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

We give an $(1.3865+\varepsilon)$-approximation for correlation clustering in complete graphs, improving the previous best factor of $1.485+\varepsilon$ of Cao et al.\ (STOC'24). Our two key contributions are independent: an efficient approximate separation oracle for the cluster-LP dual and a new rounding scheme. The dual separation task is the CCMinRatio problem: for signed vertex weights $q$, minimize $cost(S)/q(S)$ over sets with $q(S)>0$; here $cost(S)$ measures the correlation clustering disagreements attributed to $S$ in any clustering in which $S$ is a cluster. We give a randomized $(1+\varepsilon)$-approximation in time $2^{poly(1/\varepsilon)}(n+\langle q\rangle)^{O(1)}$. Via the ellipsoid method, this yields a $(1+\varepsilon)$-approximation of the fractional cluster-LP optimum, along with exactly feasible primal and dual solutions certifying its value and a per-instance certificate. The algorithm works directly on the original instance, without a global preclustering: a new localization technique restricts the search to a small universe while preserving a violation, after which weak regularity handles the resulting dense quadratic minimization. Our rounding scheme retains the cluster-based procedure of Cao et al.\ but uses a continuous conditional pivot rule whose analysis rests on a single variance inequality with explicit weights and an exact computer-assisted verification of the resulting polynomial inequalities. This also places the integrality gap of the cluster LP in the narrow range [4/3,1.3865].