Skip to main content
SpringerLink
Log in

Techniques and results on approximation algorithms for packing circles

  • Special issue commemorating the Golden Jubilee of the Institute of Mathematics and Statistics of the University of São Paulo
  • Published:
São Paulo Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

This survey provides an introductory guide to some techniques used in the design of approximation algorithms for circle packing problems. We address three such packing problems, in which the circles may have different sizes. They differ on the type of the recipient. We consider the classical bin packing and strip packing, and a variant called knapsack packing. Our aim is to discuss some techniques and basic algorithms to motivate the reader to investigate these and other related problems. We also present the ideas used on more elaborated algorithms, without going into details, and mention known results on these problems.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Baker, B.S., Schwarz, J.S.: Shelf algorithms for two-dimensional packing problems. SIAM J. Comput. 12, 508–525 (1983)

    Article  MathSciNet  Google Scholar 

  2. Basu, S., Pollack, R., Roy, M.F.: On the combinatorial and algebraic complexity of quantifier elimination. J. ACM 43(6), 1002–1045 (1996)

    Article  MathSciNet  Google Scholar 

  3. Baur, C., Fekete, S.: Approximation of geometric dispersion problems. Algorithmica 30, 451–470 (2001)

    Article  MathSciNet  Google Scholar 

  4. Becker, A., Fekete, S., Keldenich, P., Morr, S., Scheffer, C.: Packing geometric objects with optimal worst-case density. In: Proc. of the 35th International Symposium on Computational Geometry (SoCG 2019), pp. 63:1–63:6 (2019)

  5. Boll, D.W., Donovan, J., Graham, R.L., Lubachevsky, B.D.: Improving dense packings of equal disks in a square. Electron. J. Comb. 7(R46), 9 (2000)

    MathSciNet  MATH  Google Scholar 

  6. Brass, P., Moser, W., Pach, J.: Research Problems in Discrete Geometry. Springer, Berlin (2005)

    MATH  Google Scholar 

  7. 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  Google Scholar 

  8. Chung, F.R.K., Garey, M.R., Johnson, D.S.: On packing two-dimensional bins. SIAM J. Algebraic Discrete Methods 3, 66–76 (1982)

    Article  MathSciNet  Google Scholar 

  9. Coffman, E.G., Jr., Garey, M.R., Johnson, D.S.: Approximation algorithms for bin packing: an updated survey. In: Ausiello, G., Lucertini, M., Serafini, P. (eds.) Algorithms Design for Computer System Design, pp. 49–106. Springer, New York (1984)

    Google Scholar 

  10. Coffman, Jr., E.G., Garey, M.R., Johnson, D.S.: Approximation algorithms for bin packing: a survey. In: Hochbaum, D. (ed.) Approximation Algorithms for NP-hard Problems, chap. 2, pp. 46–93. PWS (1997)

  11. Coffman, E.G., Jr., Garey, M.R., Johnson, D.S., Tarjan, R.E.: Performance bounds for level oriented two-dimensional packing algorithms. SIAM J. Comput. 9, 808–826 (1980)

    Article  MathSciNet  Google Scholar 

  12. Demaine, E.D., Fekete, S.P., Lang, R.J.: Circle packing for origami design is hard. In: Proceedings of the 5th International Conference on Origami in Science, pp. 609–626 (2010)

  13. Diedrich, F., Harren, R., Jansen, K., Thöle, R., Thomas, H.: Approximation algorithms for 3d orthogonal knapsack. J. Comput. Sci. Technol. 23(5), 749–762 (2008)

    Article  MathSciNet  Google Scholar 

  14. Eisenbrand, F.: Fast Integer Programming in Fixed Dimension. In: Di Battista, G., Zwick, U. (eds.) Algorithms - ESA 2003. Lecture Notes in Computer Science, vol. 2832, pp. 196–207. Springer, Berlin (2003)

    Chapter  Google Scholar 

  15. Epstein, L.: Two-dimensional online bin packing with rotation. Theoret. Comput. Sci. 411, 2899–2911 (2010)

    Article  MathSciNet  Google Scholar 

  16. Erdős, P.: Gráfok páros körüljárású részgráfjairól. Mat. Lapok (N.S.) 18, 283–288 (1967)

    Google Scholar 

  17. Fejes-Tóth, G.: New results in the theory of packing and covering. In: Gruber, P., Wills, J. (eds.) Convexity and its Applications. Birkhäuser, Basel (1983)

