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 Algorithms for the metric ring star problem with fixed edge-cost ratio Chen, Xujin1,2; Hu, Xiaodong1,2; Jia, Xiaohua3; Tang, Zhongzheng1,2,3; Wang, Chenhao1,2,3; Zhang, Ying4 2021-10-01 Source Publication JOURNAL OF COMBINATORIAL OPTIMIZATION ISSN 1382-6905 Volume 42Issue:3Pages:499-523 Abstract We address the metric ring star problem with fixed edge-cost ratio, abbreviated as RSP. Given a complete graph G=(V,E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=(V,E)$$\end{document} with a specified depot node d is an element of V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\in V$$\end{document}, a nonnegative cost function c is an element of R+E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c\in \mathbb {R}_+<^>E$$\end{document} on E which satisfies the triangle inequality, and an edge-cost ratio M >= 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M\ge 1$$\end{document}, the RSP is to locate a ring R=(V ',E ')\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R=(V',E')$$\end{document} in G, a simple cycle through d or d itself, aiming to minimize the sum of two costs: the cost for constructing ring R, i.e., M center dot n-ary sumation e is an element of E ' c(e)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M\cdot \sum _{e\in E'}c(e)$$\end{document}, and the cost for attaching nodes in V\V '\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V{\setminus } V'$$\end{document} to their closest ring nodes (in R), i.e., n-ary sumation u is an element of V\V ' minv is an element of V ' c(uv)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{u\in V{\setminus } V'}\min _{v\in V'}c(uv)$$\end{document}. We show that the singleton ring d is an optimal solution of the RSP when M >=(|V|-1)/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M\ge (|V|-1)/2$$\end{document}. This particularly implies a |V|-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{|V|-1}$$\end{document}-approximation algorithm for the RSP with any M >= 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M\ge 1$$\end{document}. We present randomized 3-approximation algorithm and deterministic 5.06-approximation algorithm for the RSP, by adapting algorithms for the tour-connected facility location problem (tour-CFLP) and single-source rent-or-buy problem due to Eisenbrand et al. and Williamson and van Zuylen, respectively. Building on the PTAS of Eisenbrand et al. for the tour-CFLP, we give a PTAS for the RSP with |V|/M=O(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|V|/M=O(1)$$\end{document}. We also consider the capacitated RSP (CRSP) which puts an upper limit k on the number of leaf nodes that a ring node can serve, and present a (10+6M/k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(10+6M/k)$$\end{document}-approximation algorithm for this capacitated generalization. Heuristics based on some natural strategies are proposed for both the RSP and CRSP. Simulation results demonstrate that the proposed approximation and heuristic algorithms have good practical performances. Keyword Ring star Approximation algorithms Heuristics Local search Connected facility location DOI 10.1007/s10878-019-00418-w Indexed By SCI Language 英语 WOS Research Area Computer Science ; Mathematics WOS Subject Computer Science, Interdisciplinary Applications ; Mathematics, Applied WOS ID WOS:000712986900010 Publisher SPRINGER Citation statistics Document Type 期刊论文 Identifier http://ir.amss.ac.cn/handle/2S8OKBNM/59540 Collection 应用数学研究所 Corresponding Author Wang, Chenhao Affiliation 1.Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100190, Peoples R China2.Univ Chinese Acad Sci, Sch Math Sci, Beijing 100049, Peoples R China3.City Univ Hong Kong, Dept Comp Sci, Kowloon, Hong Kong, Peoples R China4.Beijing Elect Sci & Technol Inst, Beijing 100070, Peoples R China Recommended CitationGB/T 7714 Chen, Xujin,Hu, Xiaodong,Jia, Xiaohua,et al. Algorithms for the metric ring star problem with fixed edge-cost ratio[J]. JOURNAL OF COMBINATORIAL OPTIMIZATION,2021,42(3):499-523. APA Chen, Xujin,Hu, Xiaodong,Jia, Xiaohua,Tang, Zhongzheng,Wang, Chenhao,&Zhang, Ying.(2021).Algorithms for the metric ring star problem with fixed edge-cost ratio.JOURNAL OF COMBINATORIAL OPTIMIZATION,42(3),499-523. MLA Chen, Xujin,et al."Algorithms for the metric ring star problem with fixed edge-cost ratio".JOURNAL OF COMBINATORIAL OPTIMIZATION 42.3(2021):499-523.
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