Complexity and Inapproximability Results for Parallel Task Scheduling and Strip Packing

We study Parallel Task Scheduling P m | s i z e j | C max 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 12 11 $\frac {12}{11}$ for approximating pseudo-polynomial Strip Packing to 5 4 $\frac {5}{4}$ . Since the best known approximation algorithm for this problem has a ratio of 5 4 + ε $\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.