A New Approach to Online Scheduling: Approximating the Optimal Competitive Ratio

TL;DR

This paper introduces a novel scheme for online scheduling that algorithmically approximates the optimal competitive ratio, enabling near-optimal online algorithms for various scheduling problems with a high degree of precision.

Contribution

It presents the first systematic method to compute and approximate the best possible competitive ratio for online scheduling algorithms, generalizing to multiple objectives and problem settings.

Findings

Developed competitive-ratio approximation schemes for online scheduling.

Derived near-optimal deterministic and randomized algorithms.

Provided a computational method to determine the competitive ratio with arbitrary accuracy.

Abstract

We propose a new approach to competitive analysis in online scheduling by introducing the novel concept of competitive-ratio approximation schemes. Such a scheme algorithmically constructs an online algorithm with a competitive ratio arbitrarily close to the best possible competitive ratio for any online algorithm. We study the problem of scheduling jobs online to minimize the weighted sum of completion times on parallel, related, and unrelated machines, and we derive both deterministic and randomized algorithms which are almost best possible among all online algorithms of the respective settings. We also generalize our techniques to arbitrary monomial cost functions and apply them to the makespan objective. Our method relies on an abstract characterization of online algorithms combined with various simplifications and transformations. We also contribute algorithmic means to compute the actual value of the best possi- ble competitive ratio up to an arbitrary accuracy. This strongly contrasts all previous manually obtained competitiveness results for algorithms and, most importantly, it reduces the search for the optimal com- petitive ratio to a question that a computer can answer. We believe that our concept can also be applied to many other problems and yields a new perspective on online algorithms in general.