Circle Packing for Origami Design Is Hard

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.

💡 Research Summary

The paper investigates the computational complexity of a fundamental step in origami design known as the circle‑river (or tree) method. In this method a piece of paper P′ and a weighted tree T are given; each leaf of T corresponds to a circle whose radius is proportional to the incident edge weight, and each tree edge corresponds to a “river” of constant width. The design problem reduces to finding a scaling factor m that allows all circles and rivers to be placed without overlap inside P′. When the tree is a star (i.e., there are no rivers), the problem collapses to packing a set of circles of prescribed radii into a given container (a rectangle, an equilateral triangle, or a unit square).

The authors prove that this circle‑packing problem is NP‑hard by a reduction from the strongly NP‑complete 3‑Partition problem. The reduction hinges on a geometric gadget called a “symmetric 3‑pocket”. A 3‑pocket is formed by three mutually tangent unit circles, creating a triangular void. Inside each pocket a medium‑sized “plug” circle and three identical “shim” circles are placed. The radii of the shims are adjusted according to the numbers x_i of a 3‑Partition instance (specifically, r_i = r_s – x_i′/N² where x_i′ = 1/3 – x_i and N is a large integer). Lemma 4.1 shows that for any point the sum of signed distances to the three sides of the underlying triangle is constant; Lemma 4.2 proves that the three shims can fit together in a pocket if and only if the corresponding triple of numbers satisfies the 3‑Partition constraint (their adjusted sum ≤ 0). Consequently, a feasible packing of all circles exists exactly when the original 3‑Partition instance is solvable.

To embed many such gadgets into the target container, the authors use known dense packings of unit circles. For an equilateral triangle of side length 2k, Graham’s result guarantees a unique hexagonal packing of (k+2)(k+1)/2 unit circles, which automatically creates k² symmetric 3‑pockets. By inserting the plug‑shim groups into each pocket they obtain a full reduction from 3‑Partition to triangle‑paper circle packing. For rectangular and square containers they construct analogous arrangements, adding auxiliary “rock” circles to fill unwanted gaps and ensure that the only way to achieve a non‑overlapping placement is through the intended pocket structure. Throughout, they control the precision of irrational coordinates by choosing N sufficiently large, guaranteeing that the reduction runs in polynomial time and that the problem lies in NP (the coordinates of a feasible packing can be described with polynomially many bits).

In addition to the hardness results, the paper presents a positive approximation theorem: any collection of circles whose total area does not exceed 1 can always be packed into a square of side length 4/√π ≈ 2.2567. The proof uses a simple density argument combined with a recursive subdivision scheme, showing that although optimal packing may be hard to compute, a constant‑factor guarantee is achievable in polynomial time.

Overall, the work settles the complexity of several natural geometric packing problems arising from origami design, establishing that the optimization step in the circle‑river method is NP‑hard, while also providing a constructive bound for approximate packing when only the total area is constrained.