Given a fixed positive integer $k$ and a simple undirected graph $G = (V, E)$, the {\em $k^-$-path partition} problem, denoted by $k$PP for short, aims to find a minimum collection $\cal{P}$ of vertex-disjoint paths in $G$ such that each path in $\cal{P}$ has at most $k$ vertices and each vertex of $G$ appears in one path in $\cal{P}$. In this paper, we present a $\frac {k+4}5$-approximation algorithm for $k$PP when $k\in\{9,10\}$ and an improved $(\frac{\sqrt{11}-2}7 k + \frac {9-\sqrt{11}}7)$-approximation algorithm when $k \ge 11$. Our algorithms achieve the current best approximation ratios for $k \in \{ 9, 10, \ldots, 18 \}$. Our algorithms start with a maximum triangle-free path-cycle cover $\cal{F}$, which may not be feasible because of the existence of cycles or paths with more than $k$ vertices. We connect as many cycles in $\cal{F}$ with $4$ or $5$ vertices as possible by computing another maximum-weight path-cycle cover in a suitably constructed graph so that $\cal{F}$ can be transformed into a $k^-$-path partition of $G$ without losing too many edges. Keywords: $k^-$-path partition; Triangle-free path-cycle cover; $[f, g]$-factor; Approximation algorithm

翻译：给定一个固定正整数$k$和一个简单无向图$G = (V, E)$，{\em $k^-$路径划分}问题（简记为$k$PP）旨在寻找$G$中一个最小的顶点不相交路径集合$\cal{P}$，使得$\cal{P}$中的每条路径至多包含$k$个顶点，且$G$的每个顶点恰好出现在$\cal{P}$的一条路径中。本文针对$k\in\{9,10\}$的情况，提出了一个$\frac {k+4}5$近似算法；针对$k \ge 11$的情况，提出了一个改进的$(\frac{\sqrt{11}-2}7 k + \frac {9-\sqrt{11}}7)$近似算法。我们的算法在$k \in \{ 9, 10, \ldots, 18 \}$范围内达到了当前最佳的近似比。算法从一个最大无三角形路径-圈覆盖$\cal{F}$开始，该覆盖可能因存在圈或包含超过$k$个顶点的路径而不可行。我们通过在适当构造的图中计算另一个最大权路径-圈覆盖，尽可能多地连接$\cal{F}$中长度为$4$或$5$的圈，从而将$\cal{F}$转化为$G$的一个$k^-$路径划分，同时避免损失过多边。关键词：$k^-$路径划分；无三角形路径-圈覆盖；$[f, g]$因子；近似算法