We present combinatorial approximation algorithms for the weighted correlation clustering problem. In this problem, we have a set of vertices and two weight values for each pair of vertices, denoting their difference and similarity. The goal is to cluster the vertices with minimum total intra-cluster difference weights plus inter-cluster similarity weights. We present two results for weighted instances with $n$ vertices: - A randomized 3-approximation combinatorial algorithm for instances that satisfy probability constraints, running in $O(n^2)$ time. This improves the $O(n^6)$ running time of the previous best-known combinatorial approximation, a 3-approximation algorithm, introduced by Chawla et al. (2015). - A randomized 1.6-approximation combinatorial algorithm for instances that satisfy probability and triangle inequality constraints, running in $O(n^2)$ time. This improves the longstanding combinatorial 2-approximation of Ailon et al. (2008) while matching its running time.
Discrete Mathematics & Theoretical Computer Science