An 8/3 Lower Bound for Online Dynamic Bin Packing

  • Prudence W. H. Wong
  • Fencol C. C. Yung
  • Mihai Burcea
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7676)


We study the dynamic bin packing problem introduced by Coffman, Garey and Johnson. This problem is a generalization of the bin packing problem in which items may arrive and depart dynamically. The objective is to minimize the maximum number of bins used over all time. The main result is a lower bound of 8/3 ~2.666 on the achievable competitive ratio, improving the best known 2.5 lower bound. The previous lower bounds were 2.388, 2.428, and 2.5. This moves a big step forward to close the gap between the lower bound and the upper bound, which currently stands at 2.788. The gap is reduced by about 60% from 0.288 to 0.122. The improvement stems from an adversarial sequence that forces an online algorithm \({\mathcal{A}}\) to open 2s bins with items having a total size of s only and this can be adapted appropriately regardless of the current load of other bins that have already been opened by \({\mathcal{A}}\). Comparing with the previous 2.5 lower bound, this basic step gives a better way to derive the complete adversary and a better use of items of slightly different sizes leading to a tighter lower bound. Furthermore, we show that the 2.5-lower bound can be obtained using this basic step in a much simpler way without case analysis.


Competitive Ratio Total Load Online Algorithm Item Size Online Setting 
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  1. 1.
    Balogh, J., Békési, J., Galambos, G., Reinelt, G.: Lower bound for the online bin packing problem with restricted repacking. SIAM J. Comput. 38, 398–410 (2008)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    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 (2004)Google Scholar
  3. 3.
    Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press (1998)Google Scholar
  4. 4.
    Chan, J.W.-T., Lam, T.W., Wong, P.W.H.: Dynamic bin packing of unit fractions items. Theoretical Computer Science 409(3), 172–206 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chan, J.W.-T., Wong, P.W.H., Yung, F.C.C.: On dynamic bin packing: An improved lower bound and resource augmentation analysis. Algorithmica 53(2), 172–206 (2009)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    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 (1998)Google Scholar
  7. 7.
    Coffman Jr., E.G., Garey, M.R., Johnson, D.S.: Dynamic bin packing. SIAM J. Comput. 12(2), 227–258 (1983)MathSciNetMATHCrossRefGoogle 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, pp. 46–93. PWS (1996)Google Scholar
  9. 9.
    Csirik, J., Woeginger, G.J.: On-line Packing and Covering Problems. In: Fiat, A., Woeginger, G.J. (eds.) Online Algorithms 1996. LNCS, vol. 1442, pp. 147–177. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  10. 10.
    Epstein, L., Levy, M.: Dynamic multi-dimensional bin packing. Journal of Discrete Algorithms 8, 356–372 (2010)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, San Francisco (1979)MATHGoogle Scholar
  12. 12.
    Han, X., Peng, C., Ye, D., Zhang, D., Lan, Y.: Dynamic bin packing with unit fraction items revisited. Information Processing Letters 110, 1049–1054 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ivkovic, Z., Lloyd, E.L.: A fundamental restriction on fully dynamic maintenance of bin packing. Inf. Process. Lett. 59(4), 229–232 (1996)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Seiden, S.S.: On the online bin packing problem. J. ACM 49(5), 640–671 (2002)MathSciNetCrossRefGoogle Scholar
  15. 15.
    van Vliet, A.: An improved lower bound for on-line bin packing algorithms. Information Processing Letters 43(5), 277–284 (1992)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Wong, P.W.H., Yung, F.C.C.: Competitive Multi-dimensional Dynamic Bin Packing via L-Shape Bin Packing. In: Bampis, E., Jansen, K. (eds.) WAOA 2009. LNCS, vol. 5893, pp. 242–254. Springer, Heidelberg (2010)CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Prudence W. H. Wong
    • 1
  • Fencol C. C. Yung
    • 1
  • Mihai Burcea
    • 1
  1. 1.Department of Computer ScienceUniversity of LiverpoolUK

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