Improved algorithms for Correlation Clustering with local objectives

Correlation Clustering is a powerful graph partitioning model that aims to cluster items based on the notion of similarity between items. An instance of the Correlation Clustering problem consists of a graph $G$ (not necessarily complete) whose edges are labeled by a binary classifier as similar'' and dissimilar’‘. An objective which has received a lot of attention in literature is that of minimizing the number of disagreements: an edge is in disagreement if it is a similar'' edge and is present across clusters or if it is a dissimilar’’ edge and is present within a cluster. Define the disagreements vector to be an $n$ dimensional vector indexed by the vertices, where the $v$-th index is the number of disagreements at vertex $v$. Recently, Puleo and Milenkovic (ICML '16) initiated the study of the Correlation Clustering framework in which the objectives were more general functions of the disagreements vector. In this paper, we study algorithms for minimizing ${\mathrm{\ell }}_q$ norms $(q\ge 1)$ of the disagreements vector for both arbitrary and complete graphs. We present the first known algorithm for minimizing the ${\mathrm{\ell }}_q$ norm of the disagreements vector on arbitrary graphs and also provide an improved algorithm for minimizing the ${\mathrm{\ell }}_q$ norm $(q\ge 1)$ of the disagreements vector on complete graphs. We also study an alternate cluster-wise local objective introduced by Ahmadi, Khuller and Saha (IPCO '19), which aims to minimize the maximum number of disagreements associated with a cluster. We also present an improved ($2+\epsilon$) approximation algorithm for this objective. Finally, we compliment our algorithmic results for minimizing the ${\mathrm{\ell }}_q$ norm of the disagreements vector with some hardness results.