Competitive Algorithms for Generalized k-Server in Uniform Metrics

The generalized k-server problem is a far-reaching extension of the k-server problem with several applications. Here, each server s_i lies in its own metric space M_i. A request is a k-tuple r = (r₁, r₂, …, r_k) and to serve it, we need to move some server s_i to the point r_i ∊ M_i, and the goal is to minimize the total distance traveled by the servers. Despite much work, no f(k)-competitive algorithm is known for the problem for k > 2 servers, even for special cases such as uniform metrics and lines.

Here, we consider the problem in uniform metrics and give the first f(k)-competitive algorithms for general k. In particular, we obtain deterministic and randomized algorithms with competitive ratio k · 2^k and O(k³ log k) respectively. Our deterministic bound is based on a novel application of the polynomial method to online algorithms, and essentially matches the long-known lower bound of 2^k – 1. We also give a 2^2O(k)-competitive deterministic algorithm for weighted uniform metrics, which also essentially matches the recent doubly exponential lower bound for the problem.