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Universally Maximum Flow with Piecewise-Constant Capacities

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Integer Programming and Combinatorial Optimization (IPCO 1999)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1610))

Abstract

The maximum dynamic flow problem generalizes the standard maximum flow problem by introducing time. The object is to send as much flow from source to sink in T time units as possible, where capacities are interpreted as an upper bound on the rate of flow entering an arc. A related problem is the universally maximum flow, which is to send a flow from source to sink that maximizes the amount of flow arriving at the sink by time t simultaneously for all t ≤ T. We consider a further generalization of this problem that allows arc and node capacities to change over time. In particular, given a network with arc and node capacities that are piecewise constant functions of time with at most k breakpoints, and a time bound T, we show how to compute a flow that maximizes the amount of flow reaching the sink in all time intervals (0, t] simultaneously for all 0 < t ≤ T, in O(k 2 mnlog(kn 2/m)) time. The best previous algorithm requires O(nk) maximum flow computations on a network with (m+ n)k arcs and nk nodes.

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References

  1. E. J. Anderson and P. Nash. Linear Programming in Infinite-Dimensional Spaces. John Wiley & Sons, 1987.

    Google Scholar 

  2. E. J. Anderson, P. Nash, and A. B. Philpott. A class of continuous network flow problems. Mathematics of Operations Research, 7:501–14, 1982.

    MATH  MathSciNet  Google Scholar 

  3. E. J. Anderson and A. B. Philpott. A continuous-time network simplex algorithm. Networks, 19:395–425, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  4. J. E. Aronson. A survey of dynamic network flows. Annals of Operations Research, 20:1–66, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  5. R. E. Burkard, K. Dlaska, and B. Klinz. The quickest flow problem. ZOR Methods and Models of Operations Research, 37(1):31–58, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  6. L. Fleischer. Faster algorithms for the quickest transshipment problem with zero transit times. In Proceedings of the Ninth Annual ACM/SIAM Symposium on Discrete Algorithms, pages 147–156, 1998. Submitted to SIAM Journal on Optimization.

    Google Scholar 

  7. L. R. Ford and D. R. Fulkerson. Flows in Networks. Princeton University Press, 1962.

    Google Scholar 

  8. G. Gallo, M. D. Grigoriadis, and R. E. Tarjan. A fast parametric maximum flow algorithm and applications. SIAM J. Comput., 18(1):30–55, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  9. A. V. Goldberg and R. E. Tarjan. A new approach to the maximum flow problem. Journal of ACM, 35:921–940, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  10. B. Hajek and R. G. Ogier. Optimal dynamic routing in communication networks with continuous traffic. Networks, 14:457–487, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  11. B. Hoppe. Efficient Dynamic Network Flow Algorithms. PhD thesis, Cornell University, June 1995. Department of Computer Science Technical Report TR95-1524.

    Google Scholar 

  12. B. Hoppe and É. Tardos. Polynomial time algorithms for some evacuation problems. In Proc. of 5th Annual ACM-SIAM Symp. on Discrete Algorithms, pages 433–441, 1994.

    Google Scholar 

  13. B. Hoppe and É. Tardos. The quickest transshipment problem. In Proc. of 6th Annual ACM-SIAM Symp. on Discrete Algorithms, pages 512–521, 1995.

    Google Scholar 

  14. E. Minieka. Maximal, lexicographic, and dynamic network flows. Operations Research, 21:517–527, 1973.

    MATH  MathSciNet  Google Scholar 

  15. F. H. Moss and A. Segall. An optimal control approach to dynamic routing in networks. IEEE Transactions on Automatic Control, 27(2):329–339, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  16. R. G. Ogier. Minimum-delay routing in continuous-time dynamic networks with piecewise-constant capacities. Networks, 18:303–318, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  17. J. B. Orlin. Minimum convex cost dynamic network flows. Mathematics of Operations Research, 9(2):190–207, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  18. A. B. Philpott. Continuous-time flows in networks. Mathematics of Operations Research, 15(4):640–661, November 1990.

    MATH  MathSciNet  Google Scholar 

  19. A. B. Philpott and M. Craddock. An adaptive discretization method for a class of continuous network programs. Networks, 26:1–11, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  20. 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, Handbooks in Operations Research and Management Science: Networks. Elsevier Science Publishers B. V., 1995.

    Google Scholar 

  21. M. C. Pullan. An algorithm for a class of continuous linear programs. SIAM J. Control and Optimization, 31(6):1558–1577, November 1993.

    Article  MATH  MathSciNet  Google Scholar 

  22. M. C. Pullan. A study of general dynamic network programs with arc time-delays. SIAM Journal on Optimization, 7:889–912, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  23. G. I. Stassinopoulos and P. Konstantopoulos. Optimal congestion control in single destination networks. IEEE transactions on communications, 33(8):792–800, 1985.

    Article  MATH  Google Scholar 

  24. W. L. Wilkinson. An algorithm for universal maximal dynamic flows in a network. Operations Research, 19:1602–1612, 1971.

    MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Industrial Engineering and Operations Research, Columbia University, New York, NY, 10027

    Lisa Fleischer

Authors
  1. Lisa Fleischer
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Editor information

Editors and Affiliations

  1. GSIA, Carnegie Mellon University, Schenley Park, Pittsburgh, PA, 15213, USA

    Gérard Cornuéjols

  2. Institut für Mathematik, Technische Universität Graz, Steyrergasse 30, A-8010, Graz, Austria

    Rainer E. Burkard  & Gerhard J. Woeginger  & 

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© 1999 Springer-Verlag Berlin Heidelberg

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Fleischer, L. (1999). Universally Maximum Flow with Piecewise-Constant Capacities. In: Cornuéjols, G., Burkard, R.E., Woeginger, G.J. (eds) Integer Programming and Combinatorial Optimization. IPCO 1999. Lecture Notes in Computer Science, vol 1610. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48777-8_12

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  • DOI: https://doi.org/10.1007/3-540-48777-8_12

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-66019-4

  • Online ISBN: 978-3-540-48777-7

  • eBook Packages: Springer Book Archive

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