On the robustness of Fishman's bound-based method for the network reliability problem

Static network unreliability computation is an NP-hard problem, leading to the use of Monte Carlo techniques to estimate it. The latter, in turn, suffer from the rare event problem, in the frequent situation where the system's unreliability is a very small value. As a consequence, specific rare event event simulation techniques are relevant tools to provide this estimation. We focus here on a method proposed by Fishman making use of bounds on the structure function of the model. The bounds are based on the computation of (disjoint) mincuts disconnecting the set of nodes and (disjoint) minpaths ensuring that they are connected. We analyze the robustness of the method when the unreliability of links goes to zero. We show that the conditions provided by Fishman, based on a bound, are only sufficient, and we provide more insight and examples on the behavior of the method.