The Maximum Energy-Constrained Dynamic Flow Problem

  • Sándor P. Fekete
  • Alexander Hall
  • Ekkehard Köhler
  • Alexander Kröller
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5124)


We study a natural class of flow problems that occur in the context of wireless networks; the objective is to maximize the flow from a set of sources to one sink node within a given time limit, while satisfying a number of constraints. These restrictions include capacities and transit times for edges; in addition, every node has a bound on the amount of transmission it can perform, due to limited battery energy it carries. We show that this Maximum energy-constrained dynamic flow problem (ECDF) is difficult in various ways: it is NP-hard for arbitrary transit times; a solution using flow paths can have exponential-size growth; a solution using edge flow values may not exist; and finding an integral solution is NP-hard. On the positive side, we show that the problem can be solved polynomially for uniform transit times for a limited time limit; we give an FPTAS for finding a fractional flow; and, most notably, there is a distributed FPTAS that can be run directly on the network.


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  1. 1.
    Aronson, J.E.: A survey of dynamic network flows. Annals of OR 20, 1–66 (1989)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bodlaender, H., Tan, R., van Dijk, T., van Leeuwen, J.: Integer maximum flow in wireless sensor networks with energy constraint. In: Proc. SWAT (2008)Google Scholar
  3. 3.
    Burkard, R.E., Dlaska, K., Klinz, B.: The quickest flow problem. ZOR — Methods and Models of Operations Research 37, 31–58 (1993)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Busse, M., Haenselmann, T., Effelsberg, W.: A comparison of lifetime-efficient forwarding strategies for wireless sensor networks. In: Proc. PE-WASUN, pp. 33–40 (2006)Google Scholar
  5. 5.
    Chang, J.-H., Tassiulas, L.: Maximum lifetime routing in wireless sensor networks. IEEE/ACM Transactions on Networking 12(4), 609–619 (2004)CrossRefGoogle Scholar
  6. 6.
    Fekete, S.P., Kröller, A.: Geometry-based reasoning for a large sensor network. In: Proc. SoCG, pp. 475–476 (2006)Google Scholar
  7. 7.
    Fekete, S.P., Kröller, A., Pfisterer, D., Fischer, S.: Algorithmic aspects of large sensor networks. In: Proc MSWSN, pp. 141–152 (2006)Google Scholar
  8. 8.
    Fekete, S.P., Schmidt, C., Wegener, A., Fischer, S.: Hovering data clouds for recognizing traffic jams. In: Proc. IEEE-ISOLA, pp. 213–218 (2006)Google Scholar
  9. 9.
    Fleischer, L., Skutella, M.: Quickest flows over time. SIAM Journal on Computing 36, 1600–1630 (2007)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Fleischer, L.K., Tardos, É.: Efficient continuous-time dynamic network flow algorithms. Operations Research Letters 23, 71–80 (1998)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Ford, L.R., Fulkerson, D.R.: Constructing maximal dynamic flows from static flows. Operations Research 6, 419–433 (1958)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ford, L.R., Fulkerson, D.R.: Flows in Networks. Princeton University Press, Princeton (1962)MATHGoogle Scholar
  13. 13.
    Garg, N., Könemann, J.: Faster and simpler algorithms for multicommodity flow and other fractional packing problems. In: Proc. FOCS, p. 300 (1998)Google Scholar
  14. 14.
    Hall, A., Hippler, S., Skutella, M.: Multicommodity flows over time: Efficient algorithms and complexity. Theoretical Computer Science 379, 387–404 (2007)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Hoppe, B., Tardos, É.: The quickest transshipment problem. Mathematics of Operations Research 25, 36–62 (2000)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Hoppe, B.E.: Efficient dynamic network flow algorithms. PhD thesis, Cornell (1995)Google Scholar
  17. 17.
    Klinz, B., Woeginger, G.J.: Minimum cost dynamic flows: The series-parallel case. In: Balas, E., Clausen, J. (eds.) IPCO 1995. LNCS, vol. 920, pp. 329–343. Springer, Heidelberg (1995)Google Scholar
  18. 18.
    Kröller, A., Fekete, S.P., Pfisterer, D., Fischer, S.: Deterministic boundary recognition and topology extraction for large sensor networks. In: Proc. SODA, pp. 1000–1009 (2006)Google Scholar
  19. 19.
    Madan, R., Lall, S.: Distributed algorithms for maximum lifetime routing in wireless sensor networks. IEEE Transactions on Wireless Communications 5(8), 2185–2193 (2006)CrossRefGoogle Scholar
  20. 20.
    Madan, R., Luo, Z.-Q., Lall, S.: A distributed algorithm with linear convergence for maximum lifetime routing in wireless networks. In: Proc. Allerton Conference, pp. 896–905 (2005)Google Scholar
  21. 21.
    Peleg, D.: Distributed computing: a locality-sensitive approach. SIAM, Philadelphia (2000)MATHGoogle Scholar
  22. 22.
    Powell, W.B., Jaillet, P., Odoni, A.: Stochastic and dynamic networks and routing. In: Network Routing, ch. 3. Handbooks in Operations Research and Management Science, vol. 8, pp. 141–295. North–Holland, Amsterdam, The Netherlands (1995)Google Scholar
  23. 23.
    Ran, B., Boyce, D.E.: Modelling Dynamic Transportation Networks. Springer, Heidelberg (1996)Google Scholar
  24. 24.
    Sankar, A., Liu, Z.: Maximum lifetime routing in wireless ad-hoc networks. In: Proc. INFOCOM, pp. 1089–1097 (2004)Google Scholar
  25. 25.
    Schmid, S., Wattenhofer, R.: Algorithmic models for sensor networks. In: Proc. IPDPS (2006)Google Scholar
  26. 26.
    Wattenhofer, R.: Sensor networks: Distributed algorithms reloaded - or revolutions? In: Flocchini, P., Gąsieniec, L. (eds.) SIROCCO 2006. LNCS, vol. 4056, pp. 24–28. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  27. 27.
    Zussman, G., Segall, A.: Energy efficient routing in ad hoc disaster recovery networks. In: Proc. INFOCOM, pp. 682–691 (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Sándor P. Fekete
    • 1
  • Alexander Hall
    • 2
  • Ekkehard Köhler
    • 3
  • Alexander Kröller
    • 1
  1. 1.Algorithms GroupBraunschweig Institute of TechnologyBraunschweigGermany
  2. 2.EECS DepartmentUC BerkeleyUSA
  3. 3.Mathematical InstituteBrandenburg University of TechnologyCottbusGermany

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