Treffer: Approximation algorithms for 3D orthogonal knapsack

We study non-overlapping axis-parallel packings of 3D boxes with profits into a dedicated bigger box where rotation is either forbidden or permitted; 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 + and 8 + as well as an algorithm with approximation ratio 7 + 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 approxima-tion ratios 6 + and 5 + , 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 re-sult of 45/4. Topics: approximation algorithms, computational and structural complex-ity, geometric configurations. 1