Correlation clustering is a framework for partitioning datasets based on pairwise similarity and dissimilarity scores, and has been used for diverse applications in bioinformatics, social network analysis, and computer vision. Although many approximation algorithms have been designed for this problem, the best theoretical results rely on obtaining lower bounds via expensive linear programming relaxations. In this paper we prove new relationships between correlation clustering problems and edge labeling problems related to the principle of strong triadic closure. We use these connections to develop new approximation algorithms for correlation clustering that have deterministic constant factor approximation guarantees and avoid the canonical linear programming relaxation. Our approach also extends to a variant of correlation clustering called cluster deletion, that strictly prohibits placing negative edges inside clusters. Our results include 4-approximation algorithms for cluster deletion and correlation clustering, based on simplified linear programs with far fewer constraints than the canonical relaxations. More importantly, we develop faster techniques that are purely combinatorial, based on computing maximal matchings in certain auxiliary graphs and hypergraphs. This leads to a combinatorial 6-approximation for complete unweighted correlation clustering, which is the best deterministic result for any method that does not rely on linear programming. We also present the first combinatorial constant factor approximation for cluster deletion.
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