The disjoint shortest paths problem Original Research Article

The disjoint shortest paths problem is defined as follows. Given a graph G and k pairs of distinct vertices (si, ti), 1 ⩽ i ⩽ k, find whether there exist k pairwise disjoint shortest paths Pi, between si and ti for all 1 ⩽i ⩽ k. We may consider directed or undirected graphs and the paths may be vertex or edge disjoint. We show that these four problems are NP-complete when k is part of the input even for planar graphs with unit edge-lengths. We give a polynomial algorithm for the two disjoint shortest paths problem (vertex and edge disjoint paths) in undirected graphs with positive edge-lengths. We also consider the following variation of the problem. Given a graph and two distinct pairs of vertices, find whether there exist two disjoint paths P1, P2 between them such that P1 is a shortest path. We show that this problem is NP-complete for undirected graphs with unit edge-lengths. This result is surprising in view of the existence of polynomial algorithms for both the two disjoint paths problem and the two disjoint shortest paths problem for undirected graphs.