Cluster Deletion is as Hard to Approximate as Vertex Cover

Abstract

Recent breakthroughs in Cluster Editing have motivated attempts to adapt these approaches to obtain better-than-$2$ approximations for Cluster Deletion. We rule out this possibility under the Unique Games Conjecture: Cluster Deletion is NP-hard to approximate within a factor of $2-\epsilon$ for every fixed $\epsilon>0$, matching the known $2$-approximation [Veldt et al., WWW 2018]. Our approximation-preserving reduction from Vertex Cover also implies NP-hardness of approximation within $\sqrt2-\epsilon$. We also show that better-than-$2$ approximations are possible in restricted settings. We close the paper with a brief discussion of the relationship between Cluster Editing and Bad Triangle Transversal. In particular, we give a $31$-vertex graph~$G$ for which the two optimal values differ, answering an open question of Adriaens and Tatti [ICML 2026].