Metrical Task Systems on Trees via Mirror Descent and Unfair Gluing

Sébastien Bubeck,Yin Tat Lee,Michael B Cohen,James R Lee

doi:10.1137/19m1237879

Abstract

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 9 January 2019Accepted: 16 February 2021Published online: 27 May 2021Keywordsonline algorithms, convex optimization, finite metric spaceAMS Subject Headings68W27Publication DataISSN (print): 0097-5397ISSN (online): 1095-7111Publisher: Society for Industrial and Applied MathematicsCODEN: smjcat

Highlights

Let (X, d) be a finite metric space with | X| = n > 1
An online algorithm is a sequence of mappings \bfitrho = \langle \rho 1, \rho 2, . . . , \rangle where, for every t \geqslan 1, \rho t : (\BbbR X+ )t \rightar X maps a sequence of cost functions \langle c1, . . . , ct\rangle to a state
Since one can assume that D \leqslan O(log n) for an n-point hierarchically separated trees (HSTs) metric, the mirror descent framework yields an arguably simpler O((log n)2)-competitive algorithm for arbitrary HSTs that, satisfies the refined guarantees of Theorem 1.2

Summary

Introduction

The cost of the offline optimum, denoted \sansc \sanso \sans \sanst \ast (\bfitc ), is the infimum of \sum t\geqslan1[ct(\rho t) + d(\rho t - 1, \rho t)] over any sequence \langle \rho t : t \geqslan 1\rangle of states. A randomized online algorithm \bfitrho is said to be \alpha -competitive if for every \rho 0 \in X there is a constant \beta > 0 such that for all cost sequences \bfitc. There is an O(D log n)-competitive randomized algorithm for MTS on any n-point tree metric with combinatorial depth D. Since one can assume that D \leqslan O(log n) for an n-point HST metric (see [BBMN15]), the mirror descent framework yields an arguably simpler O((log n)2)-competitive algorithm for arbitrary HSTs that, satisfies the refined guarantees of Theorem 1.2. Competitive algorithms for unfair task systems are useful in constructing algorithms for HSTs, where rx is a proxy for the competitive ratio of an algorithm on MTS instances defined in a subtree rooted at x

METRICAL TASK SYSTEMS ON TREES

Lagrange multiplier corresponding to the constraint

We prove that for almost all t

Iu has measure zero for each

We conclude

We also note that