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Generalized Maximum Flows over Time

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Approximation and Online Algorithms (WAOA 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7164))

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Abstract

Flows over time and generalized flows are two advanced network flow models of utmost importance, as they incorporate two crucial features occurring in numerous real-life networks. Flows over time feature time as a problem dimension and allow to realistically model the fact that commodities (goods, information, etc.) are routed through a network over time. Generalized flows allow for gain/loss factors on the arcs that model physical transformations of a commodity due to leakage, evaporation, breeding, theft, or interest rates. Although the latter effects are usually time-bound, generalized flow models featuring a temporal dimension have never been studied in the literature.

In this paper we introduce the problem of computing a generalized maximum flow over time in networks with both gain factors and transit times on the arcs. While generalized maximum flows and maximum flows over time can be computed efficiently, our combined problem turns out to be NP-hard and even completely non-approximable. A natural special case is given by lossy networks where the loss rate per time unit is identical on all arcs. For this case we present a (practically efficient) FPTAS.

Supported by the DFG Research Center Matheon “Mathematics for key technologies” in Berlin.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Bellman, R.E.: On a routing problem. Quarterly of Applied Mathematics 16, 87–90 (1958)

    MathSciNet  MATH  Google Scholar 

  3. Beygang, K., Krumke, S.O., Zeck, C.: Generalized max flow in series-parallel graphs. Report in Wirtschaftsmathematik 125, TU Kaiserslautern (2010)

    Google Scholar 

  4. Dantzig, G.B.: Linear programming and extensions. Princeton University Press (1962)

    Google Scholar 

  5. Dijkstra, E.W.: A note on two problems in connexion with graphs. Numerische Mathematik 1, 269–271 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fleischer, L., Skutella, M.: Quickest flows over time. SIAM Journal on Computing 36, 1600–1630 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fleischer, L.K., Tardos, É.: Efficient continuous-time dynamic network flow algorithms. Operations Research Letters 23, 71–80 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fleischer, L.K., Wayne, K.D.: Fast and simple approximation schemes for generalized flow. Mathematical Programming 91, 215–238 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fleischer, L., Skutella, M.: Minimum cost flows over time without intermediate storage. In: Proceedings of the 14th Annual ACM–SIAM Symposium on Discrete Algorithms, Baltimore, MD, pp. 66–75 (2003)

    Google Scholar 

  10. Ford, L.R.: Network flow theory. Paper P-923, The Rand Corporation (1956)

    Google Scholar 

  11. Ford, L.R., Fulkerson, D.R.: Flows in Networks. Princeton University Press (1962)

    Google Scholar 

  12. Ford, L.R., Fulkerson, D.R.: Constructing maximal dynamic flows from static flows. Operations Research 6, 419–433 (1987)

    Article  MathSciNet  Google Scholar 

  13. Fredman, M.L., Tarjan, R.E.: Fibonacci heaps and their uses in improved network optimization problems. Journal of the ACM 34, 596–615 (1987)

    Article  MathSciNet  Google Scholar 

  14. Gale, D.: Transient flows in networks. Michigan Mathematical Journal 6, 59–63 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  15. Goldberg, A.V., Plotkin, S.A., Tardos, É.: Combinatorial algorithms for the generalized circulation problem. Mathematics of Operations Research 16, 351–379 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  16. Goldfarb, D., Jin, Z.: A faster combinatorial algorithm for the generalized circulation problem. Mathematics of Operations Research 21, 529–539 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  17. Goldfarb, D., Jin, Z., Orlin, J.B.: Polynomial-time highest gain augmenting path algorithms for the generalized circulation problem. Mathematics of Operations Research 22, 793–802 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gondran, M., Minoux, M.: Graphs and Algorithms. Wiley (1984)

    Google Scholar 

  19. Hoppe, B., Tardos, É.: Polynomial time algorithms for some evacuation problems. In: Proceedings of the 5th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 433–441 (1994)

    Google Scholar 

  20. Kantorovich, L.V.: Mathematical methods of organizing and planning production. Technical report, Publication House of the Leningrad State University (1939); Translated in Management Science 6, 366–422 (1960)

    Google Scholar 

  21. King, V., Rao, S., Tarjan, R.: A faster deterministic maximum flow algorithm. Journal of Algorithms 17, 447–474 (1994)

    Article  MathSciNet  Google Scholar 

  22. Klinz, B., Woeginger, G.J.: Minimum cost dynamic flows: The series parallel case. Networks 43, 153–162 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Moore, E.F.: The shortest path through a maze. In: Proceedings of the International Symposium on Switching, Part II, pp. 285–292. Harvard University Press (1959)

    Google Scholar 

  25. Onaga, K.: Dynamic programming of optimum flows in lossy communication nets. IEEE Transactions on Circuit Theory 13, 282–287 (1966)

    Google Scholar 

  26. Onaga, K.: Optimal flows in general communication networks. Journal of the Franklin Institute 283, 308–327 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  27. Powell, W.B., Jaillet, P., Odoni, A.: Stochastic and dynamic networks and routing. In: Ball, M.O., Magnanti, T.L., Monma, C.L., Nemhauser, G.L. (eds.) Network Routing, ch. 3, vol. 8, pp. 141–295. Handbooks in Operations Research and Management Science, North–Holland, Amsterdam, The Netherlands (1995)

    Chapter  Google Scholar 

  28. Radzik, T.: Faster algorithms for the generalized network flow problem. Mathematics of Operations Research 23, 69–100 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  29. Radzik, T.: Improving time bounds on maximum generalised flow computations by contracting the network. Theoretical Computer Science 312, 75–94 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  30. Restrepo, M., Williamson, D.P.: A simple gap-canceling algorithm for the generalized maximum flow problem. Mathematical Programming 118, 47–74 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Skutella, M.: An introduction to network flows over time. In: Research Trends in Combinatorial Optimization, pp. 451–482. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  32. Truemper, K.: On max flows with gains and pure min-cost flows. SIAM Journal on Applied Mathematics 32, 450–456 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  33. Wayne, K.D.: Generalized Maximum Flow Algorithms. PhD thesis, Cornell University (1999)

    Google Scholar 

  34. Wayne, K.D.: A polynomial combinatorial algorithm for generalized minimum cost flow. Mathematics of Operations Research 27, 445–459 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  36. Zadeh, N.: A bad network problem for the simplex method and other minimum cost flow algorithms. Mathematical Programming 5, 255–266 (1973)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Fakultät II – Mathematik und Naturwissenschaften, Institut für Mathematik, Sekr. MA 5-2, Technische Universität Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Martin Groß & Martin Skutella

Authors
  1. Martin Groß
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  2. Martin Skutella
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Editors and Affiliations

  1. Department of Computer Science, The University of Western Ontario, N6A 5B7, London, ON, Canada

    Roberto Solis-Oba

  2. Dipartimento di Informatica "Renato M. Capocelli", Università di Salerno, Via Ponte Don Melillo, 84081, Fisciano, SA, Italy

    Giuseppe Persiano

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Groß, M., Skutella, M. (2012). Generalized Maximum Flows over Time. In: Solis-Oba, R., Persiano, G. (eds) Approximation and Online Algorithms. WAOA 2011. Lecture Notes in Computer Science, vol 7164. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29116-6_21

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  • DOI: https://doi.org/10.1007/978-3-642-29116-6_21

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-29115-9

  • Online ISBN: 978-3-642-29116-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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