Improved approximation algorithms for the k-path partition problem

IF 1.8 3区 数学 Q1 Mathematics
Shiming Li, Wei Yu, Zhaohui Liu
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Abstract

The k-path partition problem (kPP), defined on a graph \(G=(V,E)\), is a well-known NP-hard problem when \(k\ge 3\). The goal of the kPP is to find a minimum collection of vertex-disjoint paths to cover all the vertices in G such that the number of vertices on each path is no more than k. In this paper, we give two approximation algorithms for the kPP. The first one, called Algorithm 1, uses an algorithm for the (0,1)-weighted maximum traveling salesman problem as a subroutine. When G is undirected, the approximation ratio of Algorithm 1 is \(\frac{k+12}{7} -\frac{6}{7k} \), which improves on the previous best-known approximation algorithm for every \(k\ge 7\). When G is directed, Algorithm 1 is a \(\left( \frac{k+6}{4} -\frac{3}{4k}\right) \)-approximation algorithm, which improves the existing best available approximation algorithm for every \(k\ge 10\). Our second algorithm, i.e. Algorithm 2, is a local search algorithm tailored for the kPP in undirected graphs with small k. Algorithm 2 improves on the approximation ratios of the best available algorithm for every \(k=4,5,6\). Combined with Algorithms 1 and 2, we have improved the approximation ratio for the kPP in undirected graphs for each \(k\ge 4\) as well as the approximation ratio for the kPP in directed graphs for each \(k\ge 10\). As for the negative side, we show that for any \(\epsilon >0\) it is NP-hard to approximate the kPP (with k being part of the input) within the ratio \(O(k^{1-\epsilon })\), which implies that Algorithm 1 is asymptotically optimal.

Abstract Image

K 路径分割问题的改进近似计算法
k 路径分割问题(kPP)定义在图(G=(V,E)\)上,是一个众所周知的 NP 难问题(当 \(k\ge 3\) 时)。kPP 的目标是找到覆盖 G 中所有顶点的顶点不相交路径的最小集合,使得每条路径上的顶点数不超过 k。第一种算法称为算法 1,它使用 (0,1)-weighted maximum traveling salesman 问题的算法作为子程序。当G是无向的,算法1的近似率是\(\frac{k+12}{7} -\frac{6}{7k} \),这改进了之前已知的每\(k\ge 7\)的近似算法。当G是有向的,算法1是一个((\left( \frac{k+6}{4} -\frac{3}{4k}\right) \)近似算法,它改进了现有的每一个(kge 10)的最佳近似算法。我们的第二种算法,即算法 2,是一种局部搜索算法,专为 k 较小的无向图中的 kPP 量身定制。算法 2 提高了现有最佳算法对每(k=4,5,6)个图的近似率。结合算法1和算法2,我们改进了无向图中每一个(k=4,5,6)的kPP近似率,以及有向图中每一个(k=10)的kPP近似率。至于反面,我们证明了对于任意(epsilon >0\)来说,在比率\(O(k^{1-\epsilon })\)内逼近kPP(k是输入的一部分)是NP-hard的,这意味着算法1是渐进最优的。
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来源期刊
Journal of Global Optimization
Journal of Global Optimization 数学-应用数学
CiteScore
0.10
自引率
5.60%
发文量
137
审稿时长
6 months
期刊介绍: The Journal of Global Optimization publishes carefully refereed papers that encompass theoretical, computational, and applied aspects of global optimization. While the focus is on original research contributions dealing with the search for global optima of non-convex, multi-extremal problems, the journal’s scope covers optimization in the widest sense, including nonlinear, mixed integer, combinatorial, stochastic, robust, multi-objective optimization, computational geometry, and equilibrium problems. Relevant works on data-driven methods and optimization-based data mining are of special interest. In addition to papers covering theory and algorithms of global optimization, the journal publishes significant papers on numerical experiments, new testbeds, and applications in engineering, management, and the sciences. Applications of particular interest include healthcare, computational biochemistry, energy systems, telecommunications, and finance. Apart from full-length articles, the journal features short communications on both open and solved global optimization problems. It also offers reviews of relevant books and publishes special issues.
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