Circle Packing for Origami Design Is Hard

TL;DR

This paper proves that circle packing problems into simple shapes are NP-hard, but also provides a positive packing result for circles of total area 1, with implications for origami design.

Contribution

It establishes the NP-hardness of circle packing into rectangles, triangles, and squares, and offers a universal packing bound for circles of total area 1.

Findings

Deciding circle packings into basic shapes is NP-hard.

Any set of circles with total area 1 can be packed into a square of size approximately 2.2567.

Results are motivated by and applicable to origami design problems.

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.