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Some graph optimization problems with weights satisfying linear constraints

Author

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  • Kameng Nip

    (Xiamen University)

  • Tianning Shi

    (Tsinghua University)

  • Zhenbo Wang

    (Tsinghua University)

Abstract

In this paper, we study several graph optimization problems in which the weights of vertices or edges are variables determined by several linear constraints, including maximum matching problem under linear constraints (max-MLC), minimum perfect matching problem under linear constraints (min-PMLC), shortest path problem under linear constraints (SPLC) and vertex cover problem under linear constraints (VCLC). The objective of these problems is to decide the weights that are feasible to the linear constraints, and find the optimal solutions of corresponding graph optimization problems among all feasible choices of weights. We find that these problems are NP-hard and are hard to be approximated in general. These findings suggest us to explore various special cases of them. In particular, we show that when the number of constraints is a fixed constant, all these problems are polynomially solvable. Moreover, if the total number of distinct weights is a fixed constant, then max-MLC, min-PMLC and SPLC are polynomially solvable, and VCLC has a 2-approximation algorithm. In addition, we propose approximation algorithms for various cases of max-MLC.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jcomop:v:43:y:2022:i:1:d:10.1007_s10878-021-00754-w
    DOI: 10.1007/s10878-021-00754-w
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    References listed on IDEAS

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    1. Zhenbo Wang & Kameng Nip, 2017. "Bin packing under linear constraints," Journal of Combinatorial Optimization, Springer, vol. 34(4), pages 1198-1209, November.
    2. Gusev, Vasily V., 2020. "The vertex cover game: Application to transport networks," Omega, Elsevier, vol. 97(C).
    3. Nip, Kameng & Wang, Zhenbo & Wang, Zizhuo, 2016. "Scheduling under linear constraints," European Journal of Operational Research, Elsevier, vol. 253(2), pages 290-297.
    4. H. W. Kuhn, 1955. "The Hungarian method for the assignment problem," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 2(1‐2), pages 83-97, March.
    5. Kameng Nip & Zhenbo Wang & Zizhuo Wang, 2017. "Knapsack with variable weights satisfying linear constraints," Journal of Global Optimization, Springer, vol. 69(3), pages 713-725, November.
    6. 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.
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