A shelf-based Lagrangian relaxation algorithm to schedule parallelizable tasks

This paper generalizes classical scheduling theory by removing the restriction that only a single processor can work on a given task at a particular time. Instead, it is assumed that each task can be allocated any number of identical processors from one to the maximum number available, and that a task's completion time is a nonincreasing function of the number of processors allocated. Once started, a task must run to completion without altering the number of processors given to it. Furthermore, a task can start only when no task is currently executing. The objective considered is minimization of overall completion time (make-span) for the tasks subject to the constraint of a limited number of available processors. To approximately solve this problem, an algorithm based on Lagrangian relaxation is developed, and its performance is analyzed through extensive simulations. Approximate duality gaps range from 0 to about 60% on average and are strongly a function of problem type and size.