Improved approximation algorithms for hitting 3-vertex paths

TL;DR

This paper introduces a 9/4-approximation algorithm for the weighted cluster vertex deletion problem, improving previous bounds and supporting the conjecture of a potential 2-approximation solution.

Contribution

The paper presents a novel 9/4-approximation algorithm for weighted cluster vertex deletion using local ratio, advancing the approximation bounds for this problem.

Findings

Developed a 9/4-approximation algorithm for weighted problem

Provided evidence supporting the conjecture of a 2-approximation

Contrasted with UGC-hardness of similar triangle deletion problem

Abstract

We study the problem of deleting a minimum cost set of vertices from a given vertex-weighted graph in such a way that the resulting graph has no induced path on three vertices. This problem is often called cluster vertex deletion in the literature and admits a straightforward 3-approximation algorithm since it is a special case of the vertex cover problem on a 3-uniform hypergraph. Recently, You, Wang, and Cao described an efficient 5/2-approximation algorithm for the unweighted version of the problem. Our main result is a 9/4-approximation algorithm for arbitrary weights, using the local ratio technique. We further conjecture that the problem admits a 2-approximation algorithm and give some support for the conjecture. This is in sharp contrast with the fact that the similar problem of deleting vertices to eliminate all triangles in a graph is known to be UGC-hard to approximate to within a ratio better than 3, as proved by Guruswami and Lee.