Time-Expanded Packings

  • David Adjiashvili
  • Sandro Bosio
  • Robert Weismantel
  • Rico Zenklusen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8572)

Abstract

We introduce a general model for time-expanded versions of packing problems with a variety of applications. Our notion for time-expanded packings, which we introduce in two natural variations, requires elements to be part of the solution for several consecutive time steps. Despite the fact that the time-expanded counterparts of most combinatorial optimization problems become computationally hard to solve, we present strong approximation algorithms for general dependence systems and matroids, respectively, depending on the considered variant. More precisely, for both notions of time-expanded packings that we introduce, the approximation guarantees we obtain are at most a small constant-factor worse than the best approximation algorithms for the underlying problem in its non-time-expanded version.

Keywords

Berman Summing 

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References

  1. 1.
    Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V.V., Marchetti-Spaccamela, M., Protasi, M.: Complexity and Approximation. Springer (1999)Google Scholar
  2. 2.
    Bar-Noy, A., Bar-Yehuda, R., Freund, A., Naor, J.S., Schieber, B.: A unified approach to approximating resource allocation and scheduling. J. ACM, 735–744 (2000)Google Scholar
  3. 3.
    Bar-Noy, A., Guha, S.: Approximating the throughput of multiple machines in real-time scheduling. SIAM J. Comput. 31, 331–352 (2002)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Berman, P.: A d/2 approximation for maximum weight independent set in d-claw free graphs. Nordic J. of Computing 7, 178–184 (2000)MATHGoogle Scholar
  5. 5.
    Caprara, A.: Packing 2-dimensional bins in harmony. In: Proceedings of the 43rd Symposium on Foundations of Computer Science, FOCS 2002, pp. 490–499 (2002)Google Scholar
  6. 6.
    Duffin, R.J., Karlovitz, L.A.: An infinite linear program with a duality gap. Management Science 12(1), 122–134 (1965)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Ford, L.R., Fulkerson, D.R.: Constructing maximal dynamic flows from static flows. Operations Research 6(3), 419–433 (1958)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Garey, M.R., Johnson, D.S.: Computers and Intractability. A guide to the theory of NP-completeness. W.H. Freeman and Co. (1979)Google Scholar
  9. 9.
    Ibarra, O.H., Kim, C.E.: Fast approximation algorithms for the knapsack and sum of subset problems. J. ACM 22, 463–468 (1975)MATHMathSciNetGoogle Scholar
  10. 10.
    Jansen, K., Zhang, G.: On rectangle packing: maximizing benefits. In: Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2004, pp. 204–213 (2004)Google Scholar
  11. 11.
    Koch, R., Nasrabadi, E., Skutella, M.: Continuous and discrete flows over time: A general model based on measure theory. Mathematical Methods Of Operations Research 73(3), 301–337 (2011)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Kohli, R., Krishnamurti, R.: A total-value greedy heuristic for the integer knapsack problem. Operations Research Letters 12(2), 65–71 (1992)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Lawler, E.L.: Fast approximation algorithms for knapsack problems. In: Proceedings of the 18th Symposium on Foundations of Computer Science, FOCS 1977, pp. 206–213 (1977)Google Scholar
  14. 14.
    Lee, C.C., Lee, D.T.: A simple on-line bin-packing algorithm. J. ACM 32, 562–572 (1985)MATHGoogle Scholar
  15. 15.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley (1994)Google Scholar
  16. 16.
    Pinedo, M.L.: Scheduling: Theory, Algorithms and Systems, 3rd edn. Springer (2008)Google Scholar
  17. 17.
    Schrijver, A.: Combinatorial Optimization, Polyhedra and Efficiency. Springer (2003)Google Scholar
  18. 18.
    Skutella, M.: An introduction to network flows over time. Research Trends in Combinatorial Optimization, pp. 451–482. Springer (2009)Google Scholar
  19. 19.
    Sloane, N.J.A.: Sequences A000058 and A007018. The On-Line Encyclopedia of Integer Sequences, http://oeis.org

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • David Adjiashvili
    • 1
  • Sandro Bosio
    • 1
  • Robert Weismantel
    • 1
  • Rico Zenklusen
    • 1
  1. 1.Institute for Operations ResearchETH ZürichZürichSwitzerland

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