Diameter Constrained Reliability: Complexity and Distinguished Topologies

Let G = (V, E) be a simple graph with vertical bar V vertical bar = n nodes and vertical bar E vertical bar = m links, a subset K subset of V of terminals, a vector p = (p(1),..., p(m)) is an element of [0, 1](m) and a positive integer d, called diameter. We assume nodes are perfect but links fail stochastically and independently, with probabilities q(i) = 1 - p(i). The diameter-constrained reliability (DCR for short), is the probability that the terminals of the resulting subgraph remain connected by paths composed by d links, or less. This number is denoted by R-K,G(d)(p). The general DCR computation is inside the class of NP-Hard problems, since is subsumes the complexity that a random graph is connected. The contributions of this paper are two-fold. First, the computational complexity of DCR-subproblems is discussed in terms of the number of terminal nodes k = vertical bar K vertical bar and dipameter d. Either when d = 1 or when d = 2 and k is fixed, the DCR is inside the class P of polynomial-time problems. The DCR turns NP-Hard when k >= 2 is a fixed input parameter and d >= 3. The cases where k = n or k is a free input parameter and d >= 2 is fixed are not studied in prior literature. Here, the NP-Hardness of both cases is established. Second, we extend the class of graphs that accept the DCR computation in polynomial number of elementary operations. In this class we include graphs with bounded co-rank, graphs with bounded genus, planar graphs, and, in particular, Monma graphs, which are relevant in robust network design.