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 Algorithms for the Prize-Collecting $k$-Steiner Tree Problem Han, Lu1; Wang, Changjun2; Xu, Dachuan3; Zhang, Dongmei4 2022-10-01 Source Publication TSINGHUA SCIENCE AND TECHNOLOGY ISSN 1007-0214 Volume 27Issue:5Pages:785-792 Abstract In this paper, we study the prize-collecting $k$-Steiner tree ($\mathsf{PC}k\mathsf{ST}$) problem. We are given a graph $G=(V, E)$ and an integer $k$. The graph is connected and undirected. A vertex $r\in V$ called root and a subset $R\subseteq V$ called terminals are also given. A feasible solution for the $\mathsf{PC}k\mathsf{ST}$ is a tree $F$ rooted at $r$ and connecting at least $k$ vertices in $R$. Excluding a vertex from the tree incurs a penalty cost, and including an edge in the tree incurs an edge cost. We wish to find a feasible solution with minimum total cost. The total cost of a tree $F$ is the sum of the edge costs of the edges in $F$ and the penalty costs of the vertices not in $F$. We present a simple approximation algorithm with the ratio of 5.9672 for the $\mathsf{PC}k\mathsf{ST}$. This algorithm uses the approximation algorithms for the prize-collecting Steiner tree (PCST) problem and the $k$-Steiner tree ($k\mathsf{ST}$) problem as subroutines. Then we propose a primal-dual based approximation algorithm and improve the approximation ratio to 5. Keyword Steiner trees Costs Approximation algorithms prize-collecting Steiner tree approximation algorithm DOI 10.26599/TST.2021.9010053 Indexed By SCI Language 英语 Funding Project National Natural Science Foundation of China[12001523] ; National Natural Science Foundation of China[11971046] ; National Natural Science Foundation of China[12131003] ; National Natural Science Foundation of China[11871081] ; Scientific Research Project of Beijing Municipal Education Commission[KM201910005012] ; Beijing Natural Science Foundation Project[Z200002] WOS Research Area Computer Science ; Engineering WOS Subject Computer Science, Information Systems ; Computer Science, Software Engineering ; Engineering, Electrical & Electronic WOS ID WOS:000770606800004 Publisher TSINGHUA UNIV PRESS Citation statistics Document Type 期刊论文 Identifier http://ir.amss.ac.cn/handle/2S8OKBNM/60213 Collection 中国科学院数学与系统科学研究院 Corresponding Author Xu, Dachuan Affiliation 1.Beijing Univ Posts & Telecommun, Sch Sci, Beijing 100876, Peoples R China2.Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100190, Peoples R China3.Beijing Univ Technol, Beijing Inst Sci & Engn Comp, Beijing 100124, Peoples R China4.Shandong Jianzhu Univ, Sch Comp Sci & Technol, Jinan 250101, Peoples R China Recommended CitationGB/T 7714 Han, Lu,Wang, Changjun,Xu, Dachuan,et al. Algorithms for the Prize-Collecting $k$-Steiner Tree Problem[J]. TSINGHUA SCIENCE AND TECHNOLOGY,2022,27(5):785-792. APA Han, Lu,Wang, Changjun,Xu, Dachuan,&Zhang, Dongmei.(2022).Algorithms for the Prize-Collecting $k$-Steiner Tree Problem.TSINGHUA SCIENCE AND TECHNOLOGY,27(5),785-792. MLA Han, Lu,et al."Algorithms for the Prize-Collecting $k$-Steiner Tree Problem".TSINGHUA SCIENCE AND TECHNOLOGY 27.5(2022):785-792.
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