Approximating Earliest Arrival Flows in Arbitrary Networks

  • Martin Groß
  • Jan-Philipp W. Kappmeier
  • Daniel R. Schmidt
  • Melanie Schmidt
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7501)

Abstract

The earliest arrival flow problem is motivated by evacuation planning. It asks for a flow over time in a network with supplies and demands that maximizes the satisfied demands at every point in time. Gale [1959] has shown the existence of such flows for networks with a single source and sink. For multiple sources and a single sink the existence follows from work by Minieka [1973] and an exact algorithm has been presented by Baumann and Skutella [FOCS ’06]. If multiple sinks are present, it is known that earliest arrival flows do not exist in general.

We address the open question of approximating earliest arrival flows in arbitrary networks with multiple sinks and present constructive approximations of time and value for them. We give tight bounds for the best possible approximation factor in most cases. In particular, we show that there is always a 2-value-approximation of earliest arrival flows and that no better approximation factor is possible in general. Furthermore, we describe an FPTAS for computing the best possible approximation factor (which might be better than 2) along with the corresponding flow for any given instance.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Baumann, N., Skutella, M.: Solving evacuation problems efficiently: Earliest arrival flows with multiple sources. Mathematics of Operations Research 34(2), 499–512 (2009)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Baumann, N., Köhler, E.: Approximating earliest arrival flows with flow-dependent transit times. Discrete Applied Mathematics 155(2), 161–171 (2007)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Burkard, R.E., Dlaska, K., Klinz, B.: The quickest flow problem. Mathematical Methods of Operations Research 37, 31–58 (1993)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Fleischer, L.K.: Faster algorithms for the quickest transshipment problem. SIAM Journal on Optimization 12(1), 18–35 (2001)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Fleischer, L.K., Skutella, M.: Quickest flows over time. SIAM Journal on Computing 36(6), 1600–1630 (2007)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Fleischer, L.K., Tardos, É.: Efficient continuous-time dynamic network flow algorithms. Operations Research Letters 23(3-5), 71–80 (1998)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Ford, L.R., Fulkerson, D.R.: Flows in Networks. Princeton University Press (1962)Google Scholar
  8. 8.
    Gale, D.: Transient flows in networks. Michigan Mathematical Journal 6, 59–63 (1959)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Groß, M., Skutella, M.: Generalized maximum flows over time. In: Proceedings of the 9th WAOA, pp. 247–260 (to appear, 2012)Google Scholar
  10. 10.
    Hoppe, B.: Efficient dynamic network flow algorithms. Ph.D. thesis, Cornell University (1995)Google Scholar
  11. 11.
    Hoppe, B., Tardos, É.: Polynomial time algorithms for some evacuation problems. In: Proceedings of the 5th SODA, pp. 433–441 (1994)Google Scholar
  12. 12.
    Hoppe, B., Tardos, É.: The quickest transshipment problem. Mathematics of Operations Research 25, 36–62 (2000)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Minieka, E.: Maximal, lexicographic, and dynamic network flows. Operations Research 21, 517–527 (1973)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Philpott, A.B.: Continuous-time flows in networks. Mathematics of Operations Research 15(4), 640–661 (1990)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Ruzika, S., Sperber, H., Steiner, M.: Earliest arrival flows on series-parallel graphs. Networks 57(2), 169–173 (2011)MathSciNetMATHGoogle Scholar
  16. 16.
    Schmidt, M., Skutella, M.: Earliest arrival flows in networks with multiple sinks. Discrete Applied Mathematics (2011), http://dx.doi.org/10.1016/j.dam.2011.09.023
  17. 17.
    Skutella, M.: An introduction to network flows over time. In: Cook, W., Lovász, L., Vygen, J. (eds.) Research Trends in Combinatorial Optimization, pp. 451–482. Springer (2009)Google Scholar
  18. 18.
    Tjandra, S.A.: Dynamic network optimization with application to the evacuation problem. Ph.D. thesis, Technical University of Kaiserslautern (2003)Google Scholar
  19. 19.
    Wilkinson, W.L.: An algorithm for universal maximal dynamic flows in a network. Operations Research 19, 1602–1612 (1971)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Martin Groß
    • 1
  • Jan-Philipp W. Kappmeier
    • 1
  • Daniel R. Schmidt
    • 2
  • Melanie Schmidt
    • 3
  1. 1.TU BerlinGermany
  2. 2.Universität zu KölnGermany
  3. 3.TU DortmundGermany

Personalised recommendations