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 textitDynamic 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 textitp-norm of the vector consisting of the distance of each client to its closest center at round t, for some p≥ 1 or p = ∞. We present a textitThetaleft( min(k,r) right)-regret polynomial-time online learning algorithm and show that, under some well-established computational complexity conjectures, textitconstant-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.