A Linear-Time Algorithm for Computing K-Terminal Reliability in Series-Parallel Networks

Let $G = (V,E)$ be a graph whose edges may fail with known probabilities and let $K \subseteq V$ be specified. The K-terminal reliability of G, denoted $R(G_K )$, is the probability that all vertices in K are connected. Computing $R(G_K )$ is, in general, NP-hard. For some series-parallel graphs, $R(G_K )$ can be computed in polynomial time by repeated application of well-known reliability-preserving reductions. However, for other series-parallel graphs, depending on the configuration of K, $R(G_K )$ cannot be computed in this way. Only exponential-time algorithms as used on general graphs were known for computing $R(G_K )$ for these “irreducible” series-parallel graphs. We prove that $R(G_K )$ is computable in polynomial time in the irreducible case, too. A new set of reliability-preserving “polygon-to-chain” reductions of general applicability is introduced which decreases the size of a graph, and conditions are given for a graph admitting such reductions. Combining all types of reductions, an $O(|E|)$ algorithm is presented for computing the reliability of any series-parallel graph irrespective of the vertices in K.