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Algorithmica

, Volume 81, Issue 11–12, pp 4167–4199 | Cite as

Approximation Algorithms for the Maximum Weight Internal Spanning Tree Problem

  • Zhi-Zhong ChenEmail author
  • Guohui LinEmail author
  • Lusheng Wang
  • Yong Chen
  • Dan Wang
Article
  • 85 Downloads
Part of the following topical collections:
  1. Special Issue: Computing and Combinatorics

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.

Keywords

Maximum weight internal spanning tree Maximum weight matching Approximation algorithm Performance analysis 

Notes

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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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Division of Information System DesignTokyo Denki UniversityHatoyamaJapan
  2. 2.Department of Computing ScienceUniversity of AlbertaEdmontonCanada
  3. 3.Department of Computer ScienceCity University of Hong KongKowloonChina
  4. 4.City University of Hong Kong Shenzhen Research InstituteShenzhenChina
  5. 5.Institute of Operational Research and CyberneticsHangzhou Dianzi UniversityHangzhouChina

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