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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 = 4. As a second result, we improve the lower bound of $\frac {12}{11}$ for approximating pseudo-polynomial Strip Packing to $\frac {5}{4}$. Since the best known approximation algorithm for this problem has a ratio of $\frac {5}{4} + \varepsilon $, this result almost closes the gap between approximation ratio and inapproximability result. Both results are proved by a reduction from the strongly NP-complete problem 3-Partition.

Complexity and Inapproximability Results for Parallel Task Scheduling and Strip Packing