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On Dynamic Bin Packing: An Improved Lower Bound and Resource Augmentation Analysis

  • Wun-Tat Chan
  • Prudence W. H. Wong
  • Fencol C. C. Yung
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4112)

Abstract

We study the dynamic bin packing problem introduced by Coffman, Garey and Johnson [7]. This problem is a generalization of the bin packing problem in which items may arrive and depart from the packing dynamically. The main result in this paper is a lower bound of 2.5 on the achievable competitive ratio, improving the best known 2.428 lower bound [3], and revealing that packing items of restricted form like unit fractions (i.e., of size 1/k for some integer k), which can guarantee a competitive ratio 2.4985 [3], is indeed easier.

We also investigate the resource augmentation analysis on the problem where the on-line algorithm can use bins of size b (> 1) times that of the optimal off-line algorithm. An interesting result is that we prove b = 2 is both necessary and sufficient for the on-line algorithm to match the performance of the optimal off-line algorithm, i.e., achieve 1-competitiveness. Further analysis is made to give a trade-off between the bin size multiplier b and the achievable competitive ratio.

Keywords

Competitive Ratio Item Size Resource Augmentation Harmonic Algorithm Average Case Behavior 
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.

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References

  1. 1.
    Bar-Noy, A., Ladner, R.E., Tamir, T.: Windows scheduling as a restricted version of bin packing. In: Munro, J.I. (ed.) SODA, pp. 224–233. SIAM, Philadelphia (2004)Google Scholar
  2. 2.
    Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press, Cambridge (1998)MATHGoogle Scholar
  3. 3.
    Chan, W.T., Lam, T.W., Wong, P.W.H.: Dynamic bin packing of unit fractions items. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 614–626. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  4. 4.
    Coffman Jr., E.G., Courcoubetis, C., Garey, M.R., Johnson, D.S., Shor, P.W., Weber, R.R., Yannakakis, M.: Bin packing with discrete item sizes, Part I: Perfect packing theorems and the average case behavior of optimal packings. SIAM J. Discrete Math. 13, 38–402 (2000)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Coffman Jr., E.G., Galambos, G., Martello, S., Vigo, D.: Bin packing approximation algorithms: Combinatorial analysis. In: Du, D.-Z., Pardalos, P.M. (eds.) Handbook of Combinatorial Optimization. Kluwer Academic Publishers, Dordrecht (1998)Google Scholar
  6. 6.
    Coffman Jr., E.G., Garey, M., Johnson, D.: Bin packing with divisible item sizes. Journal of Complexity 3, 405–428 (1987)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Coffman Jr., E.G., Garey, M.R., Johnson, D.S.: Dynamic bin packing. SIAM J. Comput. 12(2), 227–258 (1983)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Coffman Jr., E.G., Garey, M.R., Johnson, D.S.: Bin packing approximation algorithms: A survey. In: Hochbaum, D.S. (ed.) Approximation Algorithms for NP-Hard Problems, PWS, pp. 46–93 (1996)Google Scholar
  9. 9.
    Coffman Jr., E.G., Johnson, D.S., McGeoch, L.A., Shor, P.W., Weber, R.R.: Bin packing with discrete item sizes, Part III: Average case behavior of FFD and BFD (in preparation)Google Scholar
  10. 10.
    Coffman Jr., E.G., Johnson, D.S., Shor, P.W., Weber, R.R.: Bin packing with discrete item sizes, Part II: Tight bounds on first fit. Random Structures and Algorithms 10, 69–101 (1997)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Csirik, J., Woeginger, G.J.: Online packing and covering problems. In: Fiat, A. (ed.) Dagstuhl Seminar 1996. LNCS, vol. 1442, pp. 147–177. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  12. 12.
    Csirik, J., Woeginger, G.J.: Resource augmentation for online bounded space bin packing. J. Algorithms 44(2), 308–320 (2002)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Epstein, L., van Stee, R.: Online bin packing with resource augmentation. In: Persiano, G., Solis-Oba, R. (eds.) WAOA 2004. LNCS, vol. 3351, pp. 23–35. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  14. 14.
    Ivkovic, Z., Lloyd, E.L.: Fully dynamic algorithms for bin packing: Being (mostly) myopic helps. SIAM J. Comput. 28(2), 574–611 (1998)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kalyanasundaram, B., Pruhs, K.: Speed is as powerful as clairvoyance. In: 36th Annual Symposium on Foundations of Computer Science, pp. 214–221 (1995)Google Scholar
  16. 16.
    Seiden, S.S.: On the online bin packing problem. J. ACM 49(5), 640–671 (2002)CrossRefMathSciNetGoogle Scholar
  17. 17.
    van Vliet, A.: An improved lower bound for on-line bin packing algorithms. Inf. Process. Lett. 43(5), 277–284 (1992)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Wun-Tat Chan
    • 1
  • Prudence W. H. Wong
    • 2
  • Fencol C. C. Yung
    • 1
  1. 1.Department of Computer ScienceUniversity of Hong KongHong Kong
  2. 2.Department of Computer ScienceUniversity of LiverpoolUK

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