Efficient Segment Folding is Hard

TL;DR

This paper introduces the segment folding problem in computational origami, demonstrating its NP-hardness to determine minimal folding steps for a set of line segments, highlighting the problem's computational complexity.

Contribution

The paper formalizes the segment folding problem and proves its NP-hardness, establishing a fundamental complexity result in computational origami.

Findings

Segment folding problem is NP-hard.

Determining minimal folds for segments is computationally difficult.

Formalization of a new origami folding complexity problem.

Abstract

We introduce a computational origami problem which we call the segment folding problem: given a set of $n$ line-segments in the plane the aim is to make creases along all segments in the minimum number of folding steps. Note that a folding might alter the relative position between the segments, and a segment could split into two. We show that it is NP-hard to determine whether $n$ line segments can be folded in $n$ simple folding operations.