Monte-Carlo Approximation Algorithms for Enumeration Problems

We develop polynomial time Monte-Carlo algorithms which produce good approximate solutions to enumeration problems for which it is known that the computation of the exact solution is very hard. We start by developing a Monte-Carlo approximation algorithm for the DNF counting problem, which is the problem of counting the number of satisfying truth assignments to a formula in disjunctive normal form. The input to the algorithm is the formula and two parameters E and 8. The algorithm produces an estimate which is between 1 E and 1 + E times the number of satisfying truth assignments with probability at least 1 6. The running time of the algorithm is linear in the length of the formula times l,/~~ times ln(l/S). On the other hand, the problem of computing the exact answer for the DNF counting problem is known to be #P-complete, which implies that there is no polynomial time algorithm for the exact solution if P + NP. This paper improves and gives new applications of some of the work previously reported. Variants of an E, S approximation algorithm for the DNF counting problem have been highly tailored to be especially efficient for the network reliability problems to which they are applied. In this paper the emphasis is on the development and analysis of a much more efficient c, 6 approximation algorithm for the DNF counting problem.