Static to Dynamic Correlation Clustering
Nairen Cao, Vincent Cohen-Addad, Euiwoong Lee, Shi Li, David Rasmussen Lolck, Alantha Newman, Mikkel Thorup, Lukas Vogl, Shuyi Yan, Hanwen Zhang
TL;DR
This work tackles correlation clustering on fully-dynamic graphs under adaptive adversaries. It introduces a general framework that converts static, near-linear-time correlation-clustering algorithms into fully-dynamic ones by operating on a cluster-representation $(\mathcal{C},D)$ where $D$ tracks violated pairs, and recomputing only after controlled update intervals. The framework is instantiated with three leading static approaches—the Pivot pivot-based method, a modern 1.847-approximation local search, and the 1.437-approximation Cluster LP—yielding a fully-dynamic $1.437$-approximation with worst-case update time $O(\log 1/\delta)$ and probability $\delta$ per update, as well as linear-time variants for related subproblems. By combining preclustering, careful randomization analysis, and deamortization, the paper achieves robust dynamic guarantees against adaptive adversaries and improves upon prior dynamic clustering results that relied on oblivious adversaries or weaker approximation factors. The results have practical impact for maintaining high-quality, up-to-date clustering in rapidly changing networks, offering provable guarantees and efficient per-update work.
Abstract
Correlation clustering is a well-studied problem, first proposed by Bansal, Blum, and Chawla [BBC04]. The input is an unweighted, undirected graph. The problem is to cluster the vertices so as to minimizing the number of edges between vertices in different clusters and missing edges between vertices inside the same cluster. This problem has a wide application in data mining and machine learning. We introduce a general framework that transforms existing static correlation clustering algorithms into fully-dynamic ones that work against an adaptive adversary. We show how to apply our framework to known efficient correlation clustering algorithms, starting from the classic $3$-approximate Pivot algorithm from [ACN08]. Applied to the most recent near-linear $1.437$-approximation algorithm from [CCL+25], we get a $1.437$-approximation fully-dynamic algorithm that works with worst-case constant update time. The original static algorithm gets its approximation factor with constant probability, and we get the same against an adaptive adversary in the sense that for any given update step not known to our algorithm, our solution is a $1.437$-approximation with constant probability when we reach this update. Previous dynamic algorithms had approximation factors around $3$ in expectation, and they could only handle an oblivious adversary.