Handling Correlated Rounding Error via Preclustering: A 1.73-approximation for Correlation Clustering

We consider the classic correlation clustering problem: Given a complete graph where edges are labelled either + or −, the goal is to find a partition of the vertices that minimizes the sum of the +edges across parts plus the sum of the −edges within parts. Recently, Cohen-Addad, Lee and Newman [CLN22] gave a 1.995-approximation for the problem using the Sherali-Adams hierarchy, hence beating the integrality gap of 2 of the classic linear program. We significantly improve upon this result by providing a 1.73-approximation for the problem. Our approach brings together a new preprocessing of correlation clustering instances that enables a new LP formulation which combined with the algorithm from [CLN22] yields the improved bound.