Two-Dimensional Knapsack for Circles

  • Carla Negri Lintzmayer
  • Flávio Keidi Miyazawa
  • Eduardo Candido Xavier
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)

Abstract

In this paper we consider the Two-dimensional Knapsack for Circles problem, in which we are given a set \({\mathcal {C}}\) of circles and want to pack a subset \({\mathcal {C}}' \subseteq {\mathcal {C}}\) of them into a rectangular bin of dimensions w and h such that the sum of the area of circles in \({\mathcal {C}}'\) is maximum. By packing we mean that the circles do not overlap and they are fully contained inside the bin. We present a polynomial-time approximation scheme that, for any \(\epsilon > 0\), gives an approximation algorithm that packs a subset of the input circles into an augmented bin of dimensions w and \((1+O(\epsilon ))h\) such that the area packed is at least \((1-O(\epsilon ))\) times the area packed by an optimal solution into the regular bin of dimensions w and h. This result also extends to the multiple knapsack version of this problem.

Keywords

Circle packing Two-dimensional Knapsack Polynomial-time approximation scheme 

References

  1. 1.
    Adamaszek, A., Wiese, A.: A quasi-PTAS for the two-dimensional geometric Knapsack problem. In: Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2015), pp. 1491–1505. Society for Industrial and Applied Mathematics, Philadelphia (2015)Google Scholar
  2. 2.
    Christensen, H.I., Khan, A., Pokutta, S., Tetali, P.: Approximation and online algorithms for multidimensional bin packing: a survey. Comput. Sci. Rev. 24, 63–79 (2017)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Demaine, E.D., Fekete, S.P., Lang, R.J.: Circle Packing for Origami Design Is Hard. A K Peters/CRC Press, Singapore (2010). pp. 609–626Google Scholar
  4. 4.
    Fishkin, A.V., Gerber, O., Jansen, K., Solis-Oba, R.: On packing squares with resource augmentation: maximizing the profit. In: Proceedings of the 2005 Australasian Symposium on Theory of Computing (CATS 2005), pp. 61–67. Australian Computer Society Inc., Darlinghurst (2005)Google Scholar
  5. 5.
    Fishkin, A.V., Gerber, O., Jansen, K., Solis-Oba, R.: Packing weighted rectangles into a square. In: Jȩdrzejowicz, J., Szepietowski, A. (eds.) MFCS 2005. LNCS, vol. 3618, pp. 352–363. Springer, Heidelberg (2005).  https://doi.org/10.1007/11549345_31 CrossRefGoogle Scholar
  6. 6.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)MATHGoogle Scholar
  7. 7.
    Hifi, M., M’Hallah, R.: A literature review on circle and sphere packing problems: models and methodologies. Adv. Oper. Res. 2009, 1–22 (2009)CrossRefMATHGoogle Scholar
  8. 8.
    Hokama, P., Miyazawa, F.K., Schouery, R.C.S.: A bounded space algorithm for online circle packing. Inf. Process. Lett. 116(5), 337–342 (2016)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Jansen, K., Solis-Oba, R.: Packing squares with profits. SIAM J. Discret. Math. 26(1), 263–279 (2012)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Jansen, K., Zhang, G.: Maximizing the total profit of rectangles packed into a rectangle. Algorithmica 47(3), 323–342 (2007)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Lenstra, H.W.: Integer programming with a fixed number of variables. Math. Oper. Res. 8(4), 538–548 (1983)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Lodi, A., Martello, S., Monaci, M., Vigo, D.: Two-Dimensional Bin Packing Problems, pp. 107–129. Wiley, Hoboken (2013)MATHGoogle Scholar
  13. 13.
    Meir, A., Moser, L.: On packing of squares and cubes. J. Comb. Theory 5(2), 126–134 (1968)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Miyazawa, F.K., Pedrosa, L.L.C., Schouery, R.C.S., Sviridenko, M., Wakabayashi, Y.: Polynomial-time approximation schemes for circle and other packing problems. Algorithmica 76(2), 536–568 (2016)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Szabó, P.G., Markót, M.C., Csendes, T., Specht, E., Casado, L.G., García, I.: New Approaches to Circle Packing in a Square. Springer Optimization and its Applications. Springer, New York (2007).  https://doi.org/10.1007/978-0-387-45676-8 MATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Center for Mathematics, Computer, and CognitionFederal University of ABCSanto AndréBrazil
  2. 2.Institute of ComputingUniversity of CampinasCampinasBrazil

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