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, 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 + 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 approximation 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 result of 45/4. Research supported in part by DFG Project, Entwicklung und Analyse von Approximativen Algorithmen für Gemischte und Verallgemeinerte Packungs-und überdeckungsprobleme, JA 612/10-1, in part by the German Academic Exchange Service DAAD, in part by project AEOLUS, under EU Contract No. 015964, and in part by a Grant “DAAD Doktorandenstipendium” of the German Academic Exchange Service DAAD. Part of this work was done in duration of visit to the LIG, Grenoble University.