A simple on-line bin-packing algorithm
Journal of the ACM (JACM)
Exact Solution of the Two-Dimensional Finite Bon Packing Problem
Management Science
Packing 2-Dimensional Bins in Harmony
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
The Three-Dimensional Bin Packing Problem
Operations Research
A Combinatorial Characterization of Higher-Dimensional Orthogonal Packing
Mathematics of Operations Research
INFORMS Journal on Computing
A Set-Covering-Based Heuristic Approach for Bin-Packing Problems
INFORMS Journal on Computing
An efficient approximation scheme for the one-dimensional bin-packing problem
SFCS '82 Proceedings of the 23rd Annual Symposium on Foundations of Computer Science
On the two-dimensional Knapsack Problem
Operations Research Letters
Computers and Operations Research
A new constraint programming approach for the orthogonal packing problem
Computers and Operations Research
SIAM Journal on Computing
New Stabilization Procedures for the Cutting Stock Problem
INFORMS Journal on Computing
A new exact method for the two-dimensional bin-packing problem with fixed orientation
Operations Research Letters
A new lower bound for the non-oriented two-dimensional bin-packing problem
Operations Research Letters
Hi-index | 0.00 |
We consider geometric problems in which rectangles have to be packed in (identical) squares, that turn out to be very hard in practice and for which ILP formulations in which variables specify the coordinates in the packing perform very poorly. While most methods developed until the end of last century 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 in a square?”, that turns out to be the bottleneck of all the approaches known. Given a set of rectangles and the associated convex hull of the incidence vectors of rectangle subsets that fit in 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. Additionally, 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 integer solutions that satisfy all these inequalities generally correspond to vertices of the original convex hull. 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, that in many cases equal the lower bounds obtained by the customary set covering formulation (for which column generation is very hard) being computable within times that are orders of magnitude smaller. All our results extend immediately to the general problem of packing d-dimensional parallelepipeds in hypercubes.