Simple folding is really hard

Abstract

Simple folding (folding along one line at a time) is a practical form of origami used in manufacturing such as sheet metal bending. We prove strong NP-completeness of deciding whether a crease pattern can be simply folded, both for orthogonal paper with assigned orthogonal creases and for square paper with assigned or unassigned creases at multiples of 45°. These results settle a long standing open problem, where weak NP-hardness was established for a subset of the models considered here, leaving open the possibility of pseudopolynomial-time algorithms. We also formalize and generalize the previously proposed simple folding models, and introduce new infinite simple-fold models motivated by practical manufacturing. In the infinite models, we extend our strong NP-hardness results, as well as polynomial-time algorithms for rectangular paper with assigned or unassigned orthogonal creases (map folding). These results motivate why rectangular maps have orthogonal but not diagonal creases.