  We present approximation algorithms for some variants of center-based
clustering and related problems in the fully dynamic setting, where the
pointset evolves through an arbitrary sequence of insertions and deletions.
Specifically, we target the following problems: $k$-center (with and without
outliers), matroid-center, and diversity maximization. All algorithms employ a
coreset-based strategy and rely on the use of the cover tree data structure,
which we crucially augment to maintain, at any time, some additional
information enabling the efficient extraction of the solution for the specific
problem. For all of the aforementioned problems our algorithms yield
$(\alpha+\varepsilon)$-approximations, where $\alpha$ is the best known
approximation attainable in polynomial time in the standard off-line setting
(except for $k$-center with $z$ outliers where $\alpha = 2$ but we get a
$(3+\varepsilon)$-approximation) and $\varepsilon>0$ is a user-provided
accuracy parameter. The analysis of the algorithms is performed in terms of the
doubling dimension of the underlying metric. Remarkably, and unlike previous
works, the data structure and the running times of the insertion and deletion
procedures do not depend in any way on the accuracy parameter $\varepsilon$
and, for the two $k$-center variants, on the parameter $k$. For spaces of
bounded doubling dimension, the running times are dramatically smaller than
those that would be required to compute solutions on the entire pointset from
scratch. To the best of our knowledge, ours are the first solutions for the
matroid-center and diversity maximization problems in the fully dynamic
setting.
