  We study dynamic clustering problems from the perspective of online learning.
We consider an online learning problem, called \textit{Dynamic $k$-Clustering},
in which $k$ centers are maintained in a metric space over time (centers may
change positions) such as a dynamically changing set of $r$ clients is served
in the best possible way. The connection cost at round $t$ is given by the
\textit{$p$-norm} of the vector consisting of the distance of each client to
its closest center at round $t$, for some $p\geq 1$ or $p = \infty$. We present
a \textit{$\Theta\left( \min(k,r) \right)$-regret} polynomial-time online
learning algorithm and show that, under some well-established computational
complexity conjectures, \textit{constant-regret} cannot be achieved in
polynomial-time. In addition to the efficient solution of Dynamic
$k$-Clustering, our work contributes to the long line of research on
combinatorial online learning.
