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Latin American Symposium on Theoretical Informatics

LATIN 2018: LATIN 2018: Theoretical Informatics pp 697–711Cite as

A Tight Lower Bound for an Online Hypercube Packing Problem and Bounds for Prices of Anarchy of a Related Game

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10807))

Abstract

We prove a tight lower bound on the asymptotic performance ratio \(\rho \) of the bounded space online d-hypercube bin packing problem, solving an open question raised in 2005. In the classic d-hypercube bin packing problem, we are given a sequence of d-dimensional hypercubes and we have an unlimited number of bins, each of which is a d-dimensional unit hypercube. The goal is to pack (orthogonally) the given hypercubes into the minimum possible number of bins, in such a way that no two hypercubes in the same bin overlap. The bounded space online d-hypercube bin packing problem is a variant of the d-hypercube bin packing problem, in which the hypercubes arrive online and each one must be packed in an open bin without the knowledge of the next hypercubes. Moreover, at each moment, only a constant number of open bins are allowed (whenever a new bin is used, it is considered open, and it remains so until it is considered closed, in which case, it is not allowed to accept new hypercubes). Epstein and van Stee (SIAM J Comput 35(2):431–448, 2005) showed that \(\rho \) is \(\varOmega (\log d)\) and \(O(d/\log d)\), and conjectured that it is \(\varTheta (\log d)\). We show that \(\rho \) is in fact \(\varTheta (d/\log d)\). To obtain this result, we elaborate on some ideas presented by those authors, and go one step further showing how to obtain better (offline) packings of certain special instances for which one knows how many bins any bounded space algorithm has to use. Our main contribution establishes the existence of such packings, for large enough d, using probabilistic arguments. Such packings also lead to lower bounds for the prices of anarchy of the selfish d-hypercube bin packing game. We present a lower bound of \(\varOmega (d/\log d)\) for the pure price of anarchy of this game, and we also give a lower bound of \(\varOmega (\log d)\) for its strong price of anarchy.

The first author was partially supported by FAPESP (2013/03447-6, 2013/07699-0), CNPq (310974/2013-5, 459335/2014-6) and NUMEC/USP (Project MaCLinC). The second author was partially supported by CNPq (311499/2014-7, 425340/2016-3) and FAPESP (2015/11937-9, 2016/01860-1, 2016/23552-7). The third author was partially supported by CNPq (456792/2014-7, 306464/2016-0) and FAPESP (2015/11937-9).

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Acknowledgements

We thank the referees for helpful suggestions.

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Authors and Affiliations

  1. Institute of Mathematics and Statistics, University of São Paulo, São Paulo, Brazil

    Yoshiharu Kohayakawa & Yoshiko Wakabayashi

  2. Institute of Computing, University of Campinas, Campinas, Brazil

    Flávio Keidi Miyazawa

Authors
  1. Yoshiharu Kohayakawa
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  2. Flávio Keidi Miyazawa
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  3. Yoshiko Wakabayashi
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Correspondence to Yoshiko Wakabayashi .

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Editors and Affiliations

  1. Stony Brook University, Stony Brook, New York, USA

    Michael A. Bender

  2. Rutgers University, New Brunswick, New Jersey, USA

    Martín Farach-Colton

  3. Pace University, New York, New York, USA

    Miguel A. Mosteiro

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Kohayakawa, Y., Miyazawa, F.K., Wakabayashi, Y. (2018). A Tight Lower Bound for an Online Hypercube Packing Problem and Bounds for Prices of Anarchy of a Related Game. In: Bender, M., Farach-Colton, M., Mosteiro, M. (eds) LATIN 2018: Theoretical Informatics. LATIN 2018. Lecture Notes in Computer Science(), vol 10807. Springer, Cham. https://doi.org/10.1007/978-3-319-77404-6_51

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  • DOI: https://doi.org/10.1007/978-3-319-77404-6_51

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