  In the non-uniform $k$-center problem, the objective is to cover points in a
metric space with specified number of balls of different radii. Chakrabarty,
Goyal, and Krishnaswamy [ICALP 2016, Trans. on Algs. 2020] (CGK, henceforth)
give a constant factor approximation when there are two types of radii. In this
paper, we give a constant factor approximation for the two radii case in the
presence of outliers. To achieve this, we need to bypass the technical barrier
of bad integrality gaps in the CGK approach. We do so using "the ellipsoid
method inside the ellipsoid method": use an outer layer of the ellipsoid method
to reduce to stylized instances and use an inner layer of the ellipsoid method
to solve these specialized instances. This idea is of independent interest and
could be applicable to other problems.
  Keywords: Approximation, Clustering, Outliers, and Round-or-Cut.
