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On the minimum routing cost clustered tree problem

Author

Listed:
  • Chen-Wan Lin

    (National Chung Cheng University)

  • Bang Ye Wu

    (National Chung Cheng University)

Abstract

For an edge-weighted graph $$G=(V,E,w)$$ G = ( V , E , w ) , in which the vertices are partitioned into k clusters $$\mathcal {R}=\{R_1,R_2,\ldots ,R_k\}$$ R = { R 1 , R 2 , … , R k } , a spanning tree T of G is a clustered spanning tree if T can be cut into k subtrees by removing $$k-1$$ k - 1 edges such that each subtree is a spanning tree for one cluster. In this paper, we show the inapproximability of finding a clustered spanning tree with minimum routing cost, where the routing cost is the total distance summed over all pairs of vertices. We present a 2-approximation for the case that the input is a complete weighted graph whose edge weights obey the triangle inequality. We also study a variant in which the objective function is the total distance summed over all pairs of vertices of different clusters. We show that the problem is polynomial-time solvable when the number of clusters k is 2 and NP-hard for $$k=3$$ k = 3 . Finally, we propose a polynomial-time 2-approximation algorithm for the case of three clusters.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jcomop:v:33:y:2017:i:3:d:10.1007_s10878-016-0026-8
    DOI: 10.1007/s10878-016-0026-8
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    References listed on IDEAS

    as
    1. 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.
    2. Feremans, Corinne & Labbe, Martine & Laporte, Gilbert, 2003. "Generalized network design problems," European Journal of Operational Research, Elsevier, vol. 148(1), pages 1-13, July.
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