A Note on the Complexity of Network Reliability Problems

Let be given an undirected, simple graph G = (V, E). We associate to each vertex a number in [0, 1] its reliability, i.e. the probability that it does not fail. Furthermore, let a set S ⊆ V of servers and a set L ⊆ V of clients be given. Vertex failures are independent of each other. The network reliability asks for the probability that the graph induced by the non-failed vertices is connected. We can also ask for the probabilities of connections between vertices of S and between vertices of S and L, e.g. ‘What is the probability that all clients are connected to at least one server?’ In this note, we consider a list of such problems and prove their #P -completeness.