  The Metric $k$-median problem over a metric space $(\mathcal{X}, d)$ is
defined as follows: given a set $L \subseteq \mathcal{X}$ of facility locations
and a set $C \subseteq \mathcal{X}$ of clients, open a set $F \subseteq L$ of
$k$ facilities such that the total service cost, defined as $\Phi(F, C) \equiv
\sum_{x \in C} \min_{f \in F} d(x, f)$, is minimised. The metric $k$-means
problem is defined similarly using squared distances. In many applications
there are additional constraints that any solution needs to satisfy. This gives
rise to different constrained versions of the problem such as $r$-gather,
fault-tolerant, outlier $k$-means/$k$-median problem. Surprisingly, for many of
these constrained problems, no constant-approximation algorithm is known. We
give FPT algorithms with constant approximation guarantee for a range of
constrained $k$-median/means problems. For some of the constrained problems,
ours is the first constant factor approximation algorithm whereas for others,
we improve or match the approximation guarantee of previous works. We work
within the unified framework of Ding and Xu that allows us to simultaneously
obtain algorithms for a range of constrained problems. In particular, we obtain
a $(3+\varepsilon)$-approximation and $(9+\varepsilon)$-approximation for the
constrained versions of the $k$-median and $k$-means problem respectively in
FPT time. In many practical settings of the $k$-median/means problem, one is
allowed to open a facility at any client location, i.e., $C \subseteq L$. For
this special case, our algorithm gives a $(2+\varepsilon)$-approximation and
$(4+\varepsilon)$-approximation for the constrained versions of $k$-median and
$k$-means problem respectively in FPT time. Since our algorithm is based on
simple sampling technique, it can also be converted to a constant-pass
log-space streaming algorithm.
