  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
$\ell_q$ norms $(q \geq 1)$ of the disagreements vector for both arbitrary and
complete graphs. We present the first known algorithm for minimizing the
$\ell_q$ norm of the disagreements vector on arbitrary graphs and also provide
an improved algorithm for minimizing the $\ell_q$ norm $(q \geq 1)$ 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 ($2 + \varepsilon$) approximation
algorithm for this objective. Finally, we compliment our algorithmic results
for minimizing the $\ell_q$ norm of the disagreements vector with some hardness
results.
