This paper deals with the problem of finding a collection of vertex-disjoint paths in a given graph G=(V,E) such that each path has at least four vertices and the total number of vertices in these paths is maximized. The problem is NP-hard and admits an approximation algorithm which achieves a ratio of 2 and runs in O(|V|^8) time. The known algorithm is based on time-consuming local search, and its authors ask whether one can design a better approximation algorithm by a completely different approach. In this paper, we answer their question in the affirmative by presenting a new approximation algorithm for the problem. Our algorithm achieves a ratio of 1.874 and runs in O(min{|E|^2|V|^2, |V|^5}) time. Unlike the previously best algorithm, ours starts with a maximum matching M of G and then tries to transform M into a solution by utilizing a maximum-weight path-cycle cover in a suitably constructed graph.

翻译：本文研究在给定图G=(V,E)中寻找一组顶点不交路径的问题，要求每条路径至少包含四个顶点，并使这些路径的顶点总数最大化。该问题是NP难的，且存在一个近似比为2、运行时间为O(|V|^8)的近似算法。已知算法基于耗时的局部搜索，其作者提出能否通过完全不同的方法设计出更好的近似算法。本文通过提出该问题的新近似算法对此给出肯定回答。我们的算法近似比为1.874，运行时间为O(min{|E|^2|V|^2, |V|^5})。与先前最优算法不同，我们的算法首先求解G的最大匹配M，然后通过利用适当构造图中的最大权重路径-环覆盖将M转化为一个解。