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High-confidence estimation of small s-t reliabilities in directed acyclic networks



In the classical s-t network reliability problem, a network G is given with two designated vertices s and t. The arcs are subject to independent random failures, and the task is to compute the probability that s and t are connected in the resulting network. This probability is called the s-t reliability. We consider the problem of estimating the s-t reliability in a directed acyclic network. This problem is known to be #P-complete. Following an importance sampling idea introduced by Karp and Luby (J Complexity 1 (1985), 45–64), we design a Monte Carlo algorithm and show that by carefully exploiting the acyclicity of the network, it is possible to accurately estimate small s-t reliabilities for very large acyclic networks. For the case of equal arc failure probabilities, we give a worst-case bound on the number of samples that have to be drawn to obtain an (ε,δ)-approximation that is sharper than the upper bound presented by Karp and Luby (J Complexity 1 (1985), 45–64). Computational results on two types of randomly generated networks show the advantage of the introduced Monte Carlo approach compared to direct simulation when small reliabilities have to be estimated and demonstrate its applicability on large-scale instances. © 2010 Wiley Periodicals, Inc. NETWORKS, Vol. 57(4), 376-388 2011