Time-Expanded Graphs for Flow-Dependent Transit Times

  • Ekkehard Köhler
  • Katharina Langkau
  • Martin Skutella
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2461)

Abstract

Motivated by applications in road traffic control, we study flows in networks featuring special characteristics. In contrast to classical static flow problems, time plays a decisive role. Firstly, there are transit times on the arcs of the network which specify the amount of time it takes for flow to travel through a particular arc; more precisely, flow values on arcs may change over time. Secondly, the transit time of an arc varies with the current amount of flow using this arc. Especially the latter feature is crucial for various real-life applications of flows over time; yet, it dramatically increases the degree of difficulty of the resulting optimization problems.

Most problems dealing with flows over time and constant transit times can be translated to static flow problems in time-expanded networks. We develop an alternative time-expanded network with flow-dependent transit times to which the whole algorithmic toolbox developed for static flows can be applied. Although this approach does not entirely capture the behavior of flows over time with flow-dependent transit times, we present approximation results which provide evidence of its surprising quality.

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References

  1. [1]
    J.E. Aronson. A survey of dynamic network flows. Annals of OR, 20:1–66, 1989.MATHMathSciNetCrossRefGoogle 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.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    M. Carey. A constraint qualification for a dynamic traffic assignment model. Transp. Science, 20:55–58, 1986.Google Scholar
  4. [4]
    M. Carey. Optimal time-varying flows on congested networks. OR, 35:58–69, 1987.MATHMathSciNetGoogle Scholar
  5. [5]
    M. Carey and E. Subrahmanian. An approach for modelling time-varying flows on congested networks. Transportation Research B, 34:157–183, 2000.CrossRefGoogle Scholar
  6. [6]
    L. Fleischer and M. Skutella. The quickest multicommodity flow problem. In Proc. of IPCO’02, 2002.Google Scholar
  7. [7]
    L. Fleischer and É. Tardos. Efficient continuous-time dynamic network flow algorithms. Operations Research Letters, 23:71–80, 1998.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    L.R. Ford and D. R. Fulkerson. Constructing maximal dynamic flows from static flows. Operations Research, 6:419–433, 1958.MathSciNetGoogle Scholar
  9. [9]
    L.R. Ford and D.R. Fulkerson. Flows in Networks. Princeton University Press, Princeton, NJ, 1962.MATHGoogle Scholar
  10. [10]
    N. Gartner, C. J. Messer, and A.K. Rathi. Traffic flow theory: A state of the art report. http://www-cta.ornl.gov/cta/research/trb/tft.html, 1997.
  11. [11]
    E. Köhler and M. Skutella. Flows over time with load-dependent transit times. In Proc. of ACM-SIAM Symposium on Discrete Algorithms (SODA), 2002.Google Scholar
  12. [12]
    H. S. Mahmassani and S. Peeta. System optimal dynamic assignment for electronic route guidance in a congested traffic network. In Urban Traffic Networks. Dynamic Flow Modelling and Control, pages 3–37. Springer, Berlin, 1995.Google Scholar
  13. [13]
    D. K. Merchant and G. L. Nemhauser. A model and an algorithm for the dynamic traffic assignment problems. Transp. Science, 12:183–199, 1978.CrossRefGoogle Scholar
  14. [14]
    D.K. Merchant and G. L. Nemhauser. Optimality conditions for a dynamic traffic assignment model. Transp. Science, 12:200–207, 1978.Google Scholar
  15. [15]
    W. B. Powell, P. Jaillet, and A. Odoni. Stochastic and dynamic networks and routing. Handb. in OR and Man.Sc. vol. 8, pages 141–295. North-Holland, 1995.MathSciNetGoogle Scholar
  16. [16]
    B. Ran and D. E. Boyce. Modelling Dynamic Transportation Networks. Springer, Berlin, 1996.Google Scholar
  17. [17]
    Y. Sheffi. Urban Transportation Networks. Prentice-Hall, New Jersey, 1985.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Ekkehard Köhler
    • 1
  • Katharina Langkau
    • 1
  • Martin Skutella
    • 1
  1. 1.Fakultät II - Mathematik und Naturwissenschaften Institut für MathematikTechnische Universität BerlinBerlinGermany

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