The complexity of finding two disjoint paths with min-max objective function
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In communication networks, multiple disjoint communication paths are desirable for many applications. Such paths, however, may not exist in a network. In such a situation, paths with minimum link and/or node sharing may be considered. This paper addresses the following two related fundamental questions. First, in case of no solution of disjoint multiple paths for a given application instance, what are the criteria for finding the best solution in which paths share nodes and/or links? Second, if we know the criteria, how do we find the best solution? We propose a general framework for the answers to these two questions. This framework can be configured in a way that is suitable for a given application instance. We introduce the notion of link shareability and node shareability and consider the problem of finding minimum-cost multiple paths subject to minimum shareabilities (MCMPMS problem). We identify 65 different link/node shareability constraints, each leading to a specific version of the MCMPMS problem. In a previously published technical report, we prove that all the 65 versions are mutually inequivalent. In this paper, we show that all these versions can be solved using a unified algorithmic approach that consists of two algorithm schemes, each of which can be used to generate polynomial-time algorithms for a set of versions of MCMPMS. We also discuss some extensions where our modeling framework and algorithm schemes are applicable.