In this paper we study variants of the nonpreemptive parallel job scheduling problem in which the number of machines is polynomially bounded in the number of jobs. For this problem we show that a schedule with length at most $(1+\varepsilon)\,\mathrm{OPT}$ can be calculated in polynomial time. Unless $P=NP$, this is the best possible result (in the sense of approximation ratio), since the problem is strongly NP-hard. For the case where all jobs must be allotted to a subset of consecutive machines, a schedule with length at most $(1.5+\varepsilon)\,\mathrm{OPT}$ can be calculated in polynomial time. The previously best known results are algorithms with absolute approximation ratio 2. Furthermore, we extend both algorithms to the case of malleable jobs with the same approximation ratios.

We study non-overlapping axis-parallel packings of 3D boxes with profits into a dedicated bigger box where rotation is either forbidden or permitted, and we wish to maximize the total profit. Since this optimization problem is NP-hard, we focus on approximation algorithms. We obtain fast and simple algorithms for the non-rotational scenario with approximation ratios 9 + ϵ and 8 + ϵ, as well as an algorithm with approximation ratio 7 + ϵ that uses more sophisticated techniques; these are the smallest approximation ratios known for this problem. Furthermore, we show how the used techniques can be adapted to the case where rotation by 90° either around the z -axis or around all axes is permitted, where we obtain algorithms with approximation ratios 6 + ϵ and 5 + ϵ, respectively. Finally our methods yield a 3D generalization of a packability criterion and a strip packing algorithm with absolute approximation ratio 29/4, improving the previously best known result of 45/4.