Weighted Rectangle and Cuboid Packing

Given a set of rectangular items, all of them associated with a profit, and a single bigger rectangular bin, we can ask to find a non-rotational, non-overlapping packing of a selection of these items into the bin to maximize the profit. A detailed description of the (2 + )-approximation algorithm of Jansen and Zhang [8] for the twodimensional case is given. Furthermore we derive a (16 + )-approximation algorithm for the threedimensional case (which we call cuboid packing) and improve this algorithm in a second step to an approximation ratio of (9 + ). Finally we prove that cuboid packing does not admit an asymptotic PTAS. It turned out, that there is a mistake in Lemma 4.2 that affects the correctness of the (9+ ) algorithm. Instead of correcting the mistake here, we refer to [4]. Based on the ideas developed in this work, we derived a (9+ ) and a (8+ ) algorithm in [4].