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, each of order at most k, to cover all the vertices of V. The problem has various applications in facility location, network monitoring, transportation networks and others. Its special case on undirected graphs is NP-hard when k≥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, for any k≥3, based on a novel concept of enlarging walk to minimize the number of singletons in the k-path partition. 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≥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.