Diameter-Constrained Reliability: Complexity, Factorization and Exact computation in Weak Graphs

In this paper we address a problem from the field of network reliability, called diameter-constrained reliability. Specifically, we are given a simple graph G = (V, E) with [V] = n nodes and [E] = m links, a subset K ⊆ V of terminals, a vector p = (p1,...,pm) &epsis; [0, 1]m and a positive integer d, called diameter. We assume nodes are perfect but links fail stochastically and independently, with probabilities qi = 1 --- pi. 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 RdK,G(p). The general DCR computation is inside the class of NP-Hard problems, since is subsumes the complexity that a random graph is connected. In this paper the computational complexity of DCR-subproblems is discussed in terms of the number of terminal nodes k = [K] and diameter d. A factorization formula for exact DCR computation is provided, that runs in exponential time in the worst case. Finally, a revision of graph-classes that accept DCR computation in polynomial time is then included. In this class we have graphs with bounded co-rank, graphs with bounded genus, planar graphs, and, in particular, Monma graphs, which are relevant in robust network design. We extend this class adding arborescence graphs. A discussion of trends for future work is offered in the conclusions.