Translational Polygon Containment and Minimal Enclosure Using Mathematical Programming

We present a new algorithm for the two-dimensional translational containment problem: nd translations for k polygons which place them inside a polygonal container without overlapping. Both the polygons and the container can be nonconvex. The algorithm is based on mathematical programming principles. We generalize our containment approach to solve minimal enclosure problems. We give algorithms to nd the minimal enclosing square and the minimal area enclosing rectangle for k translating polygons. The containment algorithm consists of new algorithms for restriction, evaluation, and subdivision of two-dimensional connguration spaces. Restriction establishes lower bounds through relaxation and the solution of linear programs. Evaluation establishes upper bounds by nding potential solutions. Subdivision branches, when necessary, by introducing a cutting plane. The algorithm shows that either the upper bound of the objective (overlap) is exactly zero or the lower bound is greater than zero. Hence, it gives an exact solution to the containment problem. In our experiments, our new containment algorithm clearly outperforms purely geometric containment algorithms. For data sets from the apparel industry, it can solve containment for up to ten nonconvex polygons in practice. An implementation of the algorithm given in this paper has been licensed by Gerber Garment Technologies, the largest provider of textile CAD/CAM software in the U.S., and they are incorporating it into an existing CAD/CAM software product. Set of polygonal parts Strip of fixed width and unknown length Packing with 180 degree rotations and xy-flips allowed Figure 1: A marker making task in the apparel industry