Path cover using only short paths

We study a variant of the well-known Path Cover problem where the candidate paths in a solution have orders up to a fixed integer k. In Path Cover, one finds a minimum number of vertex-disjoint paths in an input graph to cover all the vertices; in our variant, not all paths but only those short ones, i.e., containing up to k vertices, can be used as candidates. The problem is NP-hard when $k\ge 3$; in the literature, there exist quite a number of approximation algorithms, especially for small k's. We present an improved $\frac{k}{3}$-approximation algorithm for $k\in \{6,7,8\}$, an improved $\frac{55}{31}$-approximation algorithm for $k=5$, and an improved $\frac{8}{5}$-approximation algorithm for $k=4$. The novelty inside these improved algorithms is observing a close connection between an optimal path cover and a certain polynomial-time computed edge set.