Given a graph, the general problem to cover the maximum number of vertices by a collection of vertex-disjoint long paths seemingly escapes from the literature. A path containing at least $k$ vertices is considered long. When $k \le 3$, the problem is polynomial time solvable; when $k$ is the total number of vertices, the problem reduces to the Hamiltonian path problem, which is NP-complete. For a fixed $k \ge 4$, the problem is NP-hard and the best known approximation algorithm for the weighted set packing problem implies a $k$-approximation algorithm. To the best of our knowledge, there is no approximation algorithm directly designed for the general problem; when $k = 4$, the problem admits a $4$-approximation algorithm which was presented recently. We propose the first $(0.4394 k + O(1))$-approximation algorithm for the general problem and an improved $2$-approximation algorithm when $k = 4$. Both algorithms are based on local improvement, and their theoretical performance analyses are done via amortization and their practical performance is examined through simulation studies.

翻译：从图表中可以看出,通过收集顶端脱节长路从文献中似乎可以解脱出来,覆盖顶端的最大数量。 一条至少包含美元顶端的路径被认为是很长的。 当$k\le 3美元时, 问题在于多元时间可溶化; 当 $k美元是顶端的总数时, 问题在于汉密尔顿路径问题, 这是NP- 完成的问题。 对于一个固定的 $k\ge 4 美元, 问题在于 NP 硬, 而对于加权包装问题, 最已知的近似算法意味着 $k$- opproximation 运算法。 根据我们的知识, 没有直接为一般问题设计的近似算法; 当 $k = le 3 3 时, 问题在于多元时间可溶解算法; 当 $k = 4 美元 时, 问题在于最近提出的4 $- apolxxxxx 。 我们建议对一般问题采用第一个( 0.4394 k + O(1)) ad- approcolommlomisation valation or supalislation at the supal be suppress impract imationalizationsisalizationsurview by thesisalizationalizationsalizations