On the approximability of scheduling multiprocessor tasks with time-dependent processor and time requirements

Given a set of independent dedicated multiprocessor tasks with time-dependent execution and processor requirements, our objective is to find a schedule minimizing the makespan. In fact, the time-axis is partitionned to a set of intervals, and every task has a specific processing time and processor requirement for every time-interval. We show that this problem (denoted as td-fix ) cannot be approximated by any constant ratio approximation algorithm even in the case where two processors are considered (unless ). We present a polynomial time approximation scheme (PTAS) for the related-execution-times case with release dates, where the maximum processing time, , and the minimum processing time, ,o f each task are such that , with a parameter depending only on the number of processors (the number of intervals ,a s well as are considered as fixed constants). Notice that this PTAS extends the result of [3] for the classical model with dedicated tasks ( fix ) in the case where the execution of the tasks is subject to release dates ( fix ). Furthermore, we show that for the time-dependent problem with dedicated tasks, there is no PTAS in the case where a nonfixed number of intervals is considered even in the relatedexecution-times case and with a fixed number of processors (unless ).