. Démonstration, F. Ayant-deux-configurations, G. , and S. Nbbin, il est clair, d'après la définition 5.4, que les configurations F et G ne sont pas équivalentes Maintenant si nbBin(F ) = nbBin(G), nous proposons un algorithme d'identification des configurations équivalentes (Algorithme 3) en O(n) La structure de données considérée est celle introduire au début de cette section : étant donné une configuration, ))), B(S, j) = {a j 1 , ..., a jt } est le sous-ensemble

E. Aarts and J. K. Lenstra, Local Search in Combinatorial Optimization, Series in Discrete Mathematics and Optimization, 1997.

J. E. Beasley, An Exact Two-Dimensional Non-Guillotine Cutting Tree Search Procedure, Operations Research, vol.33, issue.1, pp.49-64, 1985.
DOI : 10.1287/opre.33.1.49

B. E. Bengtsson, Packing rectangular pieces -a heuristic approach. The computer journal, pp.353-357, 1982.

J. O. Berkey and P. Y. Wang, Two-Dimensional Finite Bin-Packing Algorithms, Journal of the Operational Research Society, vol.38, issue.5, pp.423-429, 1987.
DOI : 10.1057/jors.1987.70

M. A. Boschetti and A. Mingozzi, The two-dimensional finite bin packing problem. Part I: New lower bounds for the oriented case, Quarterly Journal of the Belgian, French and Italian Operations Research Societies, vol.1, issue.1, pp.27-42, 2003.
DOI : 10.1007/s10288-002-0005-z

M. A. Boschetti and A. Mingozzi, The Two-Dimensional Finite Bin Packing Problem. Part II: New lower and upper bounds, Quarterly Journal of the Belgian, French and Italian Operations Research Societies, vol.1, issue.2, pp.135-147, 2003.
DOI : 10.1007/s10288-002-0006-y

J. M. Bourjolly and V. Rebetz, An analysis of lower bound procedures for the bin packing problem, Computers & Operations Research, vol.32, issue.3
DOI : 10.1016/S0305-0548(03)00244-2

E. K. Burke, G. Kendall, and E. G. Whitwell, A New Placement Heuristic for the Orthogonal Stock-Cutting Problem, Operations Research, vol.52, issue.4, pp.655-671, 2004.
DOI : 10.1287/opre.1040.0109

A. Caprara, M. Locatelli, and E. M. Monaci, Bidimensional Packing by Bilinear Programming, Integer Programming and Combinatorial Optimization, 11th International IPCO Conference, pp.377-391, 2005.
DOI : 10.1007/11496915_28

J. Carlier, F. Clautiaux, and E. A. Moukrim, New reduction procedures and lower bounds for the two-dimensional bin packing problem with fixed orientation, Computers & Operations Research, vol.34, issue.8, 2006.
DOI : 10.1016/j.cor.2005.08.012

B. Chazelle, The bottom-left bin-packing heuristic : an efficient implementation, IEEE Trans. Comput., C, vol.32, pp.697-707, 1983.

N. Christophides and C. Whitlock, An Algorithm for Two-Dimensional Cutting Problems, Operations Research, vol.25, issue.1, pp.30-44, 1977.
DOI : 10.1287/opre.25.1.30

F. R. Chung, M. R. Garey, and D. S. Johnson, On Packing Two-Dimensional Bins, SIAM Journal on Algebraic Discrete Methods, vol.3, issue.1, pp.66-76, 1982.
DOI : 10.1137/0603007

F. Clautiaux, J. Carlier, and E. A. Moukrim, A new exact method for the orthogonal packing problem, European Journal of Operational Research, 2006.

F. Clautiaux, J. Carlier, and E. A. Moukrim, A new exact method for the twodimensional bin-packing problem with fixed orientaion, Operations Research Letters, 2006.

F. Clautiaux and A. Jouglet, El Hayek. A new lower bound for the nonoriented two-dimensional bin-packing problem, Operations Research Letters, 2006.

E. G. Coffman, J. , M. R. Garey, and D. S. Johnson, Approximation Algorithms for Bin-Packing ??? An Updated Survey, Algorithms design for computer system design, pp.49-106, 1984.
DOI : 10.1007/978-3-7091-4338-4_3

V. Cung, M. Hifi, and E. B. Le-cun, Constrained two-dimensional cutting stock problems a best-first branch-and-bound algorithm, International Transactions in Operational Research, vol.49, issue.3, pp.185-210
DOI : 10.1016/0360-8352(89)90013-2

M. Dell-'amico, M. Martello, and E. D. Vigo, A lower bound for the non-oriented two-dimensional bin packing problems. dam, pp.13-24, 2002.

K. A. Dowsland, Some experiments with simulated annealing techniques for packing problems, European Journal of Operational Research, vol.68, issue.3, pp.389-399, 1993.
DOI : 10.1016/0377-2217(93)90195-S

U. Dyckhoff and H. Finke, Cutting and Packing in Production and Distribution : Typology and Bibliography, 1992.
DOI : 10.1007/978-3-642-58165-6

J. El-hayek, A. Moukrim, and E. S. Negre, New resolution algorithm and pretreatments for the two-dimensional bin-packing problem, Computers & Operations Research, vol.35, issue.10, 2006.
DOI : 10.1016/j.cor.2007.02.013

J. El-hayek, A. Moukrim, and E. S. Negre, Prétraitements et résolution approchée pour le problème de bin-packing en deux dimensions, le cas non orienté, 2006.

J. El-hayek, A. Moukrim, and E. S. Negre, A tabu search method for the non-oriented two-dimensional bin-packing problem, INCOM conference, 2006.

