A 7/3-Approximation for Feedback Vertex Sets in Tournaments

TL;DR

This paper introduces the first 7/3-approximation algorithm for the NP-hard minimum-weight feedback vertex set problem in tournaments, improving the previous best ratio of 5/2, with implications for cycle elimination in directed graphs.

Contribution

The paper presents a novel approximation algorithm achieving a 7/3 ratio for the feedback vertex set problem in tournaments, surpassing prior approximation bounds.

Findings

Achieved a 7/3 approximation ratio, the best known for this problem.

Improved the approximation factor from 5/2 to 7/3.

The algorithm is the first to surpass the previous ratio for this problem.

Abstract

We consider the minimum-weight feedback vertex set problem in tournaments: given a tournament with non-negative vertex weights, remove a minimum-weight set of vertices that intersects all cycles. This problem is $\mathsf{NP}$-hard to solve exactly, and Unique Games-hard to approximate by a factor better than 2. We present the first $7/3$ approximation algorithm for this problem, improving on the previously best known ratio $5/2$ given by Cai et al. [FOCS 1998, SICOMP 2001].