  Clustering problems such as $k$-Median, and $k$-Means, are motivated from
applications such as location planning, unsupervised learning among others. In
such applications, it is important to find the clustering of points that is not
``skewed'' in terms of the number of points, i.e., no cluster should contain
too many points. This is modeled by capacity constraints on the sizes of
clusters. In an orthogonal direction, another important consideration in
clustering is how to handle the presence of outliers in the data. Indeed, these
clustering problems have been generalized in the literature to separately
handle capacity constraints and outliers. To the best of our knowledge, there
has been very little work on studying the approximability of clustering
problems that can simultaneously handle both capacities and outliers.
  We initiate the study of the Capacitated $k$-Median with Outliers (C$k$MO)
problem. Here, we want to cluster all except $m$ outlier points into at most
$k$ clusters, such that (i) the clusters respect the capacity constraints, and
(ii) the cost of clustering, defined as the sum of distances of each
non-outlier point to its assigned cluster-center, is minimized.
  We design the first constant-factor approximation algorithms for C$k$MO. In
particular, our algorithm returns a (3+\epsilon)-approximation for C$k$MO in
general metric spaces, and a (1+\epsilon)-approximation in Euclidean spaces of
constant dimension, that runs in time in time $f(k, m, \epsilon) \cdot
|I_m|^{O(1)}$, where $|I_m|$ denotes the input size. We can also extend these
results to a broader class of problems, including Capacitated
k-Means/k-Facility Location with Outliers, and Size-Balanced Fair Clustering
problems with Outliers. For each of these problems, we obtain an approximation
ratio that matches the best known guarantee of the corresponding outlier-free
problem.
