The techniques for approximately counting the number of solutions to #P-complete problems are quite different from those for obtaining approximation algorithms for NP-hard optimization problems. Much of the former theory is built around the Markov chain Monte Carlo method, see Section 28.4 for references. In this chapter, we will present combinatorial algorithms (not using Markov chains) for two fundamental problems, counting the number of satisfying truth assignments for a DNF formula, and estimating the failure probability of an undirected network.
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