  Constrained clustering problems generalize classical clustering formulations,
e.g., $k$-median, $k$-means, by imposing additional constraints on the
feasibility of clustering. There has been significant recent progress in
obtaining approximation algorithms for these problems, both in the metric and
the Euclidean settings. However, the outlier version of these problems, where
the solution is allowed to leave out $m$ points from the clustering, is not
well understood. In this work, we give a general framework for reducing the
outlier version of a constrained $k$-median or $k$-means problem to the
corresponding outlier-free version with only $(1+\varepsilon)$-loss in the
approximation ratio. The reduction is obtained by mapping the original instance
of the problem to $f(k,m, \varepsilon)$ instances of the outlier-free version,
where $f(k, m, \varepsilon) = \left( \frac{k+m}{\varepsilon}\right)^{O(m)}$. As
specific applications, we get the following results:
  - First FPT (in the parameters $k$ and $m$) $(1+\varepsilon)$-approximation
algorithm for the outlier version of capacitated $k$-median and $k$-means in
Euclidean spaces with hard capacities.
  - First FPT (in the parameters $k$ and $m$) $(3+\varepsilon)$ and
$(9+\varepsilon)$ approximation algorithms for the outlier version of
capacitated $k$-median and $k$-means, respectively, in general metric spaces
with hard capacities.
  - First FPT (in the parameters $k$ and $m$) $(2-\delta)$-approximation
algorithm for the outlier version of the $k$-median problem under the Ulam
metric. Our work generalizes the known results to a larger class of constrained
clustering problems. Further, our reduction works for arbitrary metric spaces
and so can extend clustering algorithms for outlier-free versions in both
Euclidean and arbitrary metric spaces.
