  We study the consistent k-center clustering problem. In this problem, the
goal is to maintain a constant factor approximate $k$-center solution during a
sequence of $n$ point insertions and deletions while minimizing the recourse,
i.e., the number of changes made to the set of centers after each point
insertion or deletion. Previous works by Lattanzi and Vassilvitskii [ICML '12]
and Fichtenberger, Lattanzi, Norouzi-Fard, and Svensson [SODA '21] showed that
in the incremental setting, where deletions are not allowed, one can obtain $k
\cdot \textrm{polylog}(n) / n$ amortized recourse for both $k$-center and
$k$-median, and demonstrated a matching lower bound. However, no algorithm for
the fully dynamic setting achieves less than the trivial $O(k)$ changes per
update, which can be obtained by simply reclustering the full dataset after
every update.
  In this work, we give the first algorithm for consistent $k$-center
clustering for the fully dynamic setting, i.e., when both point insertions and
deletions are allowed, and improves upon a trivial $O(k)$ recourse bound.
Specifically, our algorithm maintains a constant factor approximate solution
while ensuring worst-case constant recourse per update, which is optimal in the
fully dynamic setting. Moreover, our algorithm is deterministic and is
therefore correct even if an adaptive adversary chooses the insertions and
deletions.
