Lower-dimensional Relaxations of Higher-dimensional Orthogonal Packing

Consider the feasibility problem in higher-dimensional orthogonal packing. Given a set I of d-dimensional rectangles, we need to decide whether a feasible packing in a d-dimensional rectangular container is possible. No item rotation is allowed and item edges are parallel to the coordinate axes. Typically, solution methods employ some bounds to facilitate the decision. Various bounds are known, e.g., the volume bound, Lagrangian and/or continuous relaxation of ILP models, conservative scales, dimension reduction. We generalize two types of one-dimensional bounds. The first type leads to contiguous 1D stock cutting, where items of each type are situated in consecutive bins. The second type leads to standard 1D stock cutting, we call it bar and slice relaxations. Contiguous 1D cutting leads to a new model for orthogonal packing which shares the combinatorial structure of the interval graph model of Fekete and Schepers. We compare the strength of the relaxations. Bar and slice relaxations are rather strong and have the advantage of quick computability. Moreover, we propose a method to combine these relaxations in different dimensions by probing on the packing intersection variables. This allows us to solve many infeasible benchmark instances without any branching. Moreover, we considered some benchmark strippacking instances from the literature. We could close some of them using the new bounds.