On Algorithm Design for Metrical Task Systems

Abstract

We extend the definition of Metrical Task Systems, introduced by Borodin et al. in [4]. In the extended definition, a system is described by the underlying metric space of states ${\cal M}$ as well as a set of allowable tasks ${\cal T}$ . Any request to an algorithm must be a member of ${\cal T}$ . The extension makes the model powerful enough to characterize completely many important on-line problems. We consider methods of designing competitive algorithms given the description of a system $\langle {\cal M}, {\cal T} \rangle$ . In particular, we show that it is PSPACE-hard to determine the behavior of a $c({\cal M},{\cal T})$ -competitive algorithm, where $c({\cal M},{\cal T})$ is the best possible competitive ratio on $\langle {\cal M},{\cal T} \rangle$ . In addition, we show a simple, polynomial-time algorithm for task systems $\langle {\cal U}_n,{\cal T} \rangle$ (where ${\cal U}_n$ is the uniform metric space on n nodes) that achieves a competitive ratio of $O( \log n \cdot c({\cal M},{\cal T}))$ .