  The $k$-center problem is a canonical and long-studied facility location and
clustering problem with many applications in both its symmetric and asymmetric
forms. Both versions of the problem have tight approximation factors on worst
case instances. Therefore to improve on these ratios, one must go beyond the
worst case.
  In this work, we take this approach and provide strong positive results both
for the asymmetric and symmetric $k$-center problems under a natural input
stability (promise) condition called $\alpha$-perturbation resilience [Bilu and
Linia 2012], which states that the optimal solution does not change under any
alpha-factor perturbation to the input distances. We provide algorithms that
give strong guarantees simultaneously for stable and non-stable instances: our
algorithms always inherit the worst-case guarantees of clustering approximation
algorithms, and output the optimal solution if the input is $2$-perturbation
resilient. Furthermore, we prove our result is tight by showing symmetric
$k$-center under $(2-\epsilon)$-perturbation resilience is hard unless $NP=RP$.
  The impact of our results are multifaceted. This is the first tight result
for any problem under perturbation resilience. Furthermore, our results
illustrate a surprising relationship between symmetric and asymmetric
$k$-center instances under perturbation resilience. Unlike approximation ratio,
for which symmetric $k$-center is easily solved to a factor of 2 but asymmetric
$k$-center cannot be approximated to any constant factor, both symmetric and
asymmetric $k$-center can be solved optimally under resilience to
2-perturbations. Finally, our guarantees in the setting where only part of the
data satisfies perturbation resilience makes these algorithms more applicable
to real-life instances.
