Infinite All-Layers Simple Foldability

We study the problem of deciding whether a crease pattern can be folded by simple folds (folding along one line at a time) under the infinite all-layers model introduced by Akitaya et al. (J Inform Process 25:582–589, 2017), in which each simple fold is defined by an infinite line and must fold all layers of paper that intersect this line. This model is motivated by folding in manufacturing such as sheet-metal bending. We improve on Arkin et al. (Comput Geom Theory Appl 29(1):23–46, 2014) by giving a deterministic O ( n )-time algorithm to decide simple foldability of 1D crease patterns in the all-layers model. Then we extend this 1D result to 2D, showing that simple foldability in the infinite all-layers model can be decided in linear time for both unassigned and assigned axis-aligned orthogonal crease patterns on axis-aligned 2D orthogonal paper. On the other hand, we show that simple foldability is strongly NP-complete if a subset of the creases have a mountain–valley assignment, even for axis-aligned rectangular paper.