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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
AbstractWe 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.
KeywordRing star Approximation algorithms Heuristics Local search Connected facility location
Indexed BySCI
WOS Research AreaComputer Science ; Mathematics
WOS SubjectComputer Science, Interdisciplinary Applications ; Mathematics, Applied
WOS IDWOS:000712986900010
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Document Type期刊论文
Corresponding AuthorWang, Chenhao
Affiliation1.Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100190, Peoples R China
2.Univ Chinese Acad Sci, Sch Math Sci, Beijing 100049, Peoples R China
3.City Univ Hong Kong, Dept Comp Sci, Kowloon, Hong Kong, Peoples R China
4.Beijing Elect Sci & Technol Inst, Beijing 100070, Peoples R China
Recommended Citation
GB/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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