Barely Random Algorithms and Collective Metrical Task Systems

TL;DR

It is shown that any fully randomized algorithm can be turned into a randomized algorithm that uses only $2\log n$ random bits, and achieves the same competitive ratio up to a factor of two, providing the first order-optimal barely random algorithms for metrical task systems.

Abstract

We consider metrical task systems on general metric spaces with $n$ points, and show that any fully randomized algorithm can be turned into a randomized algorithm that uses only $2\log n$ random bits, and achieves the same competitive ratio up to a factor $2$. This provides the first order-optimal barely random algorithms for metrical task systems, i.e., which use a number of random bits that does not depend on the number of requests addressed to the system. We discuss implications on various aspects of online decision-making such as: distributed systems, advice complexity, and transaction costs, suggesting broad applicability. We put forward an equivalent view that we call collective metrical task systems where $k$ agents in a metrical task system team up, and suffer the average cost paid by each agent. Our results imply that such a team can be $O(\log^2 n)$-competitive as soon as $k\geq n^2$. In comparison, a single agent is always $Ω(n)$-competitive.

Theorems & Definitions (19)

• Theorem 1.1: Section \ref{['sec:main']}
• Corollary 1.2: of Theorem \ref{['th:main']}
• Proposition 1.3: Section \ref{['sec:other']}
• Corollary 1.4: of Theorem \ref{['th:main']}
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Remark 2.3: Fractional metrical task systems, with fixed transaction costs
• Theorem 3.1