A Note on the Inapproximability of Correlation Clustering

We consider inapproximability issue for the correlation clustering problem introduced by Bansal, Blum and Chawla \cite{Ban04}. Given a graph $G = (V,E)$ where each edge is labeled either "$+$" (similar) or "$-$" (dissimilar), correlation clustering seeks to partition the vertices into clusters so that the number of pairs correctly (resp. incorrectly) classified with respected to the labels is maximized (resp. minimized). The two complementary problems are called \textsc{MaxAgree} and \textsc{MinDisagree}, respectively, and have been studied on complete graphs, where every edge is labeled, and general graphs, where some edge might not have been labeled. Natural weighted version of both problems have been studied as well. In particular, Charikar, Guruswami and Wirth \cite{Chari05} proved that for any $\epsilon > 0$ it is $\mathcal{NP}$-hard to approximate weighted (resp. unweighted) \textsc{MaxAgree} in general graphs within a factor of $79/80+\epsilon$ (resp. $115/116+\epsilon$). Our main contribution in this paper is to improve upon this result by showing that it is also hard (assuming $\mathcal{NP\neq RP}$) to approximate unweighted \textsc{MaxAgree} within a factor of $79/80+\epsilon$.