Network reliability analysis and intractability of counting diameter crystal graphs

Abstract

Consider a stochastic network, where nodes are perfect but links fail independently, ruled by failure probabilities. Additionally, we are given distinguished nodes, called terminals, and a positive integer, called diameter. The event under study is to connect terminals by paths not longer than the given diameter. The probability of this event is called diameter-constrained reliability (DCR, for short). Since the DCR subsumes connectedness probability of random graphs, its computation belongs to the class of NP-Hard problems. The computational complexity for DCR is known for fixed values of the number of terminals k n and diameter d, being n the number of nodes in the network. The contributions of this article are two-fold. First, we extend the computational complexity of the DCR when the terminal size is a function of the number of nodes, this is, when k = k(n). Second, we state counting diameter-critical graphs belongs to the class of NP-Hard problems.