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Bin packing under linear constraints

Author

Listed:
  • Zhenbo Wang

    (Tsinghua University)

  • Kameng Nip

    (Tsinghua University)

Abstract

In this paper, we study a bin packing problem in which the sizes of items are determined by k linear constraints, and the goal is to decide the sizes of items and pack them into a minimal number of unit sized bins. We first provide two scenarios that motivate this research. We show that this problem is NP-hard in general, and propose several algorithms in terms of the number of constraints. If the number of constraints is arbitrary, we propose a 2-approximation algorithm, which is based on the analysis of the Next Fit algorithm for the bin packing problem. If the number of linear constraints is a fixed constant, then we obtain a PTAS by combining linear programming and enumeration techniques, and show that it is an optimal algorithm in polynomial time when the number of constraints is one or two. It is well known that the bin packing problem is strongly NP-hard and cannot be approximated within a factor 3 / 2 unless P = NP. This result implies that the bin packing problem with a fixed number of constraints may be simper than the original bin packing problem. Finally, we discuss the case when the sizes of items are bounded.

Suggested Citation

  • Zhenbo Wang & Kameng Nip, 2017. "Bin packing under linear constraints," Journal of Combinatorial Optimization, Springer, vol. 34(4), pages 1198-1209, November.
    • Handle: RePEc:spr:jcomop:v:34:y:2017:i:4:d:10.1007_s10878-017-0140-2
    DOI: 10.1007/s10878-017-0140-2
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    References listed on IDEAS

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    1. Nip, Kameng & Wang, Zhenbo & Wang, Zizhuo, 2016. "Scheduling under linear constraints," European Journal of Operational Research, Elsevier, vol. 253(2), pages 290-297.
    2. Wansoo T. Rhee, 1988. "Optimal Bin Packing with Items of Random Sizes," Mathematics of Operations Research, INFORMS, vol. 13(1), pages 140-151, February.
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    Citations

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    Cited by:

    1. Siyun Zhang & Kameng Nip & Zhenbo Wang, 0. "Related machine scheduling with machine speeds satisfying linear constraints," Journal of Combinatorial Optimization, Springer, vol. 0, pages 1-17.
    2. Siyun Zhang & Kameng Nip & Zhenbo Wang, 2022. "Related machine scheduling with machine speeds satisfying linear constraints," Journal of Combinatorial Optimization, Springer, vol. 44(3), pages 1724-1740, October.
    3. Kameng Nip & Tianning Shi & Zhenbo Wang, 2022. "Some graph optimization problems with weights satisfying linear constraints," Journal of Combinatorial Optimization, Springer, vol. 43(1), pages 200-225, January.

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