  In the Non-Uniform k-Center problem we need to cover a finite metric space
using k balls of different radii that can be scaled uniformly. The goal is to
minimize the scaling factor. If the number of different radii is unbounded, the
problem does not admit a constant-factor approximation algorithm but it has
been conjectured that such an algorithm exists if the number of radii is
constant. Yet, this is known only for the case of two radii. Our first
contribution is a simple black box reduction which shows that if one can handle
the variant of t-1 radii with outliers, then one can also handle t radii.
Together with an algorithm by Chakrabarty and Negahbani for two radii with
outliers, this immediately implies a constant-factor approximation algorithm
for three radii, thus making further progress on the conjecture. Furthermore,
using algorithms for the k-center with outliers problem, that is the one radii
with outliers case, we also get a simple algorithm for two radii.
  The algorithm by Chakrabarty and Negahbani uses a top-down approach, starting
with the larger radius and then proceeding to the smaller one. Our reduction,
on the other hand, looks only at the smallest radius and eliminates it, which
suggests that a bottom-up approach is promising. In this spirit, we devise a
modification of the Chakrabarty and Negahbani algorithm which runs in a
bottom-up fashion, and in this way we recover their result with the advantage
of having a simpler analysis.
