Optimal Packing of High-Precision Rectangles

Abstract

The rectangle-packing problem consists of finding an enclosing rectangle of smallest area that can contain a given set of rectangles without overlap. Our new benchmark includes rectangles of successively higher precision, challenging the previous state-of-the-art, which enumerates all locations for placing rectangles, as well as all bounding box widths and heights up to the optimal box. We instead limit the rectangles’ coordinates and bounding box dimensions to the set of subset sums of the rectangles’ dimensions. We also dynamically prune values by learning from infeasible subtrees and constrain the problem by replacing rectangles and empty space with larger rectangles. Compared to the previous state-of-the-art, we test 4,500 times fewer bounding boxes on the high-precision benchmark and solve N =9 over two orders of magnitude faster. We also present all optimal solutions up to N =15, which requires 39 bits of precision to solve. Finally, on the open problem of whether or not one can pack a particular infinite series of rectangles into the unit square, we pack the first 50,000 such rectangles witha greedy heuristic and conjecture that the entire infinite series can fit..