Slaying Hydrae: Improved Bounds for Generalized k-Server in Uniform Metrics

The generalized k-server problem is an extension of the weighted k-server problem, which in turn extends the classic k-server problem. In the generalized k-server problem, each of k servers s_1, …, s_k remains in its own metric space M_i. A request is a tuple (r_1,…,r_k), where r_i ∈ M_i, and to service it, an algorithm needs to move at least one server s_i to the point r_i. The objective is to minimize the total distance traveled by all servers. In this paper, we focus on the generalized k-server problem for the case where all M_i are uniform metrics. We show an O(k^2 ·log k)-competitive randomized algorithm improving over a recent result by Bansal et al. [SODA 2018], who gave an O(k^3 ·log k)-competitive algorithm. To this end, we define an abstract online problem, called Hydra game, and we show that a randomized solution of low cost to this game implies a randomized algorithm to the generalized k-server problem with low competitive ratio. We also show that no randomized algorithm can achieve competitive ratio lower than Ω(k), thus improving the lower bound of Ω(k / log^2 k) by Bansal et al.