Computational Complexities of Folding

TL;DR

This paper establishes the computational complexity of various origami folding problems, proving hardness results and tractability conditions, and explores the limits of algorithmic approaches in origami design and reconfiguration.

Contribution

It provides new complexity results for flat-folding problems, including fixed-parameter tractability, hardness under the exponential time hypothesis, and undecidability in self-similar patterns.

Findings

Flat-folding is fixed-parameter tractable in ply and treewidth.

Certain folding problems are NP-hard, #P-complete, or PSPACE-complete.

Testing foldability of self-similar patterns is undecidable.

Abstract

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.