We study the Parallel Task Scheduling problem 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 1112 for approximating pseudo-polynomial
Strip Packing to 45. Since the best known approximation algorithm
for this problem has a ratio of 34+ε, 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