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.
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References
Brown DJ (1979) A lower bound for on-line one-dimensional bin packing algorithms. Technical report R864, Coordinated Science Laboratory, Urbana, Illinois
Coppersmith D, Paghavan P (1989) Multidimensional on-line bin packing: algorithms and worst case analysis. Oper Res Lett 8: 17–20
Csirik J, van Vliet A (1993) An on-line algorithm for multidimensional bin packing. Oper Res Lett 13: 149–158
Epstein L, van Stee R (2005a) Optimal online algorithms for multidimensional packing problems. SIAM J Comput 35(2): 431–448
Epstein L, van Stee R (2005b) Online square and cube packing. Acta Inform 41(9): 595–606
Epstein L, van Stee R (2007) Bounds for online bounded space hypercube packing. Discrete Optimization 4(2): 185–197
Johnson DS, Demers AJ, Ullman JD, Garey MR, Graham RL (1974) Worst-case performance bounds for simple one-dimensional packing algorithms. SIAM J Comput 3(4): 299–325
Liang FM (1980) A lower bound for online bin packing. Inform Process Lett 10: 76–79
Lee CC, Lee DT (1985) A simple on-line bin-packing algorithm. J ACM 32: 562–572
Miyazawa FK, Wakabayashi Y (2003) Cube packing. Theor Comput Sci 1-3(297): 355–366
Ramanan PV, Brown DJ, Lee CC, Lee DT (1989) On-line bin packing in linear time. J Algorithm 10: 305–326
Seiden SS (2002) On the online bin packing problem. J ACM 49: 640–671
Seiden SS, van Stee R (2003) New bounds for multidimensional packing. Algorithmica 36(3): 261–293
van Vliet A (1992) An improved lower bound for on-line bin packing algorithms. Inform Process Lett 43: 277–284
Yao AC-C (1980) New algorithms for bin packing. J ACM 27: 207–227
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A preliminary version of this paper appeared in Proceedings of fourth Workshop on Approximation and Online Algorithms (WAOA 2006).
The research of Deshi Ye was supported in part by State Key Development Program for Basic Research of China (“973” project, No. 2007CB310900) and NSFC (10601048).
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Han, X., Ye, D. & Zhou, Y. A note on online hypercube packing. Cent Eur J Oper Res 18, 221–239 (2010). https://doi.org/10.1007/s10100-009-0109-z
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DOI: https://doi.org/10.1007/s10100-009-0109-z