Abstract. We consider the two-dimensional bin packing and strip packing problem, where a list of rectangles has to be packed into a minimal number of rectangular bins or a strip of minimal height, respectively. All packings have to be non-overlapping and orthogonal, i.e., axisparallel. Our algorithm for strip packing has an absolute approximation ratio of 1.9396 and is the first algorithm to break the approximation ratio of 2 which was established more than a decade ago. Moreover, we present a polynomial time approximation scheme (PTAS) for strip packing where rotations by 90 degrees are permitted and 1 an algorithm for two-dimensional bin packing with an absolute worst-case ratio of 2, which is optimal provided P = N P.
We consider the problem of packing rectangles into bins that are unit squares, where the goal is to minimize the number of bins used. All rectangles have to be packed non-overlapping and orthogonal, i.e., axis-parallel. We present an algorithm with an absolute worst-case ratio of 2 for the case where the rectangles can be rotated by 90 degrees. This is optimal provided P = N P. For the case where rotation is not allowed, we prove an approximation ratio of 3 for the algorithm HYBRID FIRST FIT which was introduced by Chung et al. (SIAM J. Algebr. Discrete Methods 3(1): [66][67][68][69][70][71][72][73][74][75][76] 1982) and whose running time is slightly better than the running time of Zhang's previously known 3-approximation algorithm (Zhang in Oper. Res. Lett. 33(2):121-126, 2005).
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.
Given a list of d-dimensional cuboid items with associated profits, the \emphorthogonal knapsack problem} asks for a packing of a selection with maximal profit into the unit cube. We restrict the items to hypercube shapes and derive a (\frac{5}{4}+ε)-approximation for the two-dimensional case. In a second step we generalize our result to a (\frac{2^d+1}{2^d+ε)-approximation for d-dimensional packing
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