Weighted k-Server Bounds via Combinatorial Dichotomies

TL;DR

This paper establishes a doubly exponential lower bound on the competitive ratio for the weighted k-server problem on uniform metrics, matching known upper bounds and advancing understanding of its complexity and structure.

Contribution

It provides the first tight lower bound for the weighted k-server problem on uniform metrics and links it to a combinatorial problem using Ramsey theory, revealing structural properties.

Findings

Doubly exponential lower bound on competitive ratio

Generalized Work Function Algorithm is nearly optimal

New upper bounds for multiple weight classes

Abstract

The weighted $k$-server problem is a natural generalization of the $k$-server problem where each server has a different weight. We consider the problem on uniform metrics, which corresponds to a natural generalization of paging. Our main result is a doubly exponential lower bound on the competitive ratio of any deterministic online algorithm, that essentially matches the known upper bounds for the problem and closes a large and long-standing gap. The lower bound is based on relating the weighted $k$-server problem to a certain combinatorial problem and proving a Ramsey-theoretic lower bound for it. This combinatorial connection also reveals several structural properties of low cost feasible solutions to serve a sequence of requests. We use this to show that the generalized Work Function Algorithm achieves an almost optimum competitive ratio, and to obtain new refined upper bounds on the competitive ratio for the case of $d$ different weight classes.