## Abstract

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≥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≥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.

Original language | English |
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Article number | 105150 |

Journal | Information and Computation |

Volume | 297 |

DOIs | |

State | Published - Mar 2024 |

## Keywords

- Alternating walk
- Approximation algorithm
- Digraph
- Enlarging walk
- Path partition
- Path-cycle cover