Parameterized Tractability of Edge-Disjoint Paths on Directed Acyclic Graphs

Given a graph and pairs s i t i of terminals, the edge-disjoint paths problem is to determine whether there exist s i t i paths that do not share any edges. We consider this problem on acyclic digraphs. It is known to be NP-complete and solvable in time n O(k) where k is the number of paths. It has been a long-standing open question whether it is fixed-parameter tractable in k. We resolve this question in the negative: we show that the problem is W[1]-hard. In fact it remains W[1]-hard even if the demand graph consists of two sets of parallel edges.

[1]  Martin Skutella,et al.  The k-Splittable Flow Problem , 2005, Algorithmica.

[2]  Michael R. Fellows,et al.  Cutting Up is Hard to Do: the Parameterized Complexity of k-Cut and Related Problems , 2003, CATS.

[3]  Michel X. Goemans,et al.  On the Single-Source Unsplittable Flow Problem , 1999, Comb..

[4]  C. Lucchesi,et al.  A Minimax Theorem for Directed Graphs , 1978 .

[5]  Michel X. Goemans,et al.  On the Single-Source Unsplittable Flow Problem , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

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

[7]  D. Knuth.  Wheels within wheels , 1974 .

[8]  Tatsuya Suda,et al.  Network Border Patrol , 2000, Proceedings IEEE INFOCOM 2000. Conference on Computer Communications. Nineteenth Annual Joint Conference of the IEEE Computer and Communications Societies (Cat. No.00CH37064).

[9]  Jon M. Kleinberg,et al.  Single-source unsplittable flow , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[10]  Martin Skutella,et al.  The k-Splittable Flow Problem , 2002, Algorithmica.

[11]  Sally Floyd,et al.  Promoting the use of end-to-end congestion control in the Internet , 1999, TNET.

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

[13]  Martin Skutella.  Approximating the single source unsplittable min-cost flow problem , 2002, Math. Program..

[14]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

[15]  Jon M. Kleinberg,et al.  Approximation algorithms for disjoint paths problems , 1996 .

[16]  Clifford Stein,et al.  Improved approximation algorithms for unsplittable flow problems , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[17]  Jens Vygen,et al.  NP-completeness of Some Edge-disjoint Paths Problems , 1995, Discret. Appl. Math..

[18]  Jon M. Kleinberg,et al.  Decision algorithms for unsplittable flow and the half-disjoint paths problem , 1998, STOC '98.

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

[20]  Richard M. Karp,et al.  Reducibility among combinatorial problems" in complexity of computer computations , 1972 .

[21]  Neil Robertson,et al.  Graph Minors .XIII. The Disjoint Paths Problem , 1995, J. Comb. Theory B.

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

[23]  P. Nguyen,et al.  On the Disjoint Paths , 1998 .

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

[25]  Jean-Yves Le Boudec,et al.  Rate adaptation, Congestion Control and Fairness: A Tutorial , 2000 .