Min-Max Correlation Clustering via MultiCut

Correlation clustering is a fundamental combinatorial optimization problem arising in many contexts and applications that has been the subject of dozens of papers in the literature. In this problem we are given a general weighted graph where each edge is labeled positive or negative. The goal is to obtain a partitioning (clustering) of the vertices that minimizes disagreements - weight of negative edges trapped inside a cluster plus positive edges between different clusters. Most of the papers on this topic mainly focus on minimizing total disagreement, a global objective for this problem. In this paper, we study a cluster-wise objective function that asks to minimize the maximum number of disagreements of each cluster, which we call min-max correlation clustering. The min-max objective is a natural objective that respects the quality of every cluster. In this paper, we provide the first nontrivial approximation algorithm for this problem achieving an $\mathcal{O}(\sqrt{\log n\cdot\max{\log(|E^{-|),\log(k)}})$} approximation for general weighted graphs, where $|E^-|$ denotes the number of negative edges and $k$ is the number of clusters in the optimum solution. To do so, we also obtain a corresponding result for multicut where we wish to find a multicut solution while trying to minimize the total weight of cut edges on every component. The results are then further improved to obtain (i) $\mathcal{O}(r^{2)$-approximation} for min-max correlation clustering and min-max multicut for graphs that exclude $K_{r,r}$ minors (ii) a 14-approximation for the min-max correlation clustering on complete graphs.