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Minimum tiling of a rectangle by squares

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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.

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References

  • Beasley, J. (1985). An exact two-dimensional non-guillotine cutting tree search procedure. Operations Research, 33, 49–64.

    Article  Google Scholar 

  • Beaumont, O., Boudet, V., Rastello, F., & Robert, Y. (2002). Partitioning a square into rectangles: NP-completeness and approximation algorithms. Algorithmica, 34, 217–239.

    Article  Google Scholar 

  • Benders, J. (1962). Partitioning procedures for solving mixed-variables programming problems. Numerische Mathematik, 4, 238–252.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Brooks, R., Smith, C., Stone, A., & Tutte, W. (1940). The dissection of rectangles into squares. Duke Mathematics Journal, 7, 312–340.

    Article  Google Scholar 

  • Caprara, A., & Monaci, M. (2004). On the two-dimensional knapsack problem. Operations Research Letters, 32, 5–14.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Dolatabadi, M., Lodi, A., & Monaci, M. (2012). Exact algorithms for the two-dimensional guillotine knapsack. Computers and Operations Research, 39, 48–53.

    Article  Google Scholar 

  • Fischetti, M., & Monaci, M. (2014). Exploiting erraticism in search. Operations Research, 62, 114–122.

    Article  Google Scholar 

  • Kenyon, R. (1996). Tiling a rectangle with the fewest squares. Journal of Combinatorial Theory, 76, 272–291.

    Article  Google Scholar 

  • Kurz, S. (2012). Squaring the square with integer linear programming. Journal of Information Processing, 20, 680–685.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Lodi, A., & Monaci, M. (2013). Integer linear programming models for 2-staged two-dimensional knapsack problems. Mathematical Programming, 94, 257–278.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Lodi, M., Martello, M., Monaci, M., & Vigo, D. (2010). Two-dimensional bin packing problems. In: Paradigms of combinatorial optimization (pp. 107–129). Wiley/ISTE.

  • Lueker, G. (1975). Two NP-complete problems in nonnegative integer programming. Technical report, Report No. 178, Computer Science Laboratory, Princeton.

  • Martello, S., Monaci, M., & Vigo, D. (2003). An exact approach to the strip packing problem. INFORMS Journal on Computing, 15, 310–319.

    Article  Google Scholar 

  • Martello, S., & Toth, P. (1990). Knapsack problems: Algorithms and computer implementations. Chichester: Wiley.

    Google Scholar 

  • Miliotis, P. (1976). Integer programming approaches to the travelling salesman problem. Mathematical Programming, 10, 367–378.

    Article  Google Scholar 

  • Pietrobuoni, E. (2015). Two-dimensional bin packing problem with guillotine restrictions. Ph.D. thesis, University of Bologna, Bologna, Italy. http://amsdottorato.unibo.it/6810/.

  • 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.

    Article  Google Scholar 

  • Walters, M. (2009). Rectangles as sum of squares. Discrete Mathematics, 309, 2913–2921.

    Article  Google Scholar 

Download references

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.

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Authors and Affiliations

  1. DEI, Università di Bologna, Viale Risorgimento 2, 40136, Bologna, Italy

    Michele Monaci

  2. DPI, Universidade Federal de Viçosa, Av. P. H. Rolfs s/n, Viçosa, MG, 36570-900, Brazil

    André Gustavo dos Santos

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  1. Michele Monaci
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  2. André Gustavo dos Santos
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Correspondence to Michele Monaci.

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Monaci, M., dos Santos, A.G. Minimum tiling of a rectangle by squares. Ann Oper Res 271, 831–851 (2018). https://doi.org/10.1007/s10479-017-2746-2

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  • DOI: https://doi.org/10.1007/s10479-017-2746-2

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