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A note on online hypercube packing

Author

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  • Xin Han
  • Deshi Ye
  • Yong Zhou

Abstract

In this paper, we study an online multi-dimensional bin packing problem where all items are hypercubes. Hypercubes of different size arrive one by one, and we are asked to pack each of them without knowledge of the next pieces so that the number of bins used is minimized. Based on the techniques from one dimensional bin packing and specifically the algorithm Super Harmonic by Seiden (J ACM 49:640–671, 2002), we extend the framework for online bin packing problems developed by Seiden to the hypercube packing problem. To the best of our knowledge, this is the first paper to apply a version of Super Harmonic (and not of the Improved Harmonic algorithm) for online square packing, although the Super Harmonic has been already known before. Note that the best previous result was obtained by Epstein and van Stee (Acta Inform 41(9):595–606, 2005b) using an instance of Improved Harmonic. In this paper we show that Super Harmonic is more powerful than Improve Harmonic for online hypercube packing, and then we obtain better upper bounds on asymptotic competitive ratios. More precisely, we get an upper bound of 2.1187 for square packing and an upper bound of 2.6161 for cube packing, which improve upon the previous upper bounds 2.24437 and 2.9421 (Epstein and van Stee in Acta Inform 41(9):595–606, 2005b) for the two problems, respectively. Copyright Springer-Verlag 2010

Suggested Citation

  • Xin Han & Deshi Ye & Yong Zhou, 2010. "A note on online hypercube packing," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 18(2), pages 221-239, June.
    • Handle: RePEc:spr:cejnor:v:18:y:2010:i:2:p:221-239
    DOI: 10.1007/s10100-009-0109-z
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    Cited by:

    1. Leah Epstein, 2019. "A lower bound for online rectangle packing," Journal of Combinatorial Optimization, Springer, vol. 38(3), pages 846-866, October.
    2. Guosong Yu & Yanling Mao & Jiaoliao Xiao, 2017. "A new upper bound for the online square packing problem in a strip," Journal of Combinatorial Optimization, Springer, vol. 33(4), pages 1411-1420, May.

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