Circle Packing for Origami Design Is Hard

Abstract

We show that deciding whether a given set of circles can be packed into a rectangle, an equilateral triangle, or a unit square are NP-hard problems, settling the complexity of these natural packing problems. On the positive side, we show that any set of circles of total area 1 can be packed into a square of size 4/\sqrt{pi}=2.2567... These results are motivated by problems arising in the context of origami design.

arXiv comment17 pages, 13 figures. An abstract was presented at the 5th International Conference on Origami in Science, Mathematics and Education (5OSME). Updated to fix a numerical typo related to Figure 13, and a new result