Topology matters: Smoothed competitiveness of metrical task systems

Abstract

We consider metrical task systems, a general framework to model online problems. Borodin, Linial and Saks [3] presented a deterministic work function algorithm (WFA) for metrical task systems having a tight competitive ratio of 2n - 1. We present a smoothed competitive analysis of WFA. Given an adversarial task sequence, we smoothen the request costs by means of a symmetric additive smoothing model and analyze the competitive ratio of WFA on the smoothed task sequence. We prove upper and matching lower bounds on the smoothed competitive ratio of WFA. Our analysis reveals that the smoothed competitive ratio of WFA is much better than O(n) and that it depends on several topological parameters of the underlying graph G, such as the maximum degree D and the diameter. For example, already for small perturbations the smoothed competitive ratio of WFA reduces to O(log n) on a clique or a complete binary tree and to O(√n) on a line. We also provide the first average case analysis of WFA showing that its expected competitive ratio is O(log(D)) for various distributions. © Springer-Verlag 2004.