Cluster Deletion is as Hard to Approximate as Vertex Cover

2026-08-05

AI Summary

This study investigates the approximability limits of the Cluster Deletion problem. By constructing an approximation-preserving reduction from Vertex Cover to Cluster Deletion, it establishes for the first time an equivalence in their approximation hardness. Assuming the Unique Games Conjecture, the paper proves that no polynomial-time algorithm can achieve an approximation ratio better than \(2 - \varepsilon\) for Cluster Deletion on general graphs, thereby establishing 2 as the tight approximation threshold. Furthermore, the work demonstrates that this barrier can be surpassed on certain restricted graph classes. Additionally, it refutes a conjecture positing the equality of optimal values between Cluster Editing and Bad Triangle Transversal by presenting a counterexample based on a 31-vertex graph.

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 $2-ε$ for every fixed $ε>0$, matching the known $2$-approximation [Veldt et al., WWW 2018]. Our approximation-preserving reduction from Vertex Cover also implies NP-hardness of approximation within $\sqrt2-ε$. 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].

Problem

• Cluster Deletion
• Vertex Cover
• Approximation Hardness
• Unique Games Conjecture
• NP-hardness

Innovation

• Cluster Deletion
• Approximation Hardness
• Unique Games Conjecture
• Vertex Cover
• Bad Triangle Transversal