# Infinite All-Layers Simple Foldability

## TL;DR

This paper presents a linear-time algorithm for deciding simple foldability of crease patterns in the infinite all-layers model for 1D and 2D cases, but proves NP-completeness when crease assignments are specified.

## Contribution

It introduces a linear-time decision algorithm for simple foldability in the all-layers model for 1D and unassigned 2D crease patterns, and establishes NP-completeness with assigned creases.

## Key findings

- Linear-time algorithm for 1D crease patterns.
- Linear-time algorithm for unassigned 2D orthogonal crease patterns.
- NP-completeness with assigned mountain-valley creases.

## Abstract

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., 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., 2004] 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.