Efficient Online Learning for Dynamic k-Clustering

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{$Θ\left( \min(k,r) \right)$-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.