Approximate Dual Separation for the Cluster LP: a 1.387 approximation for Correlation Clustering

Abstract

We give a deterministic $(1.3865+\epsilon)$-approximation for correlation clustering on complete graphs, improving the previous best factor of $1.485+\epsilon$ of Cao et al. (STOC'24). Our first main contribution is an efficient weak separation oracle for the cluster-LP dual. Given signed vertex weights $q$, it either finds a set $S$ with $q(S)>cost(S)$ or certifies that $q/(1+\epsilon)$ is dual feasible, where $cost(S)$ measures the correlation clustering disagreements attributed to $S$ in any clustering in which $S$ is a cluster. Via the ellipsoid method, this yields $(1+\epsilon)$-approximate primal and dual solutions for the fractional cluster LP in deterministic time $2^{poly(1/\epsilon)}n^{O(1)}$ . The separator works directly on the original instance, without global preclustering: a localization argument restricts the search to a small universe, where weak regularity handles the resulting dense quadratic minimization. Complementing this result, we prove that exact dual separation and cluster-LP optimization are NP-hard, even for complete unweighted instances. Our second main contribution is a new rounding of the cluster LP. It combines cluster-based rounding with a continuous conditional pivot rule, and its analysis relies on single variance inequality with explicit weights. This bounds the cluster-LP integrality gap by $1.3865$, near its known lower bound of $4/3$, and gives a per-instance primal--dual certificate of approximation. Finally, we extend our separator to obtain a $(1.92+\epsilon)$-approximation for seeded correlation clustering (where each cluster may contain at most one seed from a prescribed seed set), and to bounded-weight instances, yielding a deterministic polynomial-time approximate implementation of the cluster-insertion primitive used in combinatorial correlation clustering.