Time-Optimal $k$-Server

The time-optimal $k$-server problem minimizes the time spent serving all requests instead of the distances traveled. We give a lower bound of $2k-1$ on the competitive ratio of any deterministic online algorithm for this problem, which coincides with the best known upper bound on the competitive ratio achieved by the work-function algorithm for the classical $k$-server problem. We provide further lower bounds of $k+1$ for all Euclidean spaces and $k$ for uniform metric spaces. For the latter, we give a matching $k$-competitive deterministic algorithm. Our most technical result, proven by applying Yao's principle to a suitable instance distribution on a specifically constructed metric space, is a lower bound of $k+\mathcal{O}(\log k)$ that holds even for randomized algorithms, which contrasts with the best known lower bound for the classical problem that remains polylogarithmic. With this paper, we hope to initiate a further study of this natural yet neglected problem.