A Tale of Two Dimensional Bin Packing
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Bin Packing in Multiple Dimensions: Inapproximability Results and Approximation Schemes
Mathematics of Operations Research
Packing Rectangles into 2OPT Bins Using Rotations
SWAT '08 Proceedings of the 11th Scandinavian workshop on Algorithm Theory
Improved Absolute Approximation Ratios for Two-Dimensional Packing Problems
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Approximation algorithms for orthogonal packing problems for hypercubes
Theoretical Computer Science
A Structural Lemma in 2-Dimensional Packing, and Its Implications on Approximability
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
SIAM Journal on Computing
Absolute approximation ratios for packing rectangles into bins
Journal of Scheduling
An approximation scheme for the two-stage, two-dimensional knapsack problem
Discrete Optimization
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We present an asymptotic fully polynomial time approximation scheme for the two-dimensional generalization of Bin Packing, which requires packing (or cutting) a given set of rectangles from the minimum number of square bins, with the further restriction that packing the rectangles in the bins is done in two stages, as is frequently the case in real-world applications. To the best of our knowledge, this is the first approximation scheme for a nontrivial two-dimensional (and real-world) generalization of a classical one-dimensional packing problem in which rectangles have to be packed in (finite) squares. In addition to the approximability result, we obtain as a byproduct of our analysis an interesting structural result, namely the asymptotic optimality of the lower bound provided by the natural linear programming relaxation of the problem, having one variable for each bin pattern, that can be solved effectively by column generation techniques. Besides its practical relevance, a main reason to concentrate on the two-stage version of the problem is that its study continues to shed light on the general version. In particular, we strongly conjecture that the approximation guarantee of our algorithm for two-dimensional Bin Packing without the two-stage restriction is better than the best known ones. A simplification of the method leads to an approximation scheme for the two-stage version of two-dimensional Strip Packing.