A Tight Lower Bound for an Online Hypercube Packing Problem and Bounds for Prices of Anarchy of a Related Game
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.
We thank the referees for helpful suggestions.
- 1.Aumann, R.J.: Acceptable points in general cooperative n-person games. In: Luce, R.D., Tucker, A.W. (eds.) Contribution to the Theory of Game IV, Annals of Mathematical Study, vol. 40, pp. 287–324. University Press (1959)Google Scholar
- 2.Balogh, J., Békési, J., Dósa, G., Epstein, L., Levin, A.: Lower bounds for several online variants of bin packing. http://arxiv.org/abs/1708.03228 (2017)
- 4.Bilò, V.: On the packing of selfish items. In: Proceedings of the 20th International Parallel and Distributed Processing Symposium, pp. 9–18. IEEE (2006)Google Scholar
- 5.Coffman Jr., E.G., Garey, M.R., Johnson, D.S.: Approximation algorithms for bin packing: a survey, chap. 2. In: Hochbaum, D. (ed.) Approximation Algorithms for NP-hard Problems, pp. 46–93. PWS (1997)Google Scholar
- 13.Fernandes, C.G., Ferreira, C.E., Miyazawa, F.K., Wakabayashi, Y.: Selfish square packing. In: Proceedings of the VI Latin-American Algorithms, Graphs and Optimization Symposium. Electronic Notes in Discrete Mathematics, vol. 37, pp. 369–374 (2011)Google Scholar
- 14.Fernandes, C.G., Ferreira, C.E., Miyazawa, F.K., Wakabayashi, Y.: Prices of anarchy of selfish 2D bin packing games. http://arxiv.org/abs/1707.07882 (2017)
- 15.Heydrich, S., van Stee, R.: Beating the harmonic lower bound for online bin packing. In: ICALP 2016. LIPIcs, vol. 55, pp. 41:1–41:14. Dagstuhl, Germany (2016). http://arxiv.org/abs/1511.00876v5
- 16.Heydrich, S., van Stee, R.: Improved lower bounds for online hypercube packing. http://arxiv.org/abs/1607.01229 (2016)
- 17.Kohayakawa, Y., Miyazawa, F.K., Wakabayashi, Y.: A tight lower bound for an online hypercube packing problem and bounds for prices of anarchy of a related game. http://arxiv.org/abs/1712.06763 (2017)