Box Pleating is Hard
Flat foldability of general crease patterns was first claimed to be hard for over twenty years. In this paper we prove that deciding flat foldability remains NP-complete even for box pleating, where creases form a subset of a square grid with diagonals. In addition, we provide new terminology to implicitly represent the global layer order of a flat folding, and present a new planar reduction framework for grid-aligned gadgets.
This work was begun at the 2015 Bellairs Workshop on Computational Geometry, co-organized by Erik Demaine and Godfried Toussaint. We thank the other participants of the workshop for stimulating discussions, and to Barry Hayes for helpful comments leading to some simplifications. The research of H. Akitaya was supported by NSF grant CCF-1422311 and Science without Borders. The research of T. Hull and T. Tachi were respectively supported by NSF grant EFRI-ODISSEI-1240441 and the JST Presto Program.
- [BDDO10]Benbernou, N.M., Demaine, E.D., Demaine, M.L., Ovadya, A.: Universal hinge patterns to fold orthogonal shapes. In: Origami\(^5\), pp. 405–420. A.K Peters, Singapore, July 2010Google Scholar
- [BH96]Bern, M., Hayes, B.: The complexity of flat origami. In: Proceedings of the Seventh Annual ACM-SIAM Symposium on Discrete Algorithms SODA 1996, pp. 175–183. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (1996)Google Scholar
- [DFL10]Demaine, E.D., Fekete, S.P., Lang, R.J.: Circle packing for origami design is hard. In: Origami\(^5\), pp. 609–626. A.K Peters, Singapore (2010)Google Scholar
- [Sch78]Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings of the Tenth Annual ACM Symposium on Theory of Computing, pp. 216–226. ACM (1978)Google Scholar