Compaction and separation algorithms for non-convex polygons and their applications

Given a two-dimensional, non-overlapping layout of convex and non-convex polygons, compaction can be thought of as simulating the motion of the polygons resulting from applied ‘forces’. By moving many polygons simultaneously, compaction can improve the material utilization of even tightly packed layouts. Compaction is hard: finding the tightest layout that a valid motion can reach is PSPACE-hard, and even compacting to a local optimum can require an exponential number of moves. Our first compaction algorithm uses a new velocity-based optimization model based on existing physical simulation approaches. The performance of this algorithm illustrates that these approaches cannot quickly compact tightly packed layouts. We then present a new position-based optimization model. This model represents the forces as a linear objective function, and it permits direct calculation, via linear programming, of new non-overlapping polygon positions at a local minimum of the objective. The new model yields a translational compaction algorithm that runs two orders of magnitude faster than physical simulation methods. We also consider the problem of separating overlapping polygons using the least amount of motion, prove optimal separation to be NP-complete, and show that our position-based model also yields an efficient algorithm for finding a locally optimal separation. The compaction algorithm has improved cloth utilization of human-generated layouts of trousers and other garments. Given a database of human-generated markers and with an efficient technique for matching, the separation algorithm can automatically generate layouts that approach human performance.