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 [ADK17], 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 [ABD + 04] 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 this model can be decided in linear time for unassigned 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 an axis-aligned rectangle of paper.Algorithmic results. In Section 3, we improve on [ABD + 04] by giving a deterministic O(n)time algorithm to decide simple foldability of 1D crease patterns in the all-layers model. This result removes the logarithmic factor from the best previous deterministic algorithm, or equivalently, removes the randomization from the best previous O(n) algorithm.In a recent followup to Arkin et al. [ABD + 04], Akitaya et al.[ADK17] extended the list of simple folding models, and for many models, showed strong NP-hardness for 2D paper. In particular, they introduced the infinite all-layers model of simple folds, studied here, which requires that each simple fold is defined by an infinite line, and that all layers of paper intersecting this line must be folded. This model is probably the most practical of all simple folding models; for example, Balkcom's robotic folding system [BM08] is restricted to this model. For an axis-aligned rectangle paper with axis-aligned creases (and for 1D paper), infinite and non-infinite simple fold models are equivalent [ADK17].Hardness results. In this paper, we study the complexity of one of the few remaining open problems in this area [ADK17]: infinite all-layers simple foldability on axis-aligned orthogonal paper with axis-aligned creases (henceforth referred to as orthogonal crease patterns). In Section 4, we prove that, when the creases are unassigned (can freely fold mountain or valley), this problem can be solved in polynomial (indeed, linear) time. On the other hand, we prove in Section 5 that, when the creases are partially assigned (some creases must fold mountain, some creases must fold valley, while others can freely fold mountain or valley), the problem becomes strongly NP-complete, even for an axis-aligned rectangle of paper (and thus also for the regular all-layers simple fold model [ADK17]). Remaining open problems are summarized in Section 6.
