  The Euclidean $k$-median problem is defined in the following manner: given a
set $\mathcal{X}$ of $n$ points in $\mathbb{R}^{d}$, and an integer $k$, find a
set $C \subset \mathbb{R}^{d}$ of $k$ points (called centers) such that the
cost function $\Phi(C,\mathcal{X}) \equiv \sum_{x \in \mathcal{X}} \min_{c \in
C} \|x-c\|_{2}$ is minimized. The Euclidean $k$-means problem is defined
similarly by replacing the distance with squared distance in the cost function.
Various hardness of approximation results are known for the Euclidean $k$-means
problem. However, no hardness of approximation results were known for the
Euclidean $k$-median problem. In this work, assuming the unique games
conjecture (UGC), we provide the first hardness of approximation result for the
Euclidean $k$-median problem.
  Furthermore, we study the hardness of approximation for the Euclidean
$k$-means/$k$-median problems in the bi-criteria setting where an algorithm is
allowed to choose more than $k$ centers. That is, bi-criteria approximation
algorithms are allowed to output $\beta k$ centers (for constant $\beta>1$) and
the approximation ratio is computed with respect to the optimal
$k$-means/$k$-median cost. In this setting, we show the first hardness of
approximation result for the Euclidean $k$-median problem for any $\beta <
1.015$, assuming UGC. We also show a similar bi-criteria hardness of
approximation result for the Euclidean $k$-means problem with a stronger bound
of $\beta < 1.28$, again assuming UGC.
