Advertisement

Algorithmica

, Volume 77, Issue 3, pp 867–901 | Cite as

Online Square-into-Square Packing

  • Sándor P. Fekete
  • Hella-Franziska Hoffmann
Article

Abstract

In 1967, Moon and Moser proved a tight bound on the critical density of squares in squares: any set of squares with a total area of at most 1/2 can be packed into a unit square, which is tight. The proof requires full knowledge of the set, as the algorithmic solution consists in sorting the objects by decreasing size, and packing them greedily into shelves. Since then, the online version of the problem has remained open; the best upper bound is still 1/2, while the currently best lower bound is 1/3, due to Han et al. (Theory Comput Syst 43(1):38–55, 2008). In this paper, we present a new lower bound of 11/32, based on a dynamic shelf allocation scheme, which may be interesting in itself. We also give results for the closely related problem in which the size of the square container is not fixed, but must be dynamically increased in order to accommodate online sequences of objects. For this variant, we establish an upper bound of 3/7 for the critical density, and a lower bound of 1/8. When aiming for accommodating an online sequence of squares, this corresponds to a \(2.82{\ldots }\)-competitive method for minimizing the required container size, and a lower bound of \(1.33{\ldots }\) for the achievable factor.

Keywords

Packing Online problems Packing squares Critical density 

Notes

Acknowledgments

We thank the anonymous reviewers for many helpful comments that improved the overall manuscript.

