Presenting a SODA 2010 paper by Vincenzo Bonifaci, Ho-Leung Chan, Alberto Marchetti-Spaccamela, and Nicole Megow
Abstract: We investigate the preemptive scheduling of periodic tasks with hard deadlines. We show that, even in the uniprocessor case, no polynomial time algorithm can test the feasibility of a task system within a constant speedup bound, unless P =NP. This result contrasts with recent results for sporadic task systems. For two special cases, synchronous task systems and systems with a constant number of different task types, we provide the first polynomial time constant-speedup feasibility tests for multiprocessor platforms. Furthermore, we show that the problem of testing feasibility is coNP-hard for synchronous multiprocessor task systems. The complexity of some of these problems has been open for a long time. We also propose a profit maximization variant of the feasibility problem, where every task has a non-negative profit, and the goal is to find a subset of tasks that can be scheduled feasibly with maximum profit. We give the first constant-speed, constant-approximation algorithm for the case of synchronous task systems, together with related hardness results.