Correlation clustering is a well-studied problem, first proposed by Bansal, Blum, and Chawla [Mach. Learn. '04]. The input is an unweighted, undirected graph. The problem is to cluster the vertices so as to minimize 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 Ailon, Charikar and Newman [JACM'08]. Applied to the most recent sublinear $1.485$-approximation algorithm from Cao, Cohen-Addad, Lee, Li, Lolck, Newman, Thorup, Vogl, Yan and Zhang [STOC'25], we get a $1.485$-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 $1.485$-approximation with constant probability when we reach this update. Most of previous dynamic algorithms, including the celebrated result from Behnezhad, Charikar, Ma and Tan [FOCS'19], had approximation factors around $3$ in expectation, and they could only handle an oblivious adversary. A recent algorithm by Braverman, Dharangutte, Pai, Shah, and Wang [AISTATS'25] could handle an adaptive adversary, but it has a large unspecified constant approximation ratio. This contrasts with our general transformation, which works with all the best approximation factors known for the static case.

Correlation Clustering is a fundamental and widely-studied problem in unsupervised learning and data mining. The input is a graph and the goal is to construct a clustering minimizing the number of inter-cluster edges plus the number of missing intra-cluster edges. CCL+24 introduced the cluster LP for Correlation Clustering, which they argued captures the problem much more succinctly than previous linear programming formulations. However, the cluster LP has exponential size, with a variable for every possible set of vertices in the input graph. Nevertheless, CCL+24 showed how to find a feasible solution for the cluster LP in time $O(n^{\text{poly}(1/ε)})$ with objective value at most $(1+ε)$ times the value of an optimal solution for the respective Correlation Clustering instance. Furthermore, they showed how to round a solution to the cluster LP, yielding a $(1.485+ε)$-approximation algorithm for the Correlation Clustering problem. The main technical result of this paper is a new approach to find a feasible solution for the cluster LP with objective value at most $(1+ε)$ of the optimum in time $\widetilde O(2^{\text{poly}(1/ε)} n)$, where $n$ is the number of vertices in the graph. We also show how to implement the rounding within the same time bounds, thus achieving a fast $(1.485+ε)$-approximation algorithm for the Correlation Clustering problem. This bridges the gap between state-of-the-art methods for approximating Correlation Clustering and the recent focus on fast algorithms.

Integer data is typically made differentially private by adding noise from a Discrete Laplace (or Discrete Gaussian) distribution. We study the setting where differential privacy of a counting query is achieved using bit-wise randomized response, i.e., independent, random bit flips on the encoding of the query answer.

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/єΘ(1) variables for some small є. 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 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+є-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 Õ (|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 ∼ 2−2/13 < 1.847-approximation in sub-linear (Õ(|V|)) sequential time or in sub-linear (Õ(|V|)) space in the streaming setting, and it uses only a constant number of rounds in the MPC model.