Parameterized Tractability of Edge-Disjoint Paths on Directed Acyclic Graphs

  • Aleksandrs Slivkins
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2832)


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.

On a positive side, we give an O(m+k! n) algorithm for the special case when G is acyclic and G+H is Eulerian, where H is the demand graph. We generalize this result (1) to the case when G+H is “nearly" Eulerian, (2) to an analogous special case of the unsplittable flow problem. Finally, we consider a related NP-complete routing problem when only the first edge of each path cannot be shared, and prove that it is fixed-parameter tractable on directed graphs.


Source Node Direct Acyclic Graph Sink Node Terminal Node Disjoint Path 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Baier, G., Köhler, E., Skutella, M.: On the k-Splittable Flow Problem. In: Proc. 10th Annual European Symposium on Algorithms (2002)Google Scholar
  2. 2.
    Dinitz, Y., Garg, N., Goemans, M.: On the Single-Source Unsplittable Flow Problem. In: Proc. 39th Annual Symposium on Foundations of Computer Science (1998)Google Scholar
  3. 3.
    Downey, R., Estivill-Castro, V., Fellows, M., Prieto, E., Rosamund, F.: Cutting Up is Hard to Do: the Parameterized Complexity of k-Cut and Related Problems. In: Computing: The Australasian Theory Symposium (2003)Google Scholar
  4. 4.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)Google Scholar
  5. 5.
    Even, S., Itai, A., Shamir, A.: On the complexity of timetable and multicommodity flow problems. SIAM J. Computing 5, 691–703 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Fortune, S., Hopcroft, J., Wyllie, J.: The directed subgraph homeomorphism problem. Theoretical Computer Science 10, 111–121 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Korte, B., Lovász, L., Prömel, H.-J., Schrijver, A. (eds.): Paths, Flows andVLSI-Layouts. Springer, Heidelberg (1990)Google Scholar
  8. 8.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press, NewYork (1972)Google Scholar
  9. 9.
    Kleinberg, J.: Single-source unsplittable flow. In: Proc. 37th Annual Symposium on Foundations of Computer Science (1996)Google Scholar
  10. 10.
    Kleinberg, J.: Decision algorithms for unsplittable flow and the half-disjoint paths problem. In: Proc. 30th Annual ACM Symposium on the Theory of Computing (1998)Google Scholar
  11. 11.
    Kleinberg, J.: Approximation Algorithms for Disjoint Paths Problems. Ph.D. Thesis, M.I.T. (1996)Google Scholar
  12. 12.
    Kolliopoulos, S.G., Stein, C.: Improved approximation algorithms for unsplittable flow problems. In: Proc. 38th Annual Symposium on Foundations of Computer Science (1997)Google Scholar
  13. 13.
    Lucchesi, C.L., Younger, D.H.: A minimax relation for directed graphs. J. London Mathematical Society 17, 369–374 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Robertson, N., Seymour, P.D.: Graph minors XIII. The disjoint paths problem. J. Combinatorial Theory Ser. B 63, 65–110 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Schrijver, A.: Finding k disjoint paths in a directed planar graph. SIAM J. Computing 23, 780–788 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Shiloach, Y.: A polynomial solution to the undirected two paths problem. J. of the ACM 27, 445–456 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Skutella, M.: Approximating the single source unsplittable min-cost flow problem. Mathematical Programming Ser. B 91(3), 493–514 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Vygen, J.: NP-completeness of some edge-disjoint paths problems. Discrete Appl. Math. 61, 83–90 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Vygen, J.: Disjoint paths, Rep. #94846, Research Inst. for Discrete Math., U. of Bonn (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Aleksandrs Slivkins
    • 1
  1. 1.Cornell UniversityIthacaUSA

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