Approximation algorithms for the directed path partition problems

TL;DR

This paper introduces new approximation algorithms for the directed path partition problem, achieving better approximation ratios for various values of k by novel augmentation techniques and cycle elimination methods.

Contribution

It presents the first $k/2$-approximation algorithm for the directed path partition problem and improves ratios for specific k values using innovative augmentation and cycle elimination.

Findings

First $k/2$-approximation algorithm for directed case

Improved approximation ratios for k=3 and k≥7

Novel augmentation paths and cycle elimination techniques

Abstract

Given a directed graph $G = (V, E)$, the $k$-path partition problem is 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 and others. Its special case on undirected graphs has received much attention recently, but the general directed version is seemingly untouched in the literature. We present the first $k/2$-approximation algorithm, for any $k \ge 3$, based on a novel concept of augmenting path to minimize the number of singletons in the partition. When $k \ge 7$, we present an improved $(k+2)/3$-approximation algorithm based on the maximum path-cycle cover followed by a careful $2$-cycle elimination process. When $k = 3$, we define the second novel kind of augmenting paths and propose an improved $13/9$-approximation algorithm.