A Tight Approximation Algorithm for the Cluster Vertex Deletion Problem

TL;DR

This paper presents the first 2-approximation algorithm for the cluster vertex deletion problem, matching the known hardness bound, and explores polyhedral bounds on linear programming relaxations for the problem.

Contribution

It introduces a tight 2-approximation algorithm combining local ratio and true twins, and analyzes polyhedral bounds for LP relaxations.

Findings

First 2-approximation algorithm for the problem

Proves the approximation factor is tight under UGC-hardness

Establishes bounds on LP relaxation approximability

Abstract

We give the first $2$-approximation algorithm for the cluster vertex deletion problem. This is tight, since approximating the problem within any constant factor smaller than $2$ is UGC-hard. Our algorithm combines the previous approaches, based on the local ratio technique and the management of true twins, with a novel construction of a 'good' cost function on the vertices at distance at most $2$ from any vertex of the input graph. As an additional contribution, we also study cluster vertex deletion from the polyhedral perspective, where we prove almost matching upper and lower bounds on how well linear programming relaxations can approximate the problem.