  $k$-center is one of the most popular clustering models. While it admits a
simple 2-approximation in polynomial time in general metrics, the Euclidean
version is NP-hard to approximate within a factor of 1.93, even in the plane,
if one insists the dependence on $k$ in the running time be polynomial. Without
this restriction, a classic algorithm yields a $2^{O((k\log
k)/{\epsilon})}dn$-time $(1+\epsilon)$-approximation for Euclidean $k$-center,
where $d$ is the dimension.
  We give a faster algorithm for small dimensions: roughly speaking an
$O^*(2^{O((1/\epsilon)^{O(d)} \cdot k^{1-1/d} \cdot \log k)})$-time
$(1+\epsilon)$-approximation. In particular, the running time is roughly
$O^*(2^{O((1/\epsilon)^{O(1)}\sqrt{k}\log k)})$ in the plane. We complement our
algorithmic result with a matching hardness lower bound.
  We also consider a well-studied generalization of $k$-center, called
Non-uniform $k$-center (NUkC), where we allow different radii clusters. NUkC is
NP-hard to approximate within any factor, even in the Euclidean case. We design
a $2^{O(k\log k)}n^2$ time $3$-approximation for NUkC in general metrics, and a
$2^{O((k\log k)/\epsilon)}dn$ time $(1+\epsilon)$-approximation for Euclidean
NUkC. The latter time bound matches the bound for $k$-center.
