  We present new results for LambdaCC and MotifCC, two recently introduced
variants of the well-studied correlation clustering problem. Both variants are
motivated by applications to network analysis and community detection, and have
non-trivial approximation algorithms. We first show that the standard linear
programming relaxation of LambdaCC has a $\Theta(\log n)$ integrality gap for a
certain choice of the parameter $\lambda$. This sheds light on previous
challenges encountered in obtaining parameter-independent approximation results
for LambdaCC. We generalize a previous constant-factor algorithm to provide the
best results, from the LP-rounding approach, for an extended range of
$\lambda$. MotifCC generalizes correlation clustering to the hypergraph
setting. In the case of hyperedges of degree $3$ with weights satisfying
probability constraints, we improve the best approximation factor from $9$ to
$8$. We show that in general our algorithm gives a $4(k-1)$ approximation when
hyperedges have maximum degree $k$ and probability weights. We additionally
present approximation results for LambdaCC and MotifCC where we restrict to
forming only two clusters.
