On the Robustness of Graham’s Algorithm for Online Scheduling

Abstract

While standard parallel machine scheduling is concerned with good assignments of jobs to machines, we aim to understand how the quality of an assignment is affected if the jobs’ processing times are perturbed and therefore turn out to be longer (or shorter) than declared. We focus on online scheduling with perturbations occurring at any time, such as in railway systems when trains are late. For a variety of conditions on the severity of perturbations, we present upper bounds on the worst case ratio of two makespans. For the first makespan, we let Graham’s algorithm assign jobs to machines, based on the non-perturbed processing times. We compute the makespan by replacing each job’s processing time with its perturbed version while still sticking to the computed assignment. The second is an optimal offline solution for the perturbed processing times. The deviation of this ratio from Graham’s competitive ratio (of slightly less than 2) tells us about the “price of perturbations”. For instance, we show a competitive ratio of 2 for perturbations decreasing the processing time of a job arbitrarily, and a competitive ratio of less than 2.5 for perturbations doubling the processing time of a job.

This work was partially supported by the Future and Emerging Technologies Unit of EC (IST priority - 6th FP), under contract no. FP6-021235-2 (project ARRIVAL).