Approximation algorithms for solving the heterogeneous rooted tree/path cover problems

IF 0.9 4区 数学 Q4 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS
Pengxiang Pan, Junran Lichen, Ping Yang, Jianping Li
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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.

求解异构根树/路径覆盖问题的近似算法
本文考虑了异质有根树木覆盖(HRTC)问题,进一步推广了有根树木覆盖问题。具体来说,给定一个完整图\(G=(V,E; w,f; r)\)和k个施工队,具有非均匀的施工速度\(\lambda _{1}\), \(\lambda _{2}\), \(\ldots \), \(\lambda _{k}\),其中\(r\in V\)是一个固定的公根,\(w:E\rightarrow {\mathbb {R}}^{+}\)是一个边权函数,满足三角不等式,\(f:V\rightarrow {\mathbb {R}}^{+}_{0}\)(即\({\mathbb {R}}^{+}\cup \{0\})\)是一个顶点权函数,\(f(r)=0\)),我们被要求为这k个施工队找到k棵树。每棵树都有相同的根r,并且共同覆盖V中的所有顶点,目标是最小化k个施工队的最大完工时间,其中每个施工队的完工时间等于其相关树的总施工权除以其施工速度。此外,将HRTC问题中的k棵树替换为k条路径,我们还考虑了异构根路径覆盖(HRPC)问题。我们的主要贡献如下。(1)给定任意小常数\(\delta >0\),我们首先设计了一个\(58.3286(1+\delta )\) -逼近算法来解决HRTC问题,该算法运行时间为\(O(n^{2}(n+\frac{\log n}{\delta })+\log (w(E)+f(V)))\)。同时,我们提出了一种简单的\(116.6572(1+\delta )\) -近似算法来解决HRPC问题,其时间复杂度与上述算法相同。(2)我们提出了一个\(\max \{2\rho , 2+\rho -\frac{2}{k}\}\) -近似算法来解决HRTC问题,该算法运行时间为\(O(n^{2})\),其中\(\rho \)为最大团队速度与最小团队速度之比。同时,我们可以证明上述\(\max \{2\rho , 2+\rho -\frac{2}{k}\}\) -近似算法也解决了HRPC问题。
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来源期刊
Journal of Combinatorial Optimization
Journal of Combinatorial Optimization 数学-计算机:跨学科应用
CiteScore
2.00
自引率
10.00%
发文量
83
审稿时长
6 months
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.
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