Rigid Foldability is NP-Hard

TL;DR

This paper proves that determining whether a given origami crease pattern can be rigidly folded is computationally NP-hard, highlighting the inherent complexity of origami design and reconfigurable systems.

Contribution

It establishes the NP-hardness of rigid foldability decision problems, both with all creases and with optional creases, through reductions from Partition and 1-in-3 SAT.

Findings

Rigid foldability is weakly NP-hard with all creases.

Rigid foldability with optional creases is strongly NP-hard.

Complexity arises from combinatorial configurations at vertices.

Abstract

In this paper, we show that deciding rigid foldability of a given crease pattern using all creases is weakly NP-hard by a reduction from Partition, and that deciding rigid foldability with optional creases is strongly NP-hard by a reduction from 1-in-3 SAT. Unlike flat foldability of origami or flexibility of other kinematic linkages, whose complexity originates in the complexity of the layer ordering and possible self-intersection of the material, rigid foldability from a planar state is hard even though there is no potential self-intersection. In fact, the complexity comes from the combinatorial behavior of the different possible rigid folding configurations at each vertex. The results underpin the fact that it is harder to fold from an unfolded sheet of paper than to unfold a folded state back to a plane, frequently encountered problem when realizing folding-based systems such as self-folding matter and reconfigurable robots.