Algebraic methods for stochastic minimum cut and maximum flow problems
INOC'11 Proceedings of the 5th international conference on Network optimization
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This article develops state-space partition methods for computing performance measures for stochastic networks with demands between multiple pairs of nodes. The chief concern is the evaluation of the probability that there exist separate, noninteracting flows that satisfy all demands. This relates to the multiterminal maximum flow problem discussed in the classic article of Gomory and Hu. The network arcs are assumed to have independent, discrete random capacities. We refer to the probability that all demands can be satisfied as the network reliability (with the understanding that its definition is application dependent). In addition, we also consider the calculation of secondary measures, such as the probability that a particular subset of demands can be met, and the probability that a particular arc lies on a minimum cut. The evaluation of each of these probabilities is shown to be NP-hard. The proposed methods are based on an iterative partition of the system state space, with each iteration tightening the bounds on the measure of interest. This last property allows the design of increasingly efficient Monte Carlo sampling plans that yield substantially more precise estimators than the standard Monte Carlo method that draws samples from the original capacity distribution. © 2006 Wiley Periodicals, Inc. NETWORKS, Vol. 48(2), 90–111 2006