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arXiv · 2611.04784

Optimal Approximation for Constrained Correlation Clustering via Repulsion

Yixin Cao·Ying Xu
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Abstract

The best-known approximation factor for constrained correlation clustering stood at three for nearly two decades, until a recent improvement to below $16/7$. We give a randomized polynomial-time $(2+\varepsilon)$-approximation using \emph{Repulsion}, a recursive operation that separates a metric ball from its complement. Repulsion naturally respects hard constraints and admits a short potential-based analysis; as a warm-up, radius-$2/3$ Repulsion rounds the standard metric LP into a feasible partition of cost at most three times its objective. The $(2+\varepsilon)$-approximation takes a two-stage approach, first removing must-links with arbitrarily small loss in the approximation ratio. The second stage combines lossless, constraint-aware preclustering with the bounded sub-cluster LP framework of Cao, Cohen-Addad, Lee, Li, Newman, and Vogl (STOC 2024). Repeated positive conditioning and sampling produce candidate sets, which radius-one Repulsion partitions into feasible clusters at controlled cost. We also give an approximation-preserving reduction from vertex cover to cluster deletion, a special case with only cannot-links. Apart from establishing a matching factor-two hardness threshold under the Unique Games Conjecture, it motivates our two-stage approach and Repulsion.

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Yixin Cao, Ying Xu. 2026-11-03. Optimal Approximation for Constrained Correlation Clustering via Repulsion. https://arxiv.org/abs/2611.04784

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