Approximating the directed path partition problem

Given a digraph $G=(V,E)$, the k-path partition problem aims to find a minimum collection of vertex-disjoint directed paths, of order at most k, to cover all the vertices. The problem has various applications. Its special case on undirected graphs is NP-hard when $k\ge 3$, and has received much study recently from the approximation algorithm perspective. However, the general problem on digraphs is seemingly untouched in the literature. We fill the gap with the first $k/2$-approximation algorithm, based on a novel concept of enlarging walk to minimize the number of singletons. Secondly, for $k=3$, we define a second novel kind of enlarging walks to greedily reduce the number of 2-paths in the 3-path partition and propose an improved 13/9-approximation algorithm. Lastly, for any $k\ge 7$, we present an improved $(k+2)/3$-approximation algorithm built on the maximum path-cycle cover followed by a careful 2-cycle elimination process.