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Modern computer systems distribute computation among several machines to speed up the exe- cution of programs. Yet, setup and communication costs, as well as parallelism constraints, bound the number of machines that can share the execution of a given application, and the number of machines by which it can be processed simultaneously. We study the resulting scheduling problem, stated as follows. Given a set of n jobs and m uniform machines, assign the jobs to the machines subject to parallelism and machine allotment constraints, such that the overall completion time of the schedule (or makespan) is minimized. Indeed, the multiprocessor scheduling problem(where each job can be processed by a single machine) is a special case of our problem; thus, our problem is strongly NP-hard. We present a.1Cfi/-approximation algorithm for this problem, wherefi2.0; 1) depends on the minimal number of machine allotments and the minimal parallelism allowed for any job. Also, we show that when the maximal number of machines that can share the execution of a job is some fixed constant, our problem has a polynomial time approximation scheme ; for other special cases we give optimal polynomial time algorithms. Finally, through the relation of our problem to the classic preemptivescheduling problem on multiple machines, we shed some fresh light on what is known in scheduling folklore as the power of preemption.

Multiprocessor Scheduling with Machine Allotment and Parallelism Constraints