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Partial inverse min–max spanning tree problem under the weighted bottleneck hamming distance

Author

Listed:
  • Qingzhen Dong

    (Lanzhou University)

  • Xianyue Li

    (Lanzhou University)

  • Yu Yang

    (Lanzhou University)

Abstract

Min–max spanning tree problem is a classical problem in combinatorial optimization. Its purpose is to find a spanning tree to minimize its maximum edge in a given edge weighted graph. Given a connected graph G, an edge weight vector w and a forest F, the partial inverse min–max spanning tree problem (PIMMST) is to find a new weighted vector $$w^*$$ w ∗ , so that F can be extended into a min–max spanning tree with respect to $$w^*$$ w ∗ and the gap between w and $$w^*$$ w ∗ is minimized. In this paper, we research PIMMST under the weighted bottleneck Hamming distance. Firstly, we study PIMMST with value of optimal tree restriction, a variant of PIMMST, and show that this problem can be solved in strongly polynomial time. Then, by characterizing the properties of the value of optimal tree, we present first algorithm for PIMMST under the weighted bottleneck Hamming distance with running time $$O(|E|^2\log |E|)$$ O ( | E | 2 log | E | ) , where E is the edge set of G. Finally, by giving a necessary and sufficient condition to determine the feasible solution of this problem, we present a better algorithm for this problem with running time $$O(|E|\log |E|)$$ O ( | E | log | E | ) . Moreover, we show that these algorithms can be generalized to solve these problems with capacitated constraint.

Suggested Citation

  • Qingzhen Dong & Xianyue Li & Yu Yang, 2023. "Partial inverse min–max spanning tree problem under the weighted bottleneck hamming distance," Journal of Combinatorial Optimization, Springer, vol. 46(4), pages 1-18, November.
    • Handle: RePEc:spr:jcomop:v:46:y:2023:i:4:d:10.1007_s10878-023-01093-8
    DOI: 10.1007/s10878-023-01093-8
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    References listed on IDEAS

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    1. Xianyue Li & Xichao Shu & Huijing Huang & Jingjing Bai, 2019. "Capacitated partial inverse maximum spanning tree under the weighted Hamming distance," Journal of Combinatorial Optimization, Springer, vol. 38(4), pages 1005-1018, November.
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    3. Xianyue Li & Zhao Zhang & Ding-Zhu Du, 2018. "Partial inverse maximum spanning tree in which weight can only be decreased under $$l_p$$ l p -norm," Journal of Global Optimization, Springer, vol. 70(3), pages 677-685, March.
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    6. Xiucui Guan & Xinyan He & Panos M. Pardalos & Binwu Zhang, 2017. "Inverse max $$+$$ + sum spanning tree problem under Hamming distance by modifying the sum-cost vector," Journal of Global Optimization, Springer, vol. 69(4), pages 911-925, December.
    7. Hui Wang & Xiucui Guan & Qiao Zhang & Binwu Zhang, 2021. "Capacitated inverse optimal value problem on minimum spanning tree under bottleneck Hamming distance," Journal of Combinatorial Optimization, Springer, vol. 41(4), pages 861-887, May.
    8. Xianyue Li & Zhao Zhang & Ruowang Yang & Heping Zhang & Ding-Zhu Du, 2020. "Approximation algorithms for capacitated partial inverse maximum spanning tree problem," Journal of Global Optimization, Springer, vol. 77(2), pages 319-340, June.
    9. Cai, Mao-Cheng & Duin, C.W. & Yang, Xiaoguang & Zhang, Jianzhong, 2008. "The partial inverse minimum spanning tree problem when weight increase is forbidden," European Journal of Operational Research, Elsevier, vol. 188(2), pages 348-353, July.
    10. Javad Tayyebi & Ali Reza Sepasian, 2020. "Partial inverse min–max spanning tree problem," Journal of Combinatorial Optimization, Springer, vol. 40(4), pages 1075-1091, November.
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