Complexity and inapproximability results for parallel task scheduling and strip packing

Abstract

We study Parallel Task Scheduling Pm|sizej|Cmax with a constant number of machines. This problem is known to be strongly NP-complete for each m ≥ 5, while it is solvable in pseudo-polynomial time for each m ≤ 3. We give a positive answer to the long-standing open question whether this problem is strongly NP-complete for m = 5. As a second result, we improve the lower bound of [Formula Present] for approximating pseudo-polynomial Strip Packing to [Formula Present]. Since the best known approximation algorithm for this problem has a ratio of [Formula Present], this result narrows the gap between approximation ratio and inapproximability result by a significant step. Both results are proven by a reduction from the strongly NP-complete problem 3-Partition.