    MATH  Google Scholar 

  18. Fejes-Tóth, G.: Packing and covering. In: Goodman, J., O’Rourke, J., Tóth, C.D. (eds.) Handbook of Discrete and Computational Geometry, 3rd edn. CRC Press, Boca Raton (2017)

    Google Scholar 

  19. Fejes-Tóth, G., Kuperberg, W.: A survey of recent results in the theory of packing and covering. In: Pach, J. (ed.) New Trends in Discrete and Computational Geometry, Algorithms and Combinatorics, vol. 10, pp. 251–279. Springer, Berlin (1993)

    MATH  Google Scholar 

  20. Fejes-Tóth, L.: Über einen geometrischen Satz. Math. Z. 46, 83–85 (1940)

    Article  MathSciNet  Google Scholar 

  21. Fejes-Tóth, L.: Parasites on the stem of a plant. Am. Math. Mon. 78(5), 528–529 (1971)

    Article  MathSciNet  Google Scholar 

  22. Fekete, S., Gurunathan, V., Juneja, K., Keldenich, P., Scheffer, C.: Packing squares into a disk with optimal worst-case density. In: Proc. of the 37th International Symposium on Computational Geometry (SoCG 2021), pp. 35:1–36:16 (2021)

  23. Fekete, S., Keldenich, P., Scheffer, C.: Packing disks into disks with optimal worst-case density. In: Proc. of the 35th International Symposium on Computational Geometry (SoCG 2019), pp. 35:1–35:19 (2019)

  24. Fekete, S.P., Morr, S., Scheffer, C.: Split packing: algorithms for packing circles with optimal worst-case density. Discrete Comput Geom 61, 562–594 (2019)

    Article  MathSciNet  Google Scholar 

  25. Fernandez de la Vega, W., Lueker, G.S.: Bin packing can be solved within \(1+\epsilon \) in linear time. Combinatorica 1(4), 349–355 (1981)

    Article  MathSciNet  Google Scholar 

  26. Füredi, Z.: The densest packing of equal circles into a parallel strip. Discrete Comput. Geom. 6, 95–106 (1991)

    Article  MathSciNet  Google Scholar 

  27. Garey, M.R., Graham, R.L., Ullman, J.D.: Worst-case analysis of memory allocation algorithms. In: Proc. 4th Annual ACM Symp. on the Theory of Computing, pp. 143–150 (1972)

  28. Graham, R., Lubachevsky, B.: Repeated patterns of dense packings of equal disks in a square. Electron. J. Comb. 3 (1996)

  29. Graham, R.L.: Bounds for certain multiprocessor anomalies. Bell Syst. Tech. J. 45, 1563–1581 (1966)

    Article  Google Scholar 

  30. Grigor’ev, D.Y., Vorobjov, N.N., Jr.: Solving systems of polynomial inequalities in subexponential time. J. Symb. Comput. 5(1–2), 37–64 (1988)

    Article  MathSciNet  Google Scholar 

  31. Hales, T.C.: A proof of the Kepler conjecture. Ann. Math. 162(3), 1065–1185 (2005)

    Article  MathSciNet  Google Scholar 

  32. Hales, T.C., Adams, M., Bauer, G., Dang, T.D., Harrison, J., Hoang, L.T., Kalisxyk, C., Magron, V., MClaughlin, S., Nguyen, T.T., Nguyen, Q.T., Nipkow, T., Obua, S., Pleso, J., Rute, J., Solovyev, A., Ta, T.H.A., Tran, N.T., Trieu, T.D., Urban, J., Vu, K., Zumkeller, R.: Formal Proof of the Kepler Conjecture. Forum Math Pi 5, e2 (2017)

    Article  MathSciNet  Google Scholar 

  33. Hifi, M., M’Hallah, R.: A literature review on circle and sphere packing problem: models and methodologies. Adv. Oper. Res. (2009)

  34. 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  Google Scholar 

  35. Ibarra, O.H., Kim, C.E.: Fast approximation algorithms for the knapsack and sum of subset problems. J. ACM 22, 463–468 (1975)

