A Competitive Ratio Approximation Scheme for the k-Server Problem in Fixed Finite Metrics

TL;DR

This paper develops a scheme to approximate the optimal competitive ratio for the k-server problem in fixed finite metric spaces, enabling near-optimal online algorithms with provable guarantees.

Contribution

It introduces a method to restrict analysis to finite request sequences, ensuring solutions are close to unrestricted ones, and constructs algorithms achieving optimal competitive ratios in this setting.

Findings

Restricted analysis approximates unrestricted solutions within any small factor.

Provides algorithms with optimal deterministic and randomized competitive ratios.

Establishes a competitive ratio approximation scheme for fixed finite metrics.

Abstract

We show how to restrict the analysis of a class of online problems that includes the $k$-server problem in finite metrics such that we only have to consider finite sequences of request. When applying the restrictions, both the optimal offline solutions and the best possible deterministic or randomized online solutions only differ by at most an arbitrarily small constant factor from the corresponding solutions without restrictions. Furthermore, we show how to obtain an algorithm with best possible deterministic or randomized competitive ratio for the restricted setup. Thus, for each fixed finite metrics our result qualifies as a competitive ratio approximation scheme as defined by G\"unther et al.