Complexity and Inapproximability Results for Parallel Task Scheduling and Strip Packing

Abstract

We study the Parallel Task Scheduling problem $Pm|siz{e}_j|C_{\max }$ with a constant number of machines. This problem is known to be strongly NP-complete for each $m\ge 5$, while it is solvable in pseudo-polynomial time for each $m\le 3$. 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 $\frac{4}{3}+\epsilon$, 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.