Computational Complexities of Folding

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. Findi...