Computing disjoint paths with length constraints

Abstract

We show that the problem of computing a pair of disjoint paths between nodes s and t of an undirected graph, each having at most K, K ε Z ⁺, edges is NP-complete. A heuristic for its optimization version is given whose performance is within a constant factor from the optimal. It can be generalized to compute any constant number of disjoint paths. We also generalize an algorithm in [1] to compute the maximum number of edge disjoint paths of the shortest possible length between s and t. We show that it is NP-complete to decide whether there exist at least K, K ε Z ⁺, disjoint paths that may have at most S+1 edges, where S is the minimum number of edges on any path between s and t. In addition, we examine a generalized version of the problem where disjoint paths are routed either between a node pair (s1, t1) or a node pair (s2, t2). We show that it is NP-hard to find the maximum number of disjoint paths that either connect pair (s1, t1) the shortest way or (s2, t2) the shortest way.