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Truly Asymptotic Lower Bounds for Online Vector Bin Packing

Authors János Balogh, Ilan Reuven Cohen, Leah Epstein, Asaf Levin

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Author Details

János Balogh
  • Institute of Informatics, University of Szeged, Hungary
Ilan Reuven Cohen
  • Faculty of Engineering, Bar-Ilan University, Ramat-Gan, Israel
Leah Epstein
  • Department of Mathematics, University of Haifa, Israel
Asaf Levin
  • Faculty of Industrial Engineering and Management, Technion, Haifa, Israel

Funding

The research of J. Balogh was supported by the project "Extending the activities of the HU-MATHS-IN Hungarian Industrial and Innovation Mathematical Service Network" EFOP-3.6.2-16-2017-00015, and the project "Integrated program for training new generation of scientists in the fields of computer science," EFOP-3.6.3-VEKOP-16-2017-00002, supported by the European Union and co-funded by the European Social Fund. The research of A. Levin was partially supported by ISF - Israeli Science Foundation grant number 308/18.

Acknowledgements

The results of this paper are based on the arxiv versions [J. Balogh et al., 2020; L. Babel et al., 2004]. Part of the work in [L. Babel et al., 2004] has been done while Ilan Reuven Cohen was a postdoctoral fellow at CWI Amsterdam and TU Eindhoven, he would like to thank Nikhil Bansal for discussions and suggestions related to fractional coloring and ideas from [L. Lov{á}sz, 1975].

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János Balogh, Ilan Reuven Cohen, Leah Epstein, and Asaf Levin. Truly Asymptotic Lower Bounds for Online Vector Bin Packing. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2021.8

Abstract

In this work, we consider online d-dimensional vector bin packing. It is known that no algorithm can have a competitive ratio of o(d/log² d) in the absolute sense, although upper bounds for this problem have always been presented in the asymptotic sense. Since variants of bin packing are traditionally studied with respect to the asymptotic measure, and since the two measures are different, we focus on the asymptotic measure and prove new lower bounds of the asymptotic competitive ratio. The existing lower bounds prior to this work were known to be smaller than 3, even for very large d. Here, we significantly improved on the best known lower bounds of the asymptotic competitive ratio (and as a byproduct, on the absolute competitive ratio) for online vector packing of vectors with d ≥ 3 dimensions, for every dimension d. To obtain these results, we use several different constructions, one of which is an adaptive construction with a lower bound of Ω(√d). Our main result is that the lower bound of Ω(d/log² d) on the competitive ratio holds also in the asymptotic sense. This result holds also against randomized algorithms, and requires a careful adaptation of constructions for online coloring, rather than simple black-box reductions.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Approximation algorithms
  • Mathematics of computing
Keywords
  • Bin packing
  • online algorithms
  • approximation algorithms
  • vector packing

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References

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