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New Approximability Results for Two-Dimensional Bin Packing

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Abstract

We study the two-dimensional bin packing problem: Given a list of \(n\) rectangles the objective is to find a feasible, i.e. axis-parallel and non-overlapping, packing of all rectangles into the minimum number of unit sized squares, also called bins. Our problem consists of two versions; in the first version it is not allowed to rotate the rectangles while in the other it is allowed to rotate the rectangles by \(90^{\circ }\), i.e. to exchange the widths and the heights. Two-dimensional bin packing is a generalization of its one-dimensional counterpart and is therefore strongly \(\mathcal {NP}\)-hard. Furthermore Bansal et al. (Math Oper Res 31(1):31–49, 2006) showed that even an \(\mathcal {APTAS}\) is ruled out for this problem, unless \(\mathcal {P}=\mathcal {NP}\). This lower bound of asymptotic approximability was improved by Chlebík and Chlebíková (J Discrete Algorithms 7(3):291–305, 2009) to values \(1+1/3792\) and \(1+1/2196\) for the version with and without rotations, respectively. On the positive side there is an asymptotic 1.69.. approximation by Caprara (Math Oper Res 33:203–215, 2008) without rotations and an asymptotic 1.52... approximation by Bansal et al. (SIAM J Comput 39(4):1256–1278, 2009) for both versions. We give a new asymptotic upper bound for both versions of our problem: For any fixed \(\varepsilon \) and any instance that fits optimally into \(\mathrm {OPT}\) bins, our algorithm computes a packing into \((3/2+\varepsilon )\cdot \mathrm {OPT}+69\) bins in the version without rotations and \((3/2+\varepsilon )\cdot \mathrm {OPT}+39\) bins in the version with rotations. The algorithm has polynomial running time in the input length. In our new technique we consider an optimal packing of the rectangles into the bins. We cut a small vertical or horizontal strip out of each bin and move the intersecting rectangles into additional bins. This enables us to either round the widths of all wide rectangles, or the heights of all long rectangles in this bin. After this step we round the other unrounded side of these rectangles and we achieve a solution with a simple structure and only few types of rectangles. Our algorithm initially rounds the instance and computes a solution that nearly matches the modified optimal solution.

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Authors and Affiliations

  1. Institut für Informatik, Universität Kiel, Christian-Albrechts-Platz 4, 24118, Kiel, Germany

    Klaus Jansen & Lars Prädel

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  1. Klaus Jansen
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  2. Lars Prädel
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Correspondence to Lars Prädel.

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Research supported by German Research Foundation (DFG) Project JA612/12-2, “Approximation algorithms for two- and three-dimensional packing problems and related scheduling problems”.

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Jansen, K., Prädel, L. New Approximability Results for Two-Dimensional Bin Packing. Algorithmica 74, 208–269 (2016). https://doi.org/10.1007/s00453-014-9943-z

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  • DOI: https://doi.org/10.1007/s00453-014-9943-z

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