    Article  MathSciNet  Google Scholar 

  36. Johnson, D.S.: Near-optimal bin packing algorithms. Ph.D. thesis, MIT, Cambridge, MA (1973)

  37. Johnson, D.S.: Approximation algorithms for combinatorial problems. J. Comput. Syst. Sci. 9(3), 256–278 (1974)

    Article  MathSciNet  Google Scholar 

  38. Kepler, J.: Strena Seu de Nive Sexangula (New Year’s Gift of Hexagonal Snow). Godfrey Tampach, Frankfurt-am-Main, Germany (1611)

  39. Kertész, G.: On a Problem of Parasites (in Hungarian). Master’s thesis, Eötvös University, Budapest (1982)

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

    Article  MathSciNet  Google Scholar 

  41. Lintzmayer, C.N., Miyazawa, F.K., Xavier, E.C.: Two-dimensional knapsack for circles. In: M.A. Bender, M. Farach-Colton, M.A. Mosteiro (eds.) Proc. of the LATIN 2018: Theoretical Informatics, vol. LNCS 10807, pp. 741–754 (2018)

  42. Lintzmayer, C.N., Miyazawa, F.K., Xavier, E.C.: Online circle and sphere packing. Theoret. Comput. Sci. 776, 75–94 (2019)

    Article  MathSciNet  Google Scholar 

  43. Lodi, A., Martello, S., Monaci, M.: Two-dimensional packing problems: a survey. Eur. J. Oper. Res. 141, 241–252 (2002)

    Article  MathSciNet  Google Scholar 

  44. Maranass, C.D., Floudas, C.A., Pardalos, P.M.: New results in the packing of equal circles in a square. Discrete Math. 142(1–3), 287–293 (1995)

    Article  MathSciNet  Google Scholar 

  45. Markót, M.C.: Interval methods for verifying structural optimality of circle packing configurations in the unit square. J. Comput. Appl. Math. 199(2), 353–357 (2007). Special Issue on Scientific Computing, Computer Arithmetic, and Validated Numerics (SCAN 2004)

  46. Markót, M.C.: Improved interval methods for solving circle packing problems in the unit square. J. Global Optim. 81(3), 773–803 (2021)

    Article  MathSciNet  Google Scholar 

  47. Meir, A., Moser, L.: On packing of squares and cubes. J. Comb. Theory Ser. A 5, 116–127 (1968)

    MathSciNet  MATH  Google Scholar 

  48. 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  Google Scholar 

  49. Seiden, S.S.: On the online bin packing problem. J. ACM 49(5), 640–671 (2002)

    Article  MathSciNet  Google Scholar 

  50. Specht, E.: Packomania. http://www.packomania.com (2021)

  51. Szabó, P.G., Markót, M.C., Csendes, T., Specht, E., Casado, L., García, I.: New Approaches to Circle Packing in a Square. Springer, Berlin (2007)

    MATH  Google Scholar 

  52. Thue, A.: Über die dichteste Zusammenstellung von kongruenten Kreisen in einer Ebene. Christiania [Oslo] : J. Dybwad (1910)

  53. Ullman, J.D.: The performance of a memory allocation algorithm. Tech. Rep. 100, Princeton University (1971)

Download references

Acknowledgements

The authors are grateful to the referees for many valuable suggestions, references and comments.

Funding

Research partially supported by CNPq (Proc. 314366/2018-0, 423833/2018-9, 311892/2021-3) and by FAPESP (Proc. 2015/11937-9, 2016/01860-1)

Author information

Authors and Affiliations

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

    Flávio K. Miyazawa

  2. Institute of Mathematics and Statistics, University of São Paulo, São Paulo, Brazil

    Yoshiko Wakabayashi

Authors
  1. Flávio K. Miyazawa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yoshiko Wakabayashi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Flávio K. Miyazawa.

Additional information

Communicated by Yoshiharu Kohayakawa.

Dedicated to the memory of Imre Simon (1943–2009), who continues to inspire us.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Miyazawa, F.K., Wakabayashi, Y. Techniques and results on approximation algorithms for packing circles. São Paulo J. Math. Sci. 16, 585–615 (2022). https://doi.org/10.1007/s40863-022-00301-3

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40863-022-00301-3

Keywords

Mathematics Subject Classification

Search

Navigation