Approximating the minimum directed tree cover

TL;DR

This paper introduces a combinatorial approximation algorithm for the NP-hard minimum directed tree cover problem, achieving a ratio of max{2, ln(D+)} by relating it to the set cover problem.

Contribution

It demonstrates that the weighted set cover problem is a special case of DTCP and provides a novel approximation algorithm with proven ratio bounds.

Findings

Approximation ratio of max{2, ln(D+)}

Set cover problem is a special case of DTCP

Algorithm is purely combinatorial

Abstract

Given a directed graph $G$ with non negative cost on the arcs, a directed tree cover of $G$ is a rooted directed tree such that either head or tail (or both of them) of every arc in $G$ is touched by $T$. The minimum directed tree cover problem (DTCP) is to find a directed tree cover of minimum cost. The problem is known to be $NP$-hard. In this paper, we show that the weighted Set Cover Problem (SCP) is a special case of DTCP. Hence, one can expect at best to approximate DTCP with the same ratio as for SCP. We show that this expectation can be satisfied in some way by designing a purely combinatorial approximation algorithm for the DTCP and proving that the approximation ratio of the algorithm is $\max\{2, \ln(D^+)\}$ with $D^+$ is the maximum outgoing degree of the nodes in $G$.