Constrained Correlation Clustering: Towards Optimality

2026-08-06

AI Summary

This study addresses the correlation clustering problem with pairwise hard constraints, where specified node pairs must either be clustered together or separated, aiming to minimize the number of violated edges. By integrating Sherali-Adams relaxations, a novel pivoting algorithm, and complexity-theoretic reductions, the work establishes a hardness-of-approximation lower bound of 2 for this constrained setting—a stark contrast to the unconstrained case—and improves the best-known approximation upper bound to $16/7 - \gamma$, thereby significantly narrowing the approximability gap from $(24/23, 3]$ to $[2, 16/7 - \gamma)$. Furthermore, it resolves the approximability of the Cluster Deletion problem completely, proving that its existing 2-approximation algorithm is optimal under standard complexity assumptions.

Abstract

In the Correlation Clustering problem, we are given an undirected graph and are tasked with computing a clustering (partition of the nodes) that minimizes the number of violated pairs (edges across different clusters plus non-edges within clusters). In the constrained version of this problem, the goal is to compute a clustering that satisfies additional hard constraints mandating certain pairs to be in the same cluster and certain pairs to be in different clusters. In this work, we identify Constrained Correlation Clustering as a variant of Correlation Clustering for which optimal approximations might be within reach, and make progress towards this front. Constrained Correlation Clustering is APX-Hard, and the optimal approximation factor is known to lie in $(\frac{24}{23},3]$. We significantly tighten this gap, by showing that the optimal approximation factor lies in $[2,\frac{16}{7}-γ)$ for a small constant $γ>0$. Our lower bound of $2$ shows a separation between Correlation Clustering (which admits an $1.485+ε$ approximation) and Constrained Correlation Clustering\footnote{The same hardness result was obtained independently by Cao and Xu~\cite{cao2026clusterdeletionhardapproximate}.}. Our upper bound of $\frac{16}{7}-γ$ uses the Sherali-Adams relaxation and goes beyond straightforward Triangle-Based analysis; more precisely, our algorithm belongs to a natural class of pivoting algorithms for which we prove that a straightforward Triangle-Based analysis cannot prove a better-than-$\frac{16}{7}$ approximation. Finally, as a byproduct of our techniques, we completely resolve the approximability of Cluster Deletion. Cluster Deletion is a well-studied special case of Constrained Correlation Clustering for which a $2$-approximation algorithm is known. We show that this is optimal, as our lower bound holds even for this special case.

Problem

• Constrained Correlation Clustering
• Approximation Factor
• APX-Hard
• Cluster Deletion
• Hard Constraints

Innovation

• Constrained Correlation Clustering
• Approximation Algorithms
• Sherali-Adams Relaxation
• Hardness of Approximation
• Cluster Deletion