Abstract
We consider the problem of packing a set of circles into a minimum number of unit square bins. To obtain rational solutions, we use augmented bins of height \(1+\gamma \), for some arbitrarily small number \(\gamma > 0\). For this problem, we obtain an asymptotic approximation scheme (APTAS) that is polynomial on \(\log 1/\gamma \), and thus \(\gamma \) may be given as part of the problem input. For the special case that \(\gamma \) is constant, we give a (one dimensional) resource augmentation scheme, that is, we obtain a packing into bins of unit width and height \(1+\gamma \) using no more than the number of bins in an optimal packing without resource augmentation. Additionally, we obtain an APTAS for the circle strip packing problem, whose goal is to pack a set of circles into a strip of unit width and minimum height. Our algorithms are the first approximation schemes for circle packing problems, and are based on novel ideas of iteratively separating small and large items, and may be extended to a wide range of packing problems that satisfy certain conditions. These extensions comprise problems with different kinds of items, such as regular polygons, or with bins of different shapes, such as circles and spheres. As an example, we obtain APTAS’s for the problems of packing d-dimensional spheres into hypercubes under the \(L_p\)-norm.
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Notes
To comply with the majority of works in the literature, in this paper we use the term circle, rather than disk, to refer to the interior of a region. Similarly, and for the sake of consistency with the multidimensional packing literature, we use the term sphere, rather than ball, to refer to the interior of a solid.
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Acknowledgments
We would like to thank Frank Vallentin for providing us with insights and references on the cylindric algebraic decomposition and other algebraic notions.
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This work was partially supported by CNPq (Grants 303987/2010-3, 306860/2010-4, 477203/2012-4, and 477692/2012-5), FAPESP (Grants 2010/20710-4, 2013/02434-8, 2013/03447-6, and 2013/21744-8), and Project MaClinC of NUMEC at USP, Brazil.
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Miyazawa, F.K., Pedrosa, L.L.C., Schouery, R.C.S. et al. Polynomial-Time Approximation Schemes for Circle and Other Packing Problems. Algorithmica 76, 536–568 (2016). https://doi.org/10.1007/s00453-015-0052-4
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DOI: https://doi.org/10.1007/s00453-015-0052-4