J. El-hayek, A. Moukrim, and S. Et-negre, Le problème de bin-packing en deux dimensions : nouveaux prétraitements et heuristique de résolution pour le cas non orienté, Conférence MOSIM, 2006.

O. Faroe, D. Pisinger, and E. M. Zachariasen, Guided Local Search for the Three-Dimensional Bin-Packing Problem, INFORMS Journal on Computing, vol.15, issue.3, pp.267-283, 2003.
DOI : 10.1287/ijoc.15.3.267.16080

S. Fekete and J. Schepers, New classes of fast lower bounds for bin packing problems, Mathematical Programming, pp.11-31, 2001.
DOI : 10.1007/s101070100243

S. Fekete and J. Schepers, A Combinatorial Characterization of Higher-Dimensional Orthogonal Packing, Mathematics of Operations Research, vol.29, issue.2, pp.353-368, 2004.
DOI : 10.1287/moor.1030.0079

S. Fekete and J. Schepers, A General Framework for Bounds for Higher-Dimensional Orthogonal Packing Problems, Mathematical Methods of Operational Research, vol.60, issue.2, pp.311-329, 2004.
DOI : 10.1007/s001860400376

S. Fekete, J. Schepers, and E. J. Van-der-veen, An exact algorithm for higherdimensional orthogonal packing, 2006.

M. R. Garey and D. S. Johnson, Computers and intractability, a guide to the theory of NP-completeness, 1979.

P. Gilmore and R. Gomory, A Linear Programming Approach to the Cutting-Stock Problem, Operations Research, vol.9, issue.6, pp.849-859, 1961.
DOI : 10.1287/opre.9.6.849

P. Gilmore and R. Gomory, A Linear Programming Approach to the Cutting Stock Problem???Part II, Operations Research, vol.11, issue.6, pp.863-888, 1963.
DOI : 10.1287/opre.11.6.863

P. Gilmore and R. Gomory, Multistage Cutting Stock Problems of Two and More Dimensions, Operations Research, vol.13, issue.1, pp.94-120, 1965.
DOI : 10.1287/opre.13.1.94

F. Glover and M. Laguna, Tabu search, 1997.
URL : https://hal.archives-ouvertes.fr/hal-01389283

E. Hadjiconstantinou and N. Christophides, An exact algorithm for general, orthogonal, two-dimensional knapsack problems, European Journal of Operational Research, vol.83, issue.1, pp.39-56, 1995.
DOI : 10.1016/0377-2217(93)E0278-6

M. Haouari and A. Gharbi, Fast lifting procedures for the bin packing problem, Discrete Optimization, vol.2, issue.3, pp.201-218, 2005.
DOI : 10.1016/j.disopt.2005.06.002

J. Harwig and J. W. Barnes, An adaptive tabu search approach for 2-dimensional orthogonal packing problems. to appear in Military Operations Research, 2006.

M. Hifi, An improvement of viswanathan and bagchi's exact algorithm for constrained two-dimensional cutting stock, Computers & Operations Research, vol.24, issue.8, pp.727-736, 1997.
DOI : 10.1016/S0305-0548(96)00095-0

S. Jakobs, On genetic algorithms for the packing of polygons, European Journal of Operational Research, vol.88, issue.1, pp.165-181, 1996.
DOI : 10.1016/0377-2217(94)00166-9

D. S. Johnson, Near optimal bin packing algorithms, 1973. Dissertation, Massachussetts Institute of Technology

M. Labbé, G. Laporte, and E. H. Mercure, Capacitated Vehicle Routing on Trees, Operations Research, vol.39, issue.4, pp.16-22, 1991.
DOI : 10.1287/opre.39.4.616

Z. Li and V. Milenkovic, Compaction and separation algorithms for non-convex polygons and their applications, European Journal of Operational Research, vol.84, issue.3, pp.539-561, 1995.
DOI : 10.1016/0377-2217(95)00021-H
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.45.4592

A. Lodi, S. Martello, and E. D. Vigo, Neighborhood search algorithm for the guillotine non-oriented two-dimensional bin packing problem. Meta-Heuristics : Advances and Trends in Local Search Paradigms of Optimization, pp.125-139, 1998.

A. Lodi, S. Martello, and E. D. Vigo, Heuristic and Metaheuristic Approaches for a Class of Two-Dimensional Bin Packing Problems, INFORMS Journal on Computing, vol.11, issue.4, pp.345-357, 1999.
DOI : 10.1287/ijoc.11.4.345

A. Lodi, S. Martello, and E. D. Vigo, Recent advances on two-dimensional bin packing problems, Discrete Applied Mathematics, vol.123, issue.1-3, pp.379-396, 2002.
DOI : 10.1016/S0166-218X(01)00347-X

S. Martello and P. Toth, Lower bounds and reduction procedures for the bin packing problem, Discrete Applied Mathematics, vol.28, issue.1, pp.59-70, 1990.
DOI : 10.1016/0166-218X(90)90094-S

S. Martello and D. Vigo, Exact Solution of the Two-Dimensional Finite Bin Packing Problem, Management Science, vol.44, issue.3, pp.388-399, 1998.
DOI : 10.1287/mnsc.44.3.388

S. and B. Messaoud, Caractérisation, Modélisation et Algorithmes pour des Problèmes de Découpe Guillotine, 2005.

G. Scheithauer, Equivalence and dominance for problems of optimal packing of rectangles, Ricerca Operativa, vol.83, pp.3-34, 1997.

G. Wäscher, H. Haussner, and E. H. Schumann, An improved typology of cutting and packing problems, European Journal of Operational Research, 2006.