Barely Random Algorithms and Collective Metrical Task Systems

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 $\Omega(n)$-competitive.