An efficient factoring algorithm for the quickest path multi-state flow network reliability problem

The exact evaluation of the quickest path multi-state flow network reliability parameter belongs to the family of NP-hard problems. All methods known up to date for its resolution consist of two steps. The first step generates all relevant paths with respect to the reliability under consideration. The second one, having exponential time-complexity with respect to the number of relevant paths, exploits them to compute the reliability as the probability of the union of events not mutually exclusive. As a consequence, published methods are impractical for configurations with more than thirty relevant paths. In this paper, we present a new method, called the factoring method, which finds a relevant path , uses one of its link , the total expectation theorem for transforming the given problem on two similar ones but related to smaller multi-state flow networks. This transformation, with polynomial execution-time complexity, is then recursively applied until termination conditions are met. By illustrations on versions of NSFNet, ARPANET and complete topologies, we highlight that the proposed method allows to compute quickly the reliability of multi-state flow networks with hundreds or even thousands of relevant paths and outperforms significantly the method based on the Depth-First-Search procedure (DFS-based method), which in previous literature was established as the most efficient one.