# Parametrized Metrical Task Systems

## TL;DR

This paper analyzes parametrized metrical task and service systems, revealing how the number of request types affects the competitive ratio and showing surprising equivalences between deterministic and randomized algorithms in certain cases.

## Contribution

It characterizes the competitive ratio for parametrized metrical systems on hierarchically separated trees, uncovering unexpected behaviors and equivalences between deterministic and randomized algorithms.

## Key findings

- Deterministic algorithms show no asymptotic gain beyond uniform metrics.
- Randomized algorithms show no gain even for one-level trees.
- Deterministic and randomized algorithms are similarly powerful for small m in certain cases.

## Abstract

We consider parametrized versions of metrical task systems and metrical service systems, two fundamental models of online computing, where the constrained parameter is the number of possible distinct requests $m$. Such parametrization occurs naturally in a wide range of applications. Striking examples are certain power management problems, which are modeled as metrical task systems with $m=2$. We characterize the competitive ratio in terms of the parameter $m$ for both deterministic and randomized algorithms on hierarchically separated trees. Our findings uncover a rich and unexpected picture that differs substantially from what is known or conjectured about the unparametrized versions of these problems. For metrical task systems, we show that deterministic algorithms do not exhibit any asymptotic gain beyond one-level trees (namely, uniform metric spaces), whereas randomized algorithms do not exhibit any asymptotic gain even for one-level trees. In contrast, the special case of metrical service systems (subset chasing) behaves very differently. Both deterministic and randomized algorithms exhibit gain, for $m$ sufficiently small compared to $n$, for any number of levels. Most significantly, they exhibit a large gain for uniform metric spaces and a smaller gain for two-level trees. Moreover, it turns out that in these cases (as well as in the case of metrical task systems for uniform metric spaces with $m$ being an absolute constant), deterministic algorithms are essentially as powerful as randomized algorithms. This is surprising and runs counter to the ubiquitous intuition/conjecture that, for most problems that can be modeled as metrical task systems, the randomized competitive ratio is polylogarithmic in the deterministic competitive ratio.