On the k-simple shortest paths problem in weighted directed graphs

We obtain the first approximation algorithm for finding the k-simple shortest paths connecting a pair of vertices in a weighted directed graph. Our algorithm is deterministic and has a running time of O(k(m root n + n(3/2) log n)), where in is the number of edges in the graph and n is the number of vertices. Let s, t is an element of V; the length of the i-th simple path from s to t computed by our algorithm is at most 3/2 times the length of the i-th shortest simple path from s to t. The best algorithms for computing the exact k-simple shortest paths connecting a pair of vertices in a weighted directed graph are due to Yen [19] and Lawler [13]. The running time of their algorithms, using modern data structures, is O(k(mn + n(2) log n)). Both algorithms are from the early 70's. Although this problem and other variants of the k-shortest path problem drew a lot of attention during the last three and a half decades, the O(k(mn + n(2) log n)) bound is still unbeaten.