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 $s1, \dots, sk$ remains in its own metric space $Mi$. A request is a tuple $(r1,\dots,rk)$, where $ri \in Mi$, and to service it, an algorithm needs to move at least one server $si$ to the point $ri$. 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 $Mi$ are uniform metrics. We show an $O(k² \cdot \log k)$-competitive randomized algorithm improving over a recent result by Bansal et al. [SODA 2018], who gave an $O(k³ \cdot \log k)$-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 $\Omega(k)$, thus improving the lower bound of $\Omega(k / \log² k)$ by Bansal et al.