References

  1. 1.
    Azar, Y., Epstein, L.: On two dimensional packing. J. Algorithms 25(2), 290–310 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bansal, N., Caprara, A., Jansen, K., Prädel, L., Sviridenko, M.: A structural lemma in 2-dimensional packing, and its implications on approximability. In: Algorithms and Computation, 20th International Symposium, ISAAC, pp. 77–86 (2009)Google Scholar
  3. 3.
    Bansal, N., Caprara, A., Sviridenko, M.: Improved approximation algorithms for multidimensional bin packing problems. In: 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 697–708 (2006)Google Scholar
  4. 4.
    Bansal, N., Caprara, A., Sviridenko, M.: A new approximation method for set covering problems, with applications to multidimensional bin packing. SIAM J. Comput. 39(4), 1256–1278 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bansal, N., Correa, J.R., Kenyon, C., Sviridenko, M.: Bin packing in multiple dimensions: inapproximability results and approximation schemes. Math. Oper. Res. 31(1), 31–49 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bansal, N., Han, X., Iwama, K., Sviridenko, M., Zhang, G.: A harmonic algorithm for the 3D strip packing problem. SIAM J. Comput. 42(2), 579–592 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bansal, N., Khan, A.: Improved approximation algorithm for two-dimensional bin packing. In: Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 13–25 (2014)Google Scholar
  8. 8.
    Bansal, N., Lodi, A., Sviridenko, M.: A tale of two dimensional bin packing. In: 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 657–666 (2005)Google Scholar
  9. 9.
    Bansal, N., Sviridenko, M.: New approximability and inapproximability results for 2-dimensional bin packing. In: Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 196–203 (2004)Google Scholar
  10. 10.
    Caprara, A.: Packing 2-dimensional bins in harmony. In: 43rd Symposium on Foundations of Computer Science (FOCS), pp. 490–499 (2002)Google Scholar
  11. 11.
    Caprara, A.: Packing d-dimensional bins in d stages. Math. Oper. Res. 33(1), 203–215 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Caprara, A., Lodi, A., Martello, S., Monaci, M.: Packing into the smallest square: worst-case analysis of lower bounds. Discrete Optim. 3(4), 317–326 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Caprara, A., Lodi, A., Monaci, M.: Fast approximation schemes for two-stage, two-dimensional bin packing. Math. Oper. Res. 30(1), 150–172 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Correa, J.R.: Resource augmentation in two-dimensional packing with orthogonal rotations. Oper. Res. Lett. 34(1), 85–93 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Correa, J.R., Kenyon, C.: Approximation schemes for multidimensional packing. In: Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) 2004, pp. 186–195 (2004)Google Scholar
  16. 16.
    Demaine, E.D., Fekete, S.P., Lang, R.J.: Circle packing for origami design is hard. In: Origami 5, pp. 609–626. AK Peters/CRC Press (2011)Google Scholar
  17. 17.
    Epstein, L., van Stee, R.: Optimal online bounded space multidimensional packing. In: Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2004, New Orleans, Louisiana, USA, January 11–14, 2004, pp. 214–223 (2004)Google Scholar
  18. 18.
    Epstein, L., van Stee, R.: Online square and cube packing. Acta Inform. 41(9), 595–606 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Epstein, L., van Stee, R.: Bounds for online bounded space hypercube packing. Discrete Optim. 4(2), 185–197 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Fekete, S.P., Hoffmann, H.F.: Online square-into-square packing. In: 16th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems Proceedings (APPROX), LNCS, vol. 8096, pp. 126–141 (2013)Google Scholar
  21. 21.
    Fekete, S.P., Kamphans, T., Schweer, N.: Online square packing. In: 11th International Symposium on Algorithms and Data Structures (WADS), LNCS, vol. 5664, pp. 302–314. Springer, Berlin, Heidelberg (2009)Google Scholar
  22. 22.
    Fekete, S.P., Kamphans, T., Schweer, N.: Online square packing with gravity. Algorithmica 68, 1019–1044 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Fishkin, A.V., Gerber, O., Jansen, K., Solis-Oba, R.: Packing weighted rectangles into a square. In: Mathematical Foundations of Computer Science, International Symposium (MFCS), LNCS, vol. 3618, pp. 352–363 (2005)Google Scholar
  24. 24.
    Fishkin, A.V., Gerber, O., Jansen, K., Solis-Oba, R.: On packing rectangles with resource augmentation: maximizing the profit. Algorithmic Oper. Res. 3(1), 1–12 (2008)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Han, X., Iwama, K., Zhang, G.: Online removable square packing. Theory Comput. Syst. 43(1), 38–55 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Harren, R.: Approximation algorithms for orthogonal packing problems for hypercubes. Theor. Comput. Sci. 410(44), 4504–4532 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Harren, R.: Two-dimensional packing problems. Ph.D. thesis, Saarland University (2010)Google Scholar
  28. 28.
    Harren, R., Jansen, K., Prädel, L., Schwarz, U.M., van Stee, R.: Two for one: tight approximation of 2d bin packing. Int. J. Found. Comput. Sci. 24(8), 1299–1328 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Harren, R., Jansen, K., Prädel, L., van Stee, R.: A \((5/3 + \epsilon )\)-approximation for strip packing. Comput. Geom. 47(2), 248–267 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Hougardy, S.: On packing squares into a rectangle. Comput. Geom. Theory Appl. 44(8), 456–463 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Jansen, K., Prädel, L.: New approximability results for two-dimensional bin packing. In: Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, (SODA), pp. 919–936 (2013)Google Scholar
  32. 32.
    Jansen, K., Solis-Oba, R.: New approximability results for 2-dimensional packing problems. In: Mathematical Foundations of Computer Science, International Symposium (MFCS), LNCS, vol. 4708, pp. 103–114 (2007)Google Scholar
  33. 33.
    Jansen, K., Solis-Oba, R.: A polynomial time approximation scheme for the square packing problem. In: Integer Programming and Combinatorial Optimization, 13th International Conference (IPCO), pp. 184–198 (2008)Google Scholar
  34. 34.
    Jansen, K., Solis-Oba, R.: Rectangle packing with one-dimensional resource augmentation. Discrete Optim. 6(3), 310–323 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Jansen, K., van Stee, R.: On strip packing with rotations. In: Proceedings of the 37th Annual ACM Symposium on Theory of Computing (STOC), pp. 755–761 (2005)Google Scholar
  36. 36.
    Jansen, K., Zhang, G.: Maximizing the total profit of rectangles packed into a rectangle. Algorithmica 47(3), 323–342 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Januszewski, J., Lassak, M.: On-line packing sequences of cubes in the unit cube. Geom. Dedic. 67(3), 285–293 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Kenyon, C., Rémila, E.: Approximate strip packing. In: 37th Annual Symposium on Foundations of Computer Science (FOCS), pp. 31–36 (1996)Google Scholar
  39. 39.
    Kleitman, D., Krieger, M.: Packing squares in rectangles I. Ann. N. Y. Acad. Sci. 175, 253–262 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Kleitman, D.J., Krieger, M.M.: An optimal bound for two dimensional bin packing. In: 16th Annual Symposium on Foundations of Computer Science (FOCS), pp. 163–168 (1975)Google Scholar
  41. 41.
    Leung, J.Y.T., Tam, T.W., Wong, C.S., Young, G.H., Chin, F.Y.L.: Packing squares into a square. J. Parallel Distrib. Comput. 10(3), 271–275 (1990)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Meir, A., Moser, L.: On packing of squares and cubes. J. Comb. Theory 5(2), 126–134 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Moon, J., Moser, L.: Some packing and covering theorems. Colloq. Math. 17, 103–110 (1967)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Moser, L.: Poorly formulated unsolved problems of combinatorial geometry. Mimeographed (1966)Google Scholar
  45. 45.
    Novotný, P.: A note on a packing of squares. Stud. Univ. Transp. Commun. Ilina Math. Phys. Ser. 10, 35–39 (1995)MathSciNetGoogle Scholar
  46. 46.
    Novotný, P.: On packing of squares into a rectangle. Arch. Math. (Brno) 32(2), 75–83 (1996)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Zhang, Y., Chen, J.C., Chin, F.Y.L., Han, X., Ting, H.F., Tsin, Y.H.: Improved online algorithms for 1-space bounded 2-dimensional bin packing. In: 21st International Symposium on Algorithms and Computation (ISAAC), LNCS, vol. 6507, pp. 242–253 (2010)Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Computer ScienceTU BraunschweigBraunschweigGermany
  2. 2.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada

Personalised recommendations