General Multiprocessor Task Scheduling: Approximate Solutions in Linear Time

We study the problem of scheduling $n$ independent tasks on a set of $m$ parallel processors, where the execution time of a task is a function of the subset of processors assigned to the task. For any fixed $m$, we propose a fully polynomial approximation scheme that for any fixed $\epsilon > 0$ finds a preemptive schedule of length at most $(1+\epsilon)$ times the optimum in $O(n)$ time. We also discuss the nonpreemptive variant of the problem, and present for any fixed $m$ a polynomial approximation scheme that computes an approximate solution of any fixed accuracy in linear time. In terms of the running time, this linear complexity bound gives a substantial improvement of the best previously known polynomial bound [J. Chen and A. Miranda, SIAM J. Comput., 31 (2001), pp. 1--17].