Pengxiang Pan, Junran Lichen, Ping Yang, Jianping Li
{"title":"Approximation algorithms for solving the heterogeneous rooted tree/path cover problems","authors":"Pengxiang Pan, Junran Lichen, Ping Yang, Jianping Li","doi":"10.1007/s10878-025-01278-3","DOIUrl":null,"url":null,"abstract":"<p>In this paper, we consider the heterogeneous rooted tree cover (HRTC) problem, which further generalizes the rooted tree cover problem. Specifically, given a complete graph <span>\\(G=(V,E; w,f; r)\\)</span> and <i>k</i> construction teams, having nonuniform construction speeds <span>\\(\\lambda _{1}\\)</span>, <span>\\(\\lambda _{2}\\)</span>, <span>\\(\\ldots \\)</span>, <span>\\(\\lambda _{k}\\)</span>, where <span>\\(r\\in V\\)</span> is a fixed common root, <span>\\(w:E\\rightarrow {\\mathbb {R}}^{+}\\)</span> is an edge-weight function, satisfying the triangle inequality, and <span>\\(f:V\\rightarrow {\\mathbb {R}}^{+}_{0}\\)</span> (<i>i.e., </i> <span>\\({\\mathbb {R}}^{+}\\cup \\{0\\})\\)</span> is a vertex-weight function with <span>\\(f(r)=0\\)</span>, we are asked to find <i>k</i> trees for these <i>k</i> construction teams, each tree having the same root <i>r</i>, and collectively covering all vertices in <i>V</i>, the objective is to minimize the maximum completion time of <i>k</i> construction teams, where the completion time of each team is the total construction weight of its related tree divided by its construction speed. In addition, substituting <i>k</i> paths for <i>k</i> trees in the HRTC problem, we also consider the heterogeneous rooted path cover (HRPC) problem. Our main contributions are as follows. (1) Given any small constant <span>\\(\\delta >0\\)</span>, we first design a <span>\\(58.3286(1+\\delta )\\)</span>-approximation algorithm to solve the HRTC problem, and this algorithm runs in time <span>\\(O(n^{2}(n+\\frac{\\log n}{\\delta })+\\log (w(E)+f(V)))\\)</span>. Meanwhile, we present a simple <span>\\(116.6572(1+\\delta )\\)</span>-approximation algorithm to solve the HRPC problem, whose time complexity is the same as the preceding algorithm. (2) We provide a <span>\\(\\max \\{2\\rho , 2+\\rho -\\frac{2}{k}\\}\\)</span>-approximation algorithm to resolve the HRTC problem, and that algorithm runs in time <span>\\(O(n^{2})\\)</span>, where <span>\\(\\rho \\)</span> is the ratio of the largest team speed to the smallest one. At the same time, we can prove that the preceding <span>\\(\\max \\{2\\rho , 2+\\rho -\\frac{2}{k}\\}\\)</span>-approximation algorithm also resolves the HRPC problem.</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"39 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Optimization","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10878-025-01278-3","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we consider the heterogeneous rooted tree cover (HRTC) problem, which further generalizes the rooted tree cover problem. Specifically, given a complete graph \(G=(V,E; w,f; r)\) and k construction teams, having nonuniform construction speeds \(\lambda _{1}\), \(\lambda _{2}\), \(\ldots \), \(\lambda _{k}\), where \(r\in V\) is a fixed common root, \(w:E\rightarrow {\mathbb {R}}^{+}\) is an edge-weight function, satisfying the triangle inequality, and \(f:V\rightarrow {\mathbb {R}}^{+}_{0}\) (i.e., \({\mathbb {R}}^{+}\cup \{0\})\) is a vertex-weight function with \(f(r)=0\), we are asked to find k trees for these k construction teams, each tree having the same root r, and collectively covering all vertices in V, the objective is to minimize the maximum completion time of k construction teams, where the completion time of each team is the total construction weight of its related tree divided by its construction speed. In addition, substituting k paths for k trees in the HRTC problem, we also consider the heterogeneous rooted path cover (HRPC) problem. Our main contributions are as follows. (1) Given any small constant \(\delta >0\), we first design a \(58.3286(1+\delta )\)-approximation algorithm to solve the HRTC problem, and this algorithm runs in time \(O(n^{2}(n+\frac{\log n}{\delta })+\log (w(E)+f(V)))\). Meanwhile, we present a simple \(116.6572(1+\delta )\)-approximation algorithm to solve the HRPC problem, whose time complexity is the same as the preceding algorithm. (2) We provide a \(\max \{2\rho , 2+\rho -\frac{2}{k}\}\)-approximation algorithm to resolve the HRTC problem, and that algorithm runs in time \(O(n^{2})\), where \(\rho \) is the ratio of the largest team speed to the smallest one. At the same time, we can prove that the preceding \(\max \{2\rho , 2+\rho -\frac{2}{k}\}\)-approximation algorithm also resolves the HRPC problem.
期刊介绍:
The objective of Journal of Combinatorial Optimization is to advance and promote the theory and applications of combinatorial optimization, which is an area of research at the intersection of applied mathematics, computer science, and operations research and which overlaps with many other areas such as computation complexity, computational biology, VLSI design, communication networks, and management science. It includes complexity analysis and algorithm design for combinatorial optimization problems, numerical experiments and problem discovery with applications in science and engineering.
The Journal of Combinatorial Optimization publishes refereed papers dealing with all theoretical, computational and applied aspects of combinatorial optimization. It also publishes reviews of appropriate books and special issues of journals.