Advertisement

Bin Packing with Rejection Revisited

  • Leah Epstein
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4368)

Abstract

We consider the following generalization of bin packing. Each item is associated with a size bounded by 1, as well as a rejection cost, that an algorithm must pay if it chooses not to pack this item. The cost of an algorithm is the sum of all rejection costs of rejected items plus the number of unit sized bins used for packing all other items.

We first study the offline version of the problem and design an APTAS for it. This is a non-trivial generalization of the APTAS given by Fernandez de la Vega and Lueker for the standard bin packing problem. We further give an approximation algorithm of absolute approximation ratio \(\frac 32\), this value is best possible unless P= NP.

Finally, we study an online version of the problem. For the bounded space variant, where only a constant number of bins can be open simultaneously, we design a sequence an algorithms whose competitive ratios tend to the best possible asymptotic competitive ratio. We show that our algorithms have the same asymptotic competitive ratios as these known for the standard problem, whose ratios tend to Π ∞ ≈1.691. Furthermore, we introduce an unbounded space algorithm which achieves a much smaller asymptotic competitive ratio. All our results improve upon previous results of Dósa and He.

Keywords

Competitive Ratio Online Algorithm Performance Ratio Polynomial Time Approximation Scheme Large Item 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bartal, Y., Leonardi, S., Marchetti-Spaccamela, A., Sgall, J., Stougie, L.: Multiprocessor scheduling with rejection. SIAM Journal on Discrete Mathematics 13(1), 64–78 (2000)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Caprara, A., Kellerer, H., Pferschy, U.: Approximation schemes for ordered vector packing problems. Naval Research Logistics 92, 58–69 (2003)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Coffman, E.G., Garey, M.R., Johnson, D.S.: Approximation algorithms for bin packing: A survey. In: Hochbaum, D. (ed.) Approximation algorithms. PWS Publishing Company (1997)Google Scholar
  4. 4.
    Csirik, J., Woeginger, G.J.: On-line packing and covering problems. In: Fiat, A., Woeginger, G.J. (eds.) Online Algorithms: The State of the Art, pp. 147–177 (1998)Google Scholar
  5. 5.
    de la Vega, W.F., Lueker, G.S.: Bin packing can be solved within 1 + ε in linear time. Combinatorica 1, 349–355 (1981)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Dósa, G., He, Y.: Bin packing problems with rejection penalties and their dual problems. Information and Computation 204(5), 795–815 (2006)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Engels, D.W., Karger, D.R., Kolliopoulos, S.G., Sengupta, S., Uma, R.N., Wein, J.: Techniques for scheduling with rejection. Journal of Algorithms 49(1), 175–191 (2003)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hochbaum, D.S., Shmoys, D.B.: Using dual approximation algorithms for scheduling problems: theoretical and practical results. Journal of the ACM 34(1), 144–162 (1987)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hoogeveen, H., Skutella, M., Woeginger, G.J.: Preemptive scheduling with rejection. In: Paterson, M. (ed.) ESA 2000. LNCS, vol. 1879, pp. 268–277. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  10. 10.
    Johnson, D.S., Demers, A., Ullman, J.D., Garey, M.R., Graham, R.L.: Worst-case performance bounds for simple one-dimensional packing algorithms. SIAM Journal on Computing 3, 256–278 (1974)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Johnson, D.S.: Fast algorithms for bin packing. Journal of Computer and System Sciences 8, 272–314 (1974)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Karmarkar, N., Karp, R.M.: An efficient approximation scheme for the one-dimensional bin-packing problem. In: Proceedings of the 23rd Annual Symposium on Foundations of Computer Science (FOCS 1982), pp. 312–320 (1982)Google Scholar
  13. 13.
    Lee, C.C., Lee, D.T.: A simple online bin packing algorithm. J. ACM 32(3), 562–572 (1985)CrossRefMATHGoogle Scholar
  14. 14.
    Ramanan, P., Brown, D.J., Lee, C.C., Lee, D.T.: Online bin packing in linear time. Journal of Algorithms 10, 305–326 (1989)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Seiden, S.S.: On the online bin packing problem. J. ACM 49(5), 640–671 (2002)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Simchi-Levi, D.: New worst-case results for the bin-packing problem. Naval Res. Logist. 41(4), 579–585 (1994)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Ullman, J.D.: The performance of a memory allocation algorithm. Technical Report 100, Princeton University, Princeton, NJ (1971)Google Scholar
  18. 18.
    van Vliet, A.: An improved lower bound for online bin packing algorithms. Information Processing Letters 43(5), 277–284 (1992)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Yao, A.C.C.: New algorithms for bin packing. J. ACM 27, 207–227 (1980)CrossRefMATHGoogle Scholar
  20. 20.
    Yue, M.: A simple proof of the inequality FFD(L) ≤ (11/9)OPT(L) + 1, ∀ L, for the FFD bin-packing algorithm. Acta. Math. Appl. Sinica 7, 321–331 (1991)CrossRefMATHGoogle Scholar
  21. 21.
    Zhang, G.: Private communicationGoogle Scholar
  22. 22.
    Zhang, G., Cai, X., Wong, C.K.: Linear time approximation algorithms for bin packing. Operations Research Letters 26, 217–222 (2000)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Leah Epstein
    • 1
  1. 1.Department of MathematicsUniversity of HaifaHaifaIsrael

Personalised recommendations