Advertisement

A Simple Greedy Algorithm for the k-Disjoint Flow Problem

  • Maren Martens
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5532)

Abstract

In classical network flow theory the choice of paths, on which flow is sent, is only restricted by arc capacities. This, however, is not realistic in most applications. Many problems restrict, e.g., the number of paths being used to route a commodity. One idea to increase reliability of routings, e.g., in telecommunication, is to copy a demand and send the copies along disjoint paths. Such problems theoretically result in the k-disjoint flow problem (k-DFP). This problem is a variant of the classical multicommodity flow problem with the additional requirement that the number of paths to route a commodity is bounded by a given parameter. Moreover, all paths used by the same commodity have to be arc disjoint.

We present a simple greedy algorithm for the optimization version of the k-DFP where the objective is to maximize the sum of routed demands. This algorithm generalizes a greedy algorithm by Kolman and Scheideler (2002) that approximates the corresponding unsplittable flow problem, in which every commodity may be routed along a single path only. We achieve an approximation factor of \(O(k_{\text{max}} \sqrt{m}/k_{\text{min}})\), where m is the number of arcs and \(k_{\text{max}}\) (\(k_{\text{min}}\)) is the maximum (minimum) bound on the number of paths allowed to route any of the commodities. We argue that this performance guarantee is best possible for instances where \(k_{\text{max}}/k_{\text{min}}\) is constant, unless \(\mathcal{P}=\mathcal{P}\mathcal{N}\).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aggarwal, G.C., Orlin, J.B.: On multiroute maximum flows in networks. Networks 39, 43–52 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows. Prentice-Hall, Englewood Cliffs (1993)Google Scholar
  3. 3.
    Aneja, Y.P., Chandrasekaran, R., Nair, K.P.K.: Parametric analysis of overall min-cuts and applications in undirected networks. Information Processing Letters 85, 105–109 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bagchi, A., Chaudary, A., Scheideler, C., Kolman, P.: Algorithms for fault-tolerant routing in circuit switched networks. In: Fourteenth ACM Symposium on Parallel Algorithms and Architectures (2002)Google Scholar
  5. 5.
    Baier, G.: Flows with Path Restrictions. PhD thesis, TU Berlin (2003)Google Scholar
  6. 6.
    Baier, G., Köhler, E., Skutella, M.: On the k-splittable flow problem. Algorithmica 42, 231–248 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Ford, L.R., Fulkerson, D.R.: Maximal flow through a network. Canadian Journal of Mathematics 8, 399–404 (1956)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Guruswami, V., Khanna, S., Rajaraman, R., Shepherd, B., Yannakakis, M.: Near-optimal hardness results and approximation algorithms for edge-disjoint paths and related problems. Journal of Computer and System Sciences 67, 473–496 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kishimoto, W.: A method for obtaining the maximum multiroute flows in a network. Networks 27, 279–291 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kishimoto, W., Takeuchi, M.: On m-route flows in a network. IEICE Transactions (1993) (in Japanese)Google Scholar
  11. 11.
    Kleinberg, J.M.: Approximation Algorithms for Disjoint Path Problems. PhD thesis, Massachusetts Institute of Technology (May 1996)Google Scholar
  12. 12.
    Kolliopoulos, S.G.: Edge-disjoint paths and unsplittable flow. In: Gonzalez, T.F. (ed.) Handbook of Approximation Algorithms and Metaheuristics. Chapman & Hall/CRC, Boca Raton (2007)Google Scholar
  13. 13.
    Kolman, P., Scheideler, C.: Improved bounds for the unsplittable flow problem. In: Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 184–193 (2002)Google Scholar
  14. 14.
    Korte, B., Vygen, J.: Combinatorial Optimization. Theory and Algorithms. Springer, Berlin (2000)zbMATHGoogle Scholar
  15. 15.
    Martens, M.: Path-Constrained Network Flows. PhD thesis, Universität Dortmund (2007)Google Scholar
  16. 16.
    Martens, M., Skutella, M.: Flows on few paths: Algorithms and lower bounds. Networks 48(2), 68–76 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Martens, M., Skutella, M.: Length-bounded and dynamic k-splittable flows. In: Operations Research Proceedings 2005, pp. 297–302 (2006)Google Scholar
  18. 18.
    Schrijver, A.: Combinatorial Optimization. Polyhedra and Efficiency. Springer, Berlin (2003)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Maren Martens
    • 1
  1. 1.Zuse Institute BerlinBerlinGermany

Personalised recommendations