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Annals of Operations Research

, Volume 271, Issue 2, pp 831–851 | Cite as

Minimum tiling of a rectangle by squares

  • Michele MonaciEmail author
  • André Gustavo dos Santos
Original Research
  • 106 Downloads

Abstract

We consider a two-dimensional problem in which one is required to split a given rectangular bin into the smallest number of items. The resulting items must be squares to be packed, without overlapping, into the bin so as to cover all the given rectangle. We present a mathematical model and a heuristic algorithm that is proved to find the optimal solution in some special cases. Then, we introduce a relaxation of the problem and present different exact approaches based on this relaxation. Finally, we report computational experiments on the performances of the algorithms on a large set of randomly generated instances.

Keywords

Two-dimensional packing Mathematical models Exact algorithms Computational experiments 

Notes

Acknowledgements

This research was supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), Brazil under Grant PVE 030479/2013-01. Thanks are due to an anonymous referee for helpful comments.

References

  1. Beasley, J. (1985). An exact two-dimensional non-guillotine cutting tree search procedure. Operations Research, 33, 49–64.CrossRefGoogle Scholar
  2. Beaumont, O., Boudet, V., Rastello, F., & Robert, Y. (2002). Partitioning a square into rectangles: NP-completeness and approximation algorithms. Algorithmica, 34, 217–239.CrossRefGoogle Scholar
  3. Benders, J. (1962). Partitioning procedures for solving mixed-variables programming problems. Numerische Mathematik, 4, 238–252.CrossRefGoogle Scholar
  4. Birgin, E., Lobato, R., & Morabito, R. (2010). An effective recursive partitioning approach for the packing of identical rectangles in a rectangle. Journal of the Operational Research Society, 61, 306–320.CrossRefGoogle Scholar
  5. Brooks, R., Smith, C., Stone, A., & Tutte, W. (1940). The dissection of rectangles into squares. Duke Mathematics Journal, 7, 312–340.CrossRefGoogle Scholar
  6. Caprara, A., & Monaci, M. (2004). On the two-dimensional knapsack problem. Operations Research Letters, 32, 5–14.CrossRefGoogle Scholar
  7. Cui, Y., Yang, Y., Cheng, X., & Song, P. (2008). A recursive branch-and-bound algorithm for the rectangular guillotine strip packing problem. Computers and Operations Research, 35, 1281–1291.CrossRefGoogle Scholar
  8. Dolatabadi, M., Lodi, A., & Monaci, M. (2012). Exact algorithms for the two-dimensional guillotine knapsack. Computers and Operations Research, 39, 48–53.CrossRefGoogle Scholar
  9. Fischetti, M., & Monaci, M. (2014). Exploiting erraticism in search. Operations Research, 62, 114–122.CrossRefGoogle Scholar
  10. Kenyon, R. (1996). Tiling a rectangle with the fewest squares. Journal of Combinatorial Theory, 76, 272–291.CrossRefGoogle Scholar
  11. Kurz, S. (2012). Squaring the square with integer linear programming. Journal of Information Processing, 20, 680–685.CrossRefGoogle Scholar
  12. Lodi, A., Martello, S., Monaci, M., Cicconetti, C., Lenzini, L., Mingozzi, E., et al. (2011). Efficient two-dimensional packing algorithms for mobile WiMAX. Management Science, 57, 2130–2144.CrossRefGoogle Scholar
  13. Lodi, A., & Monaci, M. (2013). Integer linear programming models for 2-staged two-dimensional knapsack problems. Mathematical Programming, 94, 257–278.CrossRefGoogle Scholar
  14. Lodi, A., Monaci, M., & Pietrobuoni, E. (2017). Partial enumeration algorithms for two-dimensional bin packing problem with guillotine constraints. Discrete Applied Mathematics, 217, 40–47.CrossRefGoogle Scholar
  15. Lodi, M., Martello, M., Monaci, M., & Vigo, D. (2010). Two-dimensional bin packing problems. In: Paradigms of combinatorial optimization (pp. 107–129). Wiley/ISTE.Google Scholar
  16. Lueker, G. (1975). Two NP-complete problems in nonnegative integer programming. Technical report, Report No. 178, Computer Science Laboratory, Princeton.Google Scholar
  17. Martello, S., Monaci, M., & Vigo, D. (2003). An exact approach to the strip packing problem. INFORMS Journal on Computing, 15, 310–319.CrossRefGoogle Scholar
  18. Martello, S., & Toth, P. (1990). Knapsack problems: Algorithms and computer implementations. Chichester: Wiley.Google Scholar
  19. Miliotis, P. (1976). Integer programming approaches to the travelling salesman problem. Mathematical Programming, 10, 367–378.CrossRefGoogle Scholar
  20. Pietrobuoni, E. (2015). Two-dimensional bin packing problem with guillotine restrictions. Ph.D. thesis, University of Bologna, Bologna, Italy. http://amsdottorato.unibo.it/6810/.
  21. Silva, E., Alvelos, F., & Valrio de Carvalho, J. (2010). An integer programming model for two- and three-stage two-dimensional cutting stock problems. European Journal of Operational Research, 205, 699–708.CrossRefGoogle Scholar
  22. Walters, M. (2009). Rectangles as sum of squares. Discrete Mathematics, 309, 2913–2921.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.DEIUniversità di BolognaBolognaItaly
  2. 2.DPIUniversidade Federal de ViçosaViçosaBrazil

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