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 \(O( \log n \cdot c({\cal M},{\cal T}))\) .