Approximation Algorithms for {3D} Orthogonal {K}napsack

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 $9+\epsilon$ and $8+\epsilon$ as well as an algorithm with approximation ratio $7+\epsilon$ 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 $90^{\circ}$ either around the $z$-axis or around all axes is permitted, where we obtain algorithms with approximation ratios $6+\epsilon$ and $5+\epsilon$, respectively. Finally our methods yield a 3D generalization of a packability criterion and a strip packing algorithm with absolute approximation ratio $\textfrac{29}{4}$, improving the previously best known result of $\textfrac{45}{4}$