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Exact Algorithms for the Bottleneck Steiner Tree Problem

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Abstract

Given n points, called terminals, in the plane ℝ2 and a positive integer k, the bottleneck Steiner tree problem is to find k Steiner points from ℝ2 and a spanning tree on the n+k points that minimizes its longest edge length. Edge length is measured by an underlying distance function on ℝ2, usually, the Euclidean or the L 1 metric. This problem is known to be NP-hard. In this paper, we study this problem in the L p metric for any 1≤p≤∞, and aim to find an exact algorithm which is efficient for small fixed k. We present the first fixed-parameter tractable algorithm running in f(k)⋅nlog 2 n time for the L 1 and the L ∞ metrics, and the first exact algorithm for the L p metric for any fixed rational p with 1<p<∞ whose time complexity is f(k)⋅(n k+nlog n), where f(k) is a function dependent only on k. Note that prior to this paper there was no known exact algorithm even for the L 2 metric.

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Authors and Affiliations

  1. Department of Computer Science, Kyonggi University, Suwon, South Korea

    Sang Won Bae

  2. Division of Computer Science, KAIST, Daejeon, South Korea

    Sunghee Choi & Chunseok Lee

  3. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan

    Shin-ichi Tanigawa

Authors
  1. Sang Won Bae
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  2. Sunghee Choi
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  3. Chunseok Lee
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  4. Shin-ichi Tanigawa
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Correspondence to Sang Won Bae.

Additional information

A preliminary version of this paper was presented at the 26th International Symposium on Algorithms and Computation (ISAAC 2009). Work by S.W. Bae was supported by the Contents Convergence Software Research Center funded by the GRRC Program of Gyeonggi Province, South Korea. Work by C. Lee and S. Choi was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2010-0024092). Work by S. Tanigawa was supported by Grant-in-Aid for JSPS Research Fellowship for Young Scientists.

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Bae, S.W., Choi, S., Lee, C. et al. Exact Algorithms for the Bottleneck Steiner Tree Problem. Algorithmica 61, 924–948 (2011). https://doi.org/10.1007/s00453-011-9553-y

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