Symmetry-breaking constraints for packing identical rectangles within polyhedra

R. Andrade, E. G. Birgin
2011 Optimization Letters  
Two problems related to packing identical rectangles within a polyhedron are tackled in the present work. Rectangles are allowed to differ only by horizontal or vertical translations and possibly ninety-degree rotations. The first considered problem consists in packing as many identical rectangles as possible within a given polyhedron, while the second problem consists in finding the smallest polyhedron of a given type that accommodates a fixed number of identical rectangles. Both problems are
more » ... odeled as mixed integer programming problems. Symmetry-breaking constraints that facilitate the solution of the MIP models are introduced. Numerical results are presented. 1 The contribution of this paper is to explore, with the help of numerical examples, the effect of three kinds of symmetry-breaking constraints: one related to the rectangles rotations and two related to how the rectangles are spatially ordered. We illustrate the impact of the symmetrybreaking constraints in the context of two problems: (i) packing as many identical rectangles within an equilateral triangle; and (ii) finding the smallest equilateral triangle within which a given set of rectangles can be packed. In both problems, the rectangles edges must be aligned with the Cartesian axes; that is, they can only be translated or rotated by ninety degrees. One slightly surprising result is that adding symmetry-breaking constraints is not always beneficial. In particular, if one is trying to determine whether k rectangles can be packed in a triangle, and if it turns out that the answer is "yes", then adding the symmetry-breaking constraints actually slows down the MIP. The main advantage of the symmetry-breaking constraints comes when proving the optimality of a solution or the infeasibility of a problem. The rest of the paper is organized as follows. Section 2 describes the models of both problems. In Section 3, the symmetry-breaking constraints are presented. Numerical experiments are shown in Section 4 and some conclusions are provided in Section 5. Mixed integer programming models Consider a set of R 1 , . . . , R k rectangles with height h and width w, centered at the origin of the Cartesian two-dimensional space and with their sides parallel to the axes. We would like to pack them within a given polyhedron Ω allowing translations and ninety-degree rotations. To model the ninety-degree rotations we introduce binary variables
doi:10.1007/s11590-011-0425-9 fatcat:kdcpcby7mrhpnaiujbxevlg4yq