  In the Correlation Clustering problem, we are given a weighted graph $G$ with
its edges labeled as "similar" or "dissimilar" by a binary classifier. The goal
is to produce a clustering that minimizes the weight of "disagreements": the
sum of the weights of "similar" edges across clusters and "dissimilar" edges
within clusters. We study the correlation clustering problem under the
following assumption: Every "similar" edge $e$ has weight
$\mathbf{w}_e\in[\alpha \mathbf{w}, \mathbf{w}]$ and every "dissimilar" edge
$e$ has weight $\mathbf{w}_e\geq \alpha \mathbf{w}$ (where $\alpha\leq 1$ and
$\mathbf{w}>0$ is a scaling parameter). We give a $(3 + 2 \log_e (1/\alpha))$
approximation algorithm for this problem. This assumption captures well the
scenario when classification errors are asymmetric. Additionally, we show an
asymptotically matching Linear Programming integrality gap of $\Omega(\log
1/\alpha)$.
