Metrical task systems on trees via mirror descent and unfair gluing

TL;DR

This paper introduces improved algorithms for metrical task systems on trees and general metrics using mirror descent, achieving better competitive ratios and removing previous bounds.

Contribution

It presents a new $O( ext{depth} imes ext{log} n)$-competitive algorithm for tree metrics and refines bounds for hierarchically separated trees using mirror descent.

Findings

Achieves $O( ext{depth} imes ext{log} n)$-competitiveness on trees.

Refines HST algorithms to $O( ext{log} n)$-competitiveness.

Provides an $O(( ext{log} n)^2)$-competitive algorithm for general metrics.

Abstract

We consider metrical task systems on tree metrics, and present an $O(\mathrm{depth} \times \log n)$-competitive randomized algorithm based on the mirror descent framework introduced in our prior work on the $k$-server problem. For the special case of hierarchically separated trees (HSTs), we use mirror descent to refine the standard approach based on gluing unfair metrical task systems. This yields an $O(\log n)$-competitive algorithm for HSTs, thus removing an extraneous $\log\log n$ in the bound of Fiat and Mendel (2003). Combined with well-known HST embedding theorems, this also gives an $O((\log n)^2)$-competitive randomized algorithm for every $n$-point metric space.