Skip to main content
Springer Nature Link
Log in

Two-Dimensional Knapsack for Circles

  • Conference paper
  • First Online:
LATIN 2018: Theoretical Informatics (LATIN 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10807))

Included in the following conference series:

  • 2845 Accesses

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.

This work was supported by São Paulo Research Foundation (grants 2016/14132-4, 2015/11937-9, 2016/23552-7, 2016/01860-1) and National Counsel of Technological and Scientific Development (grants 306358/2014-0, 311499/2014-7, and 425340/2016-3).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    Note that \(w_j = h_j\) for all \(j \ge 1\), but \(w_0\) is not necessarily equal to \(h_0\). This is the only reason we keep using \(w_j\) and \(h_j\) throughout the rest of the text.

  2. 2.

    Recall that this is a problem because this situation does not happen in a solution created by .

References

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 609–626

    Google Scholar 

  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. 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

    Chapter  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Jansen, K., Solis-Oba, R.: Packing squares with profits. SIAM J. Discret. Math. 26(1), 263–279 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Jansen, K., Zhang, G.: Maximizing the total profit of rectangles packed into a rectangle. Algorithmica 47(3), 323–342 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lenstra, H.W.: Integer programming with a fixed number of variables. Math. Oper. Res. 8(4), 538–548 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lodi, A., Martello, S., Monaci, M., Vigo, D.: Two-Dimensional Bin Packing Problems, pp. 107–129. Wiley, Hoboken (2013)

    MATH  Google Scholar 

  13. Meir, A., Moser, L.: On packing of squares and cubes. J. Comb. Theory 5(2), 126–134 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center for Mathematics, Computer, and Cognition, Federal University of ABC, Santo André, Brazil

    Carla Negri Lintzmayer

  2. Institute of Computing, University of Campinas, Campinas, Brazil

    Flávio Keidi Miyazawa & Eduardo Candido Xavier

Authors
  1. Carla Negri Lintzmayer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Flávio Keidi Miyazawa
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Eduardo Candido Xavier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Carla Negri Lintzmayer .

Editor information

Editors and Affiliations

  1. Stony Brook University, Stony Brook, New York, USA

    Michael A. Bender

  2. Rutgers University, New Brunswick, New Jersey, USA

    Martín Farach-Colton

  3. Pace University, New York, New York, USA

    Miguel A. Mosteiro

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Lintzmayer, C.N., Miyazawa, F.K., Xavier, E.C. (2018). Two-Dimensional Knapsack for Circles. In: Bender, M., Farach-Colton, M., Mosteiro, M. (eds) LATIN 2018: Theoretical Informatics. LATIN 2018. Lecture Notes in Computer Science(), vol 10807. Springer, Cham. https://doi.org/10.1007/978-3-319-77404-6_54

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-77404-6_54

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-77403-9

  • Online ISBN: 978-3-319-77404-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics