The Directed Disjoint Shortest Paths Problem

In the k disjoint shortest paths problem (k-DSPP), we are given a graph and its vertex pairs (s1, t1), . . . , (sk, tk), and the objective is to find k pairwise disjoint paths P1, . . . , Pk such that each path Pi is a shortest path from si to ti, if they exist. If the length of each edge is equal to zero, then this problem amounts to the disjoint paths problem, which is one of the well-studied problems in algorithmic graph theory and combinatorial optimization. Eilam-Tzoreff [5] focused on the case when the length of each edge is positive, and showed that the undirected version of 2-DSPP can be solved in polynomial time. Polynomial solvability of the directed version was posed as an open problem in [5]. In this paper, we solve this problem affirmatively, that is, we give a first polynomial time algorithm for the directed version of 2-DSPP when the length of each edge is positive. Note that the 2 disjoint paths problem in digraphs is NP-hard, which implies that the directed 2-DSPP is NP-hard if the length of each edge can be zero. We extend our result to the case when the instance has two terminal pairs and the number of paths is a fixed constant greater than two. We also show that the undirected k-DSPP and the vertex-disjoint version of the directed k-DSPP can be solved in polynomial time if the input graph is planar and k is a fixed constant. 1998 ACM Subject Classification G.2.2 [Graph Theory] Graph Algorithms