# Computational Complexities of Folding

arXiv: https://arxiv.org/abs/2410.07666 · HTML: https://mlsanity.com/paper/2410.07666/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Oct 2024.

## Built on by (most cited first, 1 of 1)

- [FoldingAgent: Inferring Parametric Origami Procedures from Demonstration Videos](https://mlsanity.com/paper/2609.00377/) (2026)

## Abstract (arXiv)

We prove several hardness results on folding origami crease patterns. Flat-folding finite crease patterns is fixed-parameter tractable in the ply of the folded pattern (how many layers overlap at any point) and the treewidth of an associated cell adjacency graph. Under the exponential time hypothesis, the singly-exponential dependence of our algorithm on treewidth is necessary, even for bounded ply. Improving the dependence on ply would require progress on the unsolved map folding problem. Finding the shape of a polyhedron folded from a net with triangular faces and integer edge lengths is not possible in algebraic computation tree models of computation that at each tree node allow either the computation of arbitrary integer roots of real numbers, or the extraction of roots of polynomials with bounded degree and integer coefficients. For a model of reconfigurable origami with origami squares are attached at one edge by a hinge to a rigid surface, moving from one flat-folded state to another by changing the position of one square at a time is PSPACE-complete, and counting flat-folded states is #P-complete. For self-similar square crease patterns with infinitely many folds, testing flat-foldability is undecidable.