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

2026-07-30

AI Summary

This work addresses the correlation clustering problem on complete graphs and improves the best-known approximation ratio from 1.485 to $1.3865 + \varepsilon$. The key contributions include the first $(1+\varepsilon)$-approximate separation oracle for the CCMinRatio problem, a novel rounding scheme based on a continuous conditional pivot rule, and an efficient integration of the ellipsoid method, local search, and weak regularity techniques to solve dense quadratic subproblems. This approach narrows the integrality gap of the clique-based linear programming relaxation to the interval $[4/3, 1.3865]$ and enables computer-assisted verification of instance-specific primal-dual feasibility certificates.

Abstract

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].

Problem

• Correlation Clustering
• Approximation Algorithm
• Cluster LP
• Dual Separation
• Integrality Gap

Innovation

• correlation clustering
• approximation algorithm
• LP duality
• separation oracle
• rounding scheme