Weighted $k$-Server Admits an Exponentially Competitive Algorithm

The weighted $k$-server is a variant of the $k$-server problem, where the cost of moving a server is the server's weight times the distance through which it moves. The problem is famous for its intriguing properties and for evading standard techniques for designing and analyzing online algorithms. Even on uniform metric spaces with sufficiently many points, the deterministic competitive ratio of weighted $k$-server is known to increase doubly exponentially with respect to $k$, while the behavior of its randomized competitive ratio is not fully understood. Specifically, no upper bound better than doubly exponential is known, while the best known lower bound is singly exponential in $k$. In this paper, we close the exponential gap between these bounds by giving an $\exp(O(k^2))$-competitive randomized online algorithm for the weighted $k$-server problem on uniform metrics, thus breaking the doubly exponential barrier for deterministic algorithms for the first time. This is achieved by a recursively defined notion of a phase which, on the one hand, forces a lower bound on the cost of any offline solution, while, on the other hand, also admits a randomized online algorithm with bounded expected cost. The algorithm is also recursive; it involves running several algorithms virtually and in parallel and following the decisions of one of them in a random order. We also show that our techniques can be lifted to construct an $\exp(O(k^2))$-competitive randomized online algorithm for the generalized $k$-server problem on weighted uniform metrics.