Computing disjoint paths with length constraints
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  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.
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