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 $s_1, \dots, s_k$ remains in its own metric space $M_i$. A request is a tuple $(r_1,\dots,r_k)$, where $r_i \in M_i$, 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 $O(k^2 \cdot \log k)$-competitive randomized algorithm improving over a recent result by Bansal et al. [SODA 2018], who gave an $O(k^3 \cdot \log k)$-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.