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On the clustered Steiner tree problem

Author

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  • Bang Ye Wu

    (National Chung Cheng University)

  • Chen-Wan Lin

    (National Chung Cheng University)

Abstract

We investigate the Clustered Steiner tree problem on metric graphs, which is a variant of the Steiner minimum tree problem. In this problem, the required vertices are partitioned into clusters, and the subtrees spanning different clusters must be disjoint in a feasible clustered Steiner tree. In this paper, it is shown that the problem is NP-hard even if the inter-cluster tree and all the local topologies are given, where a local topology specifies the tree structure of required vertices in the same cluster. We show that the Steiner ratio of this problem is lower and upper bounded by three and four, respectively. We also propose a $$(\rho +2)$$ ( ρ + 2 ) -approximation algorithm, where $$\rho $$ ρ is the approximation ratio for the Steiner minimum tree problem, and the approximation ratio can be improved to $$\rho +1$$ ρ + 1 if the local topologies are given. Two variants of this problem are also studied. When the goal is to minimize the inter-cluster cost and ignore the cost of local trees, the problem can be solved in polynomial time. But it is NP-hard if we ask for the minimum cost of local trees among all solutions with minimum inter-cluster cost.

Suggested Citation

  • Bang Ye Wu & Chen-Wan Lin, 2015. "On the clustered Steiner tree problem," Journal of Combinatorial Optimization, Springer, vol. 30(2), pages 370-386, August.
    • Handle: RePEc:spr:jcomop:v:30:y:2015:i:2:d:10.1007_s10878-014-9772-7
    DOI: 10.1007/s10878-014-9772-7
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    References listed on IDEAS

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    1. Tsan-Sheng Hsu & Kuo-Hui Tsai & Da-Wei Wang & D. T. Lee, 2005. "Two Variations of the Minimum Steiner Problem," Journal of Combinatorial Optimization, Springer, vol. 9(1), pages 101-120, February.
    2. Wei Ding & Guoliang Xue, 2014. "Minimum diameter cost-constrained Steiner trees," Journal of Combinatorial Optimization, Springer, vol. 27(1), pages 32-48, January.
    3. Feng Zou & Xianyue Li & Suogang Gao & Weili Wu, 2009. "Node-weighted Steiner tree approximation in unit disk graphs," Journal of Combinatorial Optimization, Springer, vol. 18(4), pages 342-349, November.
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    Cited by:

    1. Mattia D’Emidio & Luca Forlizzi & Daniele Frigioni & Stefano Leucci & Guido Proietti, 2019. "Hardness, approximability, and fixed-parameter tractability of the clustered shortest-path tree problem," Journal of Combinatorial Optimization, Springer, vol. 38(1), pages 165-184, July.
    2. Chen-Wan Lin & Bang Ye Wu, 2017. "On the minimum routing cost clustered tree problem," Journal of Combinatorial Optimization, Springer, vol. 33(3), pages 1106-1121, April.

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