Complexity and Inapproximability Results for Parallel Task Scheduling and Strip Packing

Abstract

We study Parallel Task Scheduling \(Pm|size_j|C_{\max }\) 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 \(\frac{4}{3} + \varepsilon \), 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.