Approximation Algorithms for the Maximum Weight Internal Spanning Tree Problem
- 85 Downloads
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.
KeywordsMaximum weight internal spanning tree Maximum weight matching Approximation algorithm Performance analysis
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.
- 7.Knauer, M., Spoerhase, J.: Better approximation algorithms for the maximum internal spanning tree problem. In: Proceedings of WADS 2009, LNCS 5664, pp. 489–470 (2009)Google Scholar
- 8.Li, W., Chen, J., Wang, J.: Deeper local search for better approximation on maximum internal spanning tree. In: Proceedings of ESA 2014, LNCS 8737, pp. 642–653 (2014)Google Scholar
- 9.Li, W., Wang, J., Chen, J., Cao, Y.: A \(2k\)-vertex kernel for maximum internal spanning tree. In: Proceedings of WADS 2015, LNCS 9214, pp. 495–505 (2015)Google Scholar
- 10.Li, X., Jiang, H., Feng, H.: Polynomial time for finding a spanning tree with maximum number of internal vertices on interval graphs. In: Proceedings of FAW 2016, LNCS 9711, pp. 92–101 (2016)Google Scholar
- 11.Li, X., Zhu, D.: A \(4/3\)-approximation algorithm for the maximum internal spanning tree problem. CoRR, arXiv:1409.3700 (2014)
- 12.Li, X., Zhu, D.: Approximating the maximum internal spanning tree problem via a maximum path-cycle cover. In: Proceedings of ISAAC 2014, LNCS 8889, pp. 467–478 (2014)Google Scholar
- 13.Prieto, E.: Systematic kernelization in FPT algorithm design. In: Ph.D. Thesis, The University of Newcastle, Australia (2005)Google Scholar
- 14.Prieto, E., Sloper, C.: Either/or: using vertex cover structure in designing FPT-algorithms—the case of \(k\)-internal spanning tree. In: Proceedings of WADS 2003, LNCS 2748, pp. 474–483 (2003)Google Scholar
- 17.Salamon, G.: Degree-based spanning tree optimization. In: Ph.D. Thesis, Budapest University of Technology and Economics, Hungary (2009)Google Scholar