Multiprocessor Scheduling with Availability Constraints

Abstract

When scheduling on parallel machines, these may exhibit periods of unavailability, due to maintenance or failures, or due to jobs that must be executed at certain predefined times. We consider the problem of non-preemptively scheduling a given set of tasks on identical processors with periods of unavailability to minimize the maximum completion time. This problem is strongly NP-hard, thus polynomial approximation algorithms are being studied for its solution. Often considered approximation algorithms for multiprocessor scheduling and generalizations thereof are LPT (largest processing time first) and Multifit with their variants. We give a simple polynomial Multifit-based algorithm, the schedules of which end within 1.5 the maximum between the end of the optimal schedule and the latest end of a downtime when there are at most two downtimes on each machine. Even when there is at most one downtime on each machine, no polynomial algorithm can insure a better worst-case bound for this problem unless P=NP. For the case when there is at most one downtime on each machine we also present a simple LPT-based algorithm which has the same property. We also give details of the upper bound proofs.