Two-path subsets: Efficient counting and applications to performability analysis

AbstractThe problem of computing performability probabilities in stochastic PERT and flow networks is studied when the network is “minimally designed” to withstand any two component failures. Polynomial-time algorithms to compute performability when the network is planar — the non-planar versions being NP-hard — solve related “two-path subset” problems. Given an acyclic graph with weights on the arcs, the algorithms compute the total weight of all subsets of arcs that are contained in 1.(1) two source-sink paths, or2.(2) two arc-disjoint source-sink paths. A polynomial algorithm is given for (1), and for (2) in the case where the graph is a source-sink planar k-flow graph, that is, edge-minimal with respect to supporting k units of flow