  We consider inapproximability of the correlation clustering problem defined
as follows: 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 respect to the labels is maximized (resp.
minimized). The two complementary problems are called MaxAgree and 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 edge-weighted versions of both problems have been studied as well. Let
S-MaxAgree denote the weighted problem where all weights are taken from set S,
we show that S-MaxAgree with weights bounded by $O(|V|^{1/2-\delta})$
essentially belongs to the same hardness class in the following sense: if there
is a polynomial time algorithm that approximates S-MaxAgree within a factor of
$\lambda = O(\log{|V|})$ with high probability, then for any choice of S',
S'-MaxAgree can be approximated in polynomial time within a factor of $(\lambda
+ \epsilon)$, where $\epsilon > 0$ can be arbitrarily small, with high
probability. A similar statement also holds for $S-MinDisagree. This result
implies it is hard (assuming $NP \neq RP$) to approximate unweighted MaxAgree
within a factor of $80/79-\epsilon$, improving upon a previous known factor of
$116/115-\epsilon$ by Charikar et. al. \cite{Chari05}.
