On the Complexity of the Diameter Constrained Reliability

Let G = (V,E) be a simple graph with |V | = n nodes and |E| = m links, a subset K ⊆ V of terminals, a vector p = (p1, . . . , pm) ∈ [0, 1] 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 R K,G(p). The general DCR computation is inside the class of NP-Hard problems, since it 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. 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 the case k = n (or k free) and d ≥ 3 is established. The complexity of the case k = n and d = 2 requires further research.