Linking-Centers and Reliable-Trees of a Network

We address synthesis problems in the general context of network reliability, and describe two procedures ¿ CENTERTREE and EXCENTER ¿ whose running times never grow more than cubically with the number of vertices of the network. The first procedure solves two basic related problems ¿ the most linking center and the most linking tree. The first problem requires locating a point on a network through which every connection between two given sets of vertices has to pass. This location maximizes the minimum reliability of the most reliable path connecting through the center every pair of vertices of the two given sets. The second problem asks for a most reliable tree connecting the two sets of vertices. The second procedure generalizes the first one and solves a constrained version of the most linking center problem that involves two (instead of one) reliabilities assigned to each network element and four (instead of two) sets of vertices to be connected. In general, the similarly extended, constrained version of the most linking tree problem is NP-hard, and only in a restricted case is it solvable by procedure EXCENTER. We prove that two extensions of the most linking tree problem, involving edge costs, are NP-hard.