Abstract
Given a set of rectangular items of different sizes and a rectangular container, the aim of the bi-dimensional Orthogonal Packing Problem (OPP-2 for short) is to decide whether there exists a non-overlapping packing of the items in this container. The rotation of items is not allowed. In this paper we present a new exact algorithm for solving OPP-2, based upon the characterization of solutions using interval graphs proposed by Fekete and Schepers. The algorithm uses MPQ-trees, which were introduced by Korte and Möhring to recognize interval graphs.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Baldacci, R., Boschetti, M.: A cutting plane approach for the two-dimensional orthogonal non-guillotine cutting stock problem. Eur. J. Oper. Res 183(3), 1136–1149 (2007)
Beasley, J.: Algorithms for unconstrained two-dimensional guillotine cutting. J. Oper. Res. Soc. 36(4), 297–306 (1985)
Beasley, J.: An exact two-dimensional non-guillotine cutting tree search procedure. Oper. Res 33(1), 49–64 (1985)
Booth, K., Lueker, G.: Linear algorithms to recognize interval graphs and test for the consecutive ones property. In: Proceedings of the seventh Annual ACM Symposium on Theory of Computing (STOC’75), pp. 255–265 (1975)
Caprara, A., Monaci, M.: On the two-dimensional knapsack problem. Oper. Res. Lett. 32(1), 5–14 (2004)
Carlier, J., Clautiaux, F., Moukrim, A.: New reduction procedures and lower bounds for the two-dimensional bin packing problem with fixed orientation. Comput. Oper. Res. 34(8), 2223–2250 (2007)
Christofides, N., Hadjiconstantinou, E.: An exact algorithm for orthogonal 2-d cutting problems using guillotine cuts. Eur. J. Oper. Res. 83(1), 21–38 (1995)
Clautiaux, F., Carlier, J., Moukrim, A.: A new exact method for the orthogonal packing problem. Eur. J. Oper. Res. 183(3), 1196–1211 (2007)
Clautiaux, F., Jouglet, A., Carlier, J., Moukrim, A.: A new constraint programming approach for the orthogonal packing problem. Comput. Oper. Res. 35(3), 944–959 (2008)
Fekete, S., Schepers, J.: On more-dimensional packing I: Modeling. Tech. rep., University of Köln, Germany (1997)
Fekete, S., Schepers, J.: On more-dimensional packing III: Exact algorithms. Tech. rep., University of Köln, Germany (1997)
Fekete, S.P., Schepers, J., van der Veen, J.: An exact algorithm for higher-dimensional orthogonal packing. Oper. Res. 55, 569–587 (2007)
Ferreira, E., Oliveira, J.: A note on fekete and schepers’ algorithm for the non-guillotinable two-dimensional packing problem. Technical Report (2005). http://paginas.fe.up.pt/~jfo/techreports/Fekete%20and%20Schepers%20OPP%%20degeneracy.pdf
Fulkerson, D., Gross, O.: Incidence matrices and interval graphs. Pac. J. Math. 15(3), 835–855 (1965)
Ghouila-Houri, A.: Caractérisation des graphes non orientes dont on peut orienter les aretes de maniere à obtenir le graphe d’une rélation d’ordre. C. R. Math. Acad. Sci. 254, 1370–1371 (1962)
Gilmore, P., Hoffman, A.: A characterization of comparability graphs and of interval graphs. Can. J. Math. 16, 539–548 (1964)
Korte, N., Möhring, R.: An incremental linear-time algorithm for recognizing interval graphs. SIAM J. Comput. 18(1), 68–81 (1989)
Rose, D., Tarjan, R., Lueker, G.: Algorithmic aspects of vertex eliminationon graphs. SIAM J. Comput. 5, 266–283 (1976)
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was partially supported by the ANR Project GraTel ANR-09-blan-0373-01.
Rights and permissions
About this article
Cite this article
Joncour, C., Pêcher, A. & Valicov, P. MPQ-trees for the Orthogonal Packing Problem. J Math Model Algor 11, 3–22 (2012). https://doi.org/10.1007/s10852-011-9159-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10852-011-9159-z