Advertisement

The Quickest Multicommodity Flow Problem

  • Lisa Fleischer
  • Martin Skutella
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2337)

Abstract

Traditionally, flows over time are solved in time-expanded networks which contain one copy of the original network for each discrete time step. While this method makes available the whole algorithmic toolbox developed for static flows, its main and often fatal drawback is the enormous size of the time-expanded network. In particular, this approach usually does not lead to efficient algorithms with running time polynomial in the input size since the size of the time-expanded network is only pseudo-polynomial.

We present two different approaches for coping with this difficulty. Firstly, inspired by the work of Ford and Fulkerson on maximal s-t-flows over time (or ‘maximal dynamic s-t-flows’), we show that static, length-bounded flows lead to provably good multicommodity flows over time. These solutions not only feature a simple structure but can also be computed very efficiently in polynomial time.

Secondly, we investigate ‘condensed’ time-expanded networks which rely on a rougher discretization of time. Unfortunately, there is a natural tradeoff between the roughness of the discretization and the quality of the achievable solutions. However, we prove that a solution of arbitrary precision can be computed in polynomial time through an appropriate discretization leading to a condensed time expanded network of polynomial size. In particular, this approach yields a fully polynomial time approximation scheme for the quickest multicommodity flow problem and also for more general problems.

Keywords

Polynomial Time Transit Time Intermediate Node Discrete Time Model Input Size 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. E. Aronson. A survey of dynamic network flows. Annals of Operations Research, 20:1–66, 1989.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    R. E. Burkard, K. Dlaska, and B. Klinz. The quickest flow problem. ZOR-Methods and Models of Operations Research, 37:31–58, 1993.MathSciNetzbMATHGoogle Scholar
  3. 3.
    L. K. Fleischer. Approximating fractional multicommodity flows independent of the number of commodities. SIAM Journal on Discrete Mathematics, 13:505–520, 2000.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    L. K. Fleischer and É Tardos. Efficient continuous-time dynamic network flow algorithms. Operations Research Letters, 23:71–80, 1998.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    L. R. Ford and D. R. Fulkerson. Constructing maximal dynamic flows from static flows. Operations Research, 6:419–433, 1958.MathSciNetCrossRefGoogle Scholar
  6. 6.
    L. R. Ford and D. R. Fulkerson. Flows in Networks. Princeton University Press, Princeton, NJ, 1962.zbMATHGoogle Scholar
  7. 7.
    D. Gale. Transient flows in networks. Michigan Mathematical Journal, 6:59–63, 1959.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    N. Garg and J. Könemann. Faster and simpler algorithms for multicommodity flow and other fractional packing problems. In Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science, pages 300–309, Palo Alto, CA, 1998.Google Scholar
  9. 9.
    M. Grötschel, L. Lovász, and A. Schrijver. Geometric Algorithms and Combinatorial Optimization, volume 2 of Algorithms and Combinatorics. Springer, Berlin, 1988.zbMATHCrossRefGoogle Scholar
  10. 10.
    G. Handler and I. Zang. A dual algorithm for the constrained shortest path problem. Networks, 10:293–310, 1980.MathSciNetCrossRefGoogle Scholar
  11. 11.
    R. Hassin. Approximation schemes for the restricted shortest path problem. Mathematics of Operations Research, 17:36–42, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    B. Hoppe and É Tardos. Polynomial time algorithms for some evacuation problems. In Proceedings of the 5th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 433–441, Arlington, VA, 1994.Google Scholar
  13. 13.
    B. Hoppe and É Tardos. The quickest transshipment problem. Mathematics of Operations Research, 25:36–62, 2000.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    B. Klinz and G. J. Woeginger. Minimum cost dynamic flows: The series-parallel case. In E. Balas and J. Clausen, editors, Integer Programming and Combinatorial Optimization, volume 920 of Lecture Notes in Computer Science, pages 329–343. Springer, Berlin, 1995.CrossRefGoogle Scholar
  15. 15.
    D. H. Lorenz and D. Raz. A simple efficient approximation scheme for the restricted shortest path problem. Operations Research Letters, 28:213–219, 2001.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    E. Minieka. Maximal, lexicographic, and dynamic network flows. Operations Research, 21:517–527, 1973.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    C. A. Phillips. The network inhibition problem. In Proceedings of the 25th Annual ACM Symposium on the Theory of Computing, pages 776–785, San Diego, CA, 1993.Google Scholar
  18. 18.
    W. B. Powell, P. Jaillet, and A. Odoni. Stochastic and dynamic networks and routing. In M. O. Ball, T. L. Magnanti, C. L. Monma, and G. L. Nemhauser, editors, Network Routing, volume 8 of Handbooks in Operations Research and Management Science, chapter 3, pages 141–295. North-Holland, Amsterdam, The Netherlands, 1995.CrossRefGoogle Scholar
  19. 19.
    W. L. Wilkinson. An algorithm for universal maximal dynamic flows in a network. Operations Research, 19:1602–1612, 1971.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Lisa Fleischer
    • 1
  • Martin Skutella
    • 2
  1. 1.Graduate School of Industrial AdministrationCarnegie Mellon UniversityPittsburghUSA
  2. 2.Institut für Mathematik, MA 6-1Technische Universität BerlinBerlinGermany

Personalised recommendations