A Randomized Fully Polynomial Time Approximation Scheme for the All-Terminal Network Reliability Problem

Abstract

The classic all-terminal network reliability problem posits a graph, each of whose edges fails independently with some given probability. The goal is to determine the probability that the network becomes disconnected due to edge failures. This problem has obvious applications in the design of communication networks. Since the problem is ${\sharp {\cal P}}$-complete and thus believed hard to solve exactly, a great deal of research has been devoted to estimating the failure probability. In this paper, we give a fully polynomial randomized approximation scheme that, given any n-vertex graph with specified failure probabilities, computes in time polynomial in n and $1/\epsilon$ an estimate for the failure probability that is accurate to within a relative error of $1\pm\epsilon$ with high probability. We also give a deterministic polynomial approximation scheme for the case of small failure probabilities. Some extensions to evaluating probabilities of $k$-connectivity, strong connectivity in directed Eulerian graphs and $r$-way disconnection, and to evaluating the Tutte polynomial are also described. This version of the paper corrects several errata that appeared in the previous journal publication [D. R. Karger, SIAM J. Comput., 29 (1999), pp. 492--514].