where there is a considerable body of work on analytic (i.e., closed form) counting estimates, such results are rare in the kinds of interesting cases identified in Section 1.5. Ideally, we would like to have available some algorithmic paradigms with reasonably wide applicability. To this end, we investigate here a very general approach to generation problems. The approach is based on simulating a simple dynamic stochastic process, namely a finite Markov chain, which moves around a space containing the structures of interest and converges to some desired distribution on them. Since the relations we consider will almost always be self-reducible, the results we obtain carry over directly to the corresponding counting problems by virtue of the observations of Section 1.4.