The complexity of approximate counting

There are several computational problems that can be formulated as problems of counting the number of objects having a certain property. Valiant [22] has introduced the class #P which includes a variety of counting problems such as counting the number of perfect matchings in a graph, computing the permanent of a matrix [22], finding the size of a backtrack search tree [14], and computing the probability that a network remains connected when links can fail with a certain probability [23].