The Disjoint Shortest Paths Problem

Abstract The disjoint shortest paths problem is defined as follows. Given a graph G and k pairs of distinct vertices ( s i , t i ), 1 ⩽ i ⩽ k , find whether there exist k pairwise disjoint shortest paths P i , between s i and t i 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 P 1 , P 2 between them such that P 1 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.

[1]  MATTHIAS MIDDENDORF,et al.  On the complexity of the disjoint paths problem , 1993, Comb..

[2]  Alexander Schrijver,et al.  Finding k Disjoint Paths in a Directed Planar Graph , 1994, SIAM J. Comput..

[3]  Alon Itai,et al.  The complexity of finding maximum disjoint paths with length constraints , 1982, Networks.

[4]  James F. Lynch,et al.  The equivalence of theorem proving and the interconnection problem , 1975, SIGD.

[5]  Alon Itai,et al.  On the Complexity of Timetable and Multicommodity Flow Problems , 1976, SIAM J. Comput..

[6]  Yossi Shiloach,et al.  A Polynomial Solution to the Undirected Two Paths Problem , 1980, JACM.

[7]  Paul D. Seymour,et al.  Graph minors. VI. Disjoint paths across a disc , 1986, J. Comb. Theory, Ser. B.

[8]  Neil Robertson,et al.  Disjoint Paths—A Survey , 1985 .

[9]  Richard M. Karp,et al.  On the Computational Complexity of Combinatorial Problems , 1975, Networks.

[10]  Yehoshua Perl,et al.  Finding Two Disjoint Paths Between Two Pairs of Vertices in a Graph , 1978, JACM.

[11]  Nimrod Megiddo,et al.  On orientations and shortest paths , 1989 .

[12]  Paul D. Seymour,et al.  Graph minors. VII. Disjoint paths on a surface , 1988, J. Comb. Theory, Ser. B.

[13]  Paul D. Seymour,et al.  Graph minors. IX. Disjoint crossed paths , 1990, J. Comb. Theory, Ser. B.

[14]  Philip N. Klein,et al.  Faster Shortest-Path Algorithms for Planar Graphs , 1997, J. Comput. Syst. Sci..

[15]  Chung-Lun Li,et al.  Finding disjoint paths with different path-costs: Complexity and algorithms , 1992, Networks.

[16]  András Sebő.  Integer plane multiflows with a fixed number of demands , 1993 .

[17]  John E. Hopcroft,et al.  The Directed Subgraph Homeomorphism Problem , 1978, Theor. Comput. Sci..

[18]  David Lichtenstein,et al.  Planar Formulae and Their Uses , 1982, SIAM J. Comput..

[19]  P. Seymour.  On Odd Cuts and Plane Multicommodity Flows , 1981 .

[20]  Chung-Lun Li,et al.  The complexity of finding two disjoint paths with min-max objective function , 1989, Discret. Appl. Math..

[21]  Alon Itai,et al.  On the complexity of time table and multi-commodity flow problems , 1975, 16th Annual Symposium on Foundations of Computer Science (sfcs 1975).