Local correlation clustering

Correlation clustering is perhaps the most natural formulation of clustering. Given n objects and a pairwise similarity measure, the goal is to cluster the objects so that, to the best possible extent, similar objects are put in the same cluster and dissimilar objects are put in different clusters. Despite its theoretical appeal, the practical relevance of correlation clustering still remains largely unexplored. This is mainly due to the fact that correlation clustering requires the Θ(n) pairwise similarities as input. In large datasets this is infeasible to compute or even only to store. In this paper we initiate the investigation into local algorithms for correlation clustering, laying the theoretical foundations for clustering “big data”. In local correlation clustering we are given the identifier of a single object and we want to return the cluster to which it belongs in some globally consistent near-optimal clustering, using a small number of similarity queries. Local algorithms for correlation clustering open the door to sublinear-time algorithms, which are particularly useful when the similarity between items is costly to compute, as it is often the case in many practical application domains. They also imply (i) distributed and streaming clustering algorithms, (ii) constant-time estimators and testers for cluster edit distance, and (iii) property-preserving parallel reconstruction algorithms for clusterability. Specifically, we devise a local clustering algorithm attaining a (3, ε)-approximation (a solution with cost at most 3 · OPT + εn, where OPT is the optimal cost). Its running time is O(1/ε) independently of the dataset size. If desired, an explicit approximate clustering for all n objects can be produced in time O(n/ε) (which is provably optimal). We also provide a fully additive (1, ε)-approximation with local query complexity poly(1/ε) and time complexity 2. The explicit clustering can be found in time n · poly(1/ε) + 2. The latter yields the fastest polynomial-time approximation scheme for correlation clustering known to date.