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 number of the +edges across parts plus the number of the -edges within parts. Recently, Cohen-Addad, Lee and Newman [CLN22] presented a 1.994-approximation algorithm for the problem using the Sherali-Adams hierarchy, hence breaking through the integrality gap of 2 for the classic linear program and improving upon the 2.06-approximation of Chawla, Makarychev, Schramm and Yaroslavtsev [CMSY15].

We significantly improve the state-of-the-art by providing a 1.73-approximation for the problem. Our approach introduces a preclustering of Correlation Clustering instances that allows us to essentially ignore the error arising from the correlated rounding used by [CLN22]. This additional power simplifies the previous algorithm and analysis. More importantly, it enables a new set-based rounding that complements the previous roundings. A combination of these two rounding algorithms yields the improved bound.