Direct Applications

Abstract

We have seen in Chapter 2 that the Markov chain simulation paradigm provides an elegant general approach to generation problems, and have developed some theoretical machinery for analysing the efficiency of the resulting algorithms. The purpose of this chapter is to demonstrate the utility of the approach by applying it to some concrete and non-trivial examples. We shall show how to generate various combinatorial structures by constructing suitable ergodic Markov chains having the structures as states and transitions corresponding to simple local perturbations of the structures. The rate of convergence will be investigated using the techniques of Chapter 2, and in particular the rapid mixing characterisation of Corollary 2.8. In each case, the detailed structure of the Markov chain will enable us to estimate the conductance of its underlying graph, and we develop a useful general methodology for doing this. Our results constitute apparently the first demonstrations of rapid mixing for Markov chains with genuinely complex structure. As corollaries, we deduce the existence of efficient approximation algorithms for two significant #P-complete counting problems.