Combinatorial Correlation Clustering
Vincent Cohen-Addad, David Rasmussen Lolck, Marcin Pilipczuk, Mikkel Thorup, Shuyi Yan, Hanwen Zhang
TL;DR
The paper advances Correlation Clustering by presenting a combinatorial local-search framework augmented with flips and iterative weighting that beats the long-standing 3-approximation barrier. It introduces a robust preclustering step and pivot-based strategies to escape poor local minima, achieving a $2-\frac{2}{13}$-approximation in near-linear time for sequential/semi-streaming settings and a $2-\frac{1}{8}$-approximation in a constant-round MPC model, with further sublinear and streaming implementations. The results hinge on cost-estimation via sampling, careful concentration bounds, and a multi-layered local-search scheme that remains efficient across models (sublinear, streaming, MPC) while maintaining rigorous guarantees. These advances suggest practical, fast, scalable correlation clustering in real-world data while offering a path toward even better constants in broader computation models. Overall, the work significantly narrows the gap between theory and scalable practice for correlation clustering. $\,$
Abstract
Correlation Clustering is a classic clustering objective arising in numerous machine learning and data mining applications. Given a graph $G=(V,E)$, the goal is to partition the vertex set into clusters so as to minimize the number of edges between clusters plus the number of edges missing within clusters. The problem is APX-hard and the best known polynomial time approximation factor is 1.73 by Cohen-Addad, Lee, Li, and Newman [FOCS'23]. They use an LP with $|V|^{1/ε^{Θ(1)}}$ variables for some small $ε$. However, due to the practical relevance of correlation clustering, there has also been great interest in getting more efficient sequential and parallel algorithms. The classic combinatorial \emph{pivot} algorithm of Ailon, Charikar and Newman [JACM'08] provides a 3-approximation in linear time. Like most other algorithms discussed here, this uses randomization. Recently, Behnezhad, Charikar, Ma and Tan [FOCS'22] presented a $3+ε$-approximate solution for solving problem in a constant number of rounds in the Massively Parallel Computation (MPC) setting. Very recently, Cao, Huang, Su [SODA'24] provided a 2.4-approximation in a polylogarithmic number of rounds in the MPC model and in $\tilde{O} (|E|^{1.5})$ time in the classic sequential setting. They asked whether it is possible to get a better than 3-approximation in near-linear time? We resolve this problem with an efficient combinatorial algorithm providing a drastically better approximation factor. It achieves a $\sim 2-2/13 < 1.847$-approximation in sub-linear ($\tilde O(|V|)$) sequential time or in sub-linear ($\tilde O(|V|)$) space in the streaming setting. In the MPC model, we give an algorithm using only a constant number of rounds that achieves a $\sim 2-1/8 < 1.876$-approximation.