Ramsey-type theorems for metric spaces with applications to online problems

TL;DR

This paper establishes nearly logarithmic lower bounds for randomized online algorithms in metric space problems, using Ramsey-type theorems that identify large ultrametric subspaces within any metric space.

Contribution

It introduces Ramsey-type theorems for metric spaces, showing the existence of large ultrametric subspaces, and applies these to derive lower bounds for online problems like metrical task systems and k-server.

Findings

Nearly logarithmic lower bound for randomized competitive ratio

Existence of large ultrametric subspaces in any metric space

Ramsey-type theorems may be of independent interest

Abstract

A nearly logarithmic lower bound on the randomized competitive ratio for the metrical task systems problem is presented. This implies a similar lower bound for the extensively studied k-server problem. The proof is based on Ramsey-type theorems for metric spaces, that state that every metric space contains a large subspace which is approximately a hierarchically well-separated tree (and in particular an ultrametric). These Ramsey-type theorems may be of independent interest.