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Improved lower bounds for the online bin stretching problem

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Abstract

We use game theory techniques to automatically compute improved lower bounds on the competitive ratio for the bin stretching problem. Using these techniques, we improve the best lower bound for this problem to 19/14. We explain the technique and show that it can be generalized to compute lower bounds for any online or semi-online packing or scheduling problem.

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Notes

  1. https://github.com/mgabay/Bin-Stretching-Lower-Bounds.

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Acknowledgments

This research has been partially supported by Project ICS No 5379 and Belarusian BRFFI Grant (Project F13K-078). The research of the first and the second author has been partially supported by the LabEx PERSYVAL-Lab (ANR–11-LABX-0025).

Author information

Authors and Affiliations

  1. Univ. Grenoble Alpes, CNRS, G-SCOP, 38000, Grenoble, France

    Michaël Gabay & Nadia Brauner

  2. FPMI DMA Department, Belarusian State University, 4 Nezavisimosti Avenue, 220030, Minsk, Belarus

    Vladimir Kotov

Authors
  1. Michaël Gabay
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  2. Nadia Brauner
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  3. Vladimir Kotov
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Corresponding author

Correspondence to Nadia Brauner.

Appendix: Proof of the lower bound

Appendix: Proof of the lower bound

The following tree proves the 19/14 lower bound for the bin stretching problem with 3 bins. This lower bound was obtained using our algorithm with parameters \(m=3\) and \(C=14\).

In the proof, a single decision of the adversary is provided for each decision of the algorithm. We do not explore branches where the algorithm packs the item in a bin, making it larger than or equal to 19. Moreover, we stop exploring a branch when there is a feasible item making all algorithms fail. We denote these latter nodes by “cut: Wmin+ \(w_j \texttt {>=UB}\)”. We recall the input sequence on the leaves. The next items are not added to this sequence. For instance, for a leaf “input: [2,1,7] / cut: Wmin + 3>= UB” the whole input sequence is (2,1,7,3).

In order to make the proof easier to read, we divide the tree in two levels: the first level is a root tree and the second level is a set of subtrees, one for each leaf of the root tree (Figs. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13).

Fig. 2
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Root tree

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Subtree 1, layout [6, 0, 0], items (2,1,3)

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Fig. 4
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Subtree 2, layout [7, 3, 0], items (2,1,3,4)

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Fig. 5
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Subtree 3, layout [3, 3, 4], items (2,1,3,4)

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Subtree 4, layout [7, 3, 0], items (2,1,1,2,4)

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Fig. 7
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Subtree 5, layout [3, 3, 4], items (2,1,1,2,4)

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Fig. 8
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Subtree 6, layout [5, 1, 0], items (2,1,1,2)

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Fig. 9
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Subtree 7, layout [7, 1, 2], items (2,1,1,2,4)

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Fig. 10
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Subtree 8, layout [3, 1, 6], items (2,1,1,2,4)

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Fig. 11
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Subtree 11, layout [2, 1, 1], items (2,1,1)

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Fig. 12
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Subtree 9, layout [3, 5, 2], items (2,1,1,2,4)

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Fig. 13
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Subtree 10, layout [2, 2, 0], items (2,1,1)

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Gabay, M., Brauner, N. & Kotov, V. Improved lower bounds for the online bin stretching problem. 4OR-Q J Oper Res 15, 183–199 (2017). https://doi.org/10.1007/s10288-016-0330-2

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  • DOI: https://doi.org/10.1007/s10288-016-0330-2

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