Bidimensional packing by bilinear programming

We consider geometric problems in which rectangles have to be packed into (identical) squares. These problems turn out to be very hard in practice, and ILP formulations in which variables specify the coordinates in the packing perform very poorly. While most methods developed so far are based on simple geometric considerations, a recent landmark result of Fekete and Schepers suggests to put these geometric aspects aside and use the most advanced tools for the 1-dimensional case. In this paper we make additional progress in this direction, especially on the basic question "Does a given set of rectangles fit into a square?'', which turns out to be the bottleneck of all the approaches known. Given a set of rectangles and the associated convex hull of rectangle subsets that fit into a square, we derive a wide class of valid inequalities for this convex hull from a complete description of the two knapsack polytopes associated with the widths and the heights of the rectangles, respectively. In addition, we illustrate how to solve the associated separation problem as a bilinear program, for which we develop a solution method that turns out to be fast in practice, and show that the integer solutions that satisfy all these constraints generally correspond to vertices of the original convex hull for the benchmark instances in the literature. The same tools are used to derive lower bounds for the 2-dimensional bin packing problem, corresponding to the determination of an optimal pair of so-called dual feasible functions. These lower bounds in many cases equal those obtained by the customary set covering formulation (for which column generation is very hard), but are computable within a time that is smaller by some orders of magnitude. This allows us to close a few of the benchmark instances in the literature.