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Approximation Algorithms for the Maximum Weight Internal Spanning Tree Problem

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Abstract

Given a vertex-weighted connected graph \(G = (V, E)\), the maximum weight internal spanning tree (MwIST for short) problem asks for a spanning tree T of G such that the total weight of internal vertices in T is maximized. The unweighted variant, denoted as MIST, is NP-hard and APX-hard, and the currently best approximation algorithm has a proven performance ratio of 13 / 17. The currently best approximation algorithm for MwIST only has a performance ratio of \(1/3 - \epsilon \), for any \(\epsilon > 0\). In this paper, we present a simple algorithm based on a novel relationship between MwIST and maximum weight matching, and show that it achieves a significantly better approximation ratio of 1/2. When restricted to claw-free graphs, a special case previously studied, we design a 7/12-approximation algorithm.

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Acknowledgements

The authors are grateful to the reviewers for their insightful comments and for their suggested changes that improve the presentation greatly. ZZC was supported in part by the Grant-in-Aid for Scientific Research of the Ministry of Education, Culture, Sports, Science and Technology of Japan, under Grant No. 24500023. GL was supported by the NSERC Canada and the NSFC Grant No. 61672323; most of his work was done while visiting ZZC at the Tokyo Denki University at Hatoyama. LW is fully supported by a Grant from the Research Grants Council of the Hong Kong Special Administrative Region, China, CityU 11256116. YC was supported in part by the NSERC Canada, the NSFC Grants Nos. 11771114 and 11571252, and the China Scholarship Council Grant No. 201508330054.

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Authors and Affiliations

  1. Division of Information System Design, Tokyo Denki University, Hatoyama, Saitama, 350-0394, Japan

    Zhi-Zhong Chen

  2. Department of Computing Science, University of Alberta, Edmonton, AB, T6G 2E8, Canada

    Guohui Lin & Yong Chen

  3. Department of Computer Science, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, China

    Lusheng Wang & Dan Wang

  4. City University of Hong Kong Shenzhen Research Institute, Shenzhen Hi-Tech Industrial Park, Nanshan District, Shenzhen, China

    Lusheng Wang

  5. Institute of Operational Research and Cybernetics, Hangzhou Dianzi University, Hangzhou, 310018, Zhejiang, China

    Yong Chen

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  1. Zhi-Zhong Chen
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  2. Guohui Lin
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  3. Lusheng Wang
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  4. Yong Chen
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  5. Dan Wang
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Correspondence to Zhi-Zhong Chen or Guohui Lin.

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An extended abstract appears in the Proceedings of COCOON 2017. LNCS 10392, pp. 124–136.

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Chen, ZZ., Lin, G., Wang, L. et al. Approximation Algorithms for the Maximum Weight Internal Spanning Tree Problem. Algorithmica 81, 4167–4199 (2019). https://doi.org/10.1007/s00453-018-00533-w

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