Folding a Paper Strip to Minimize Thickness

TL;DR

This paper investigates the computational complexity of folding origami patterns to minimize thickness and paper usage, proving NP-completeness for these optimization problems and identifying fixed-parameter tractability in one case.

Contribution

It provides the first complexity analysis of folding optimization problems related to thickness and paper consumption, including NP-completeness proofs and fixed-parameter tractability results.

Findings

Both optimization problems are strongly NP-complete in 1D.

The problem of minimizing layers in 1D is fixed-parameter tractable.

The study advances understanding of computational limits in origami design.

Abstract

In this paper, we study how to fold a specified origami crease pattern in order to minimize the impact of paper thickness. Specifically, origami designs are often expressed by a mountain-valley pattern (plane graph of creases with relative fold orientations), but in general this specification is consistent with exponentially many possible folded states. We analyze the complexity of finding the best consistent folded state according to two metrics: minimizing the total number of layers in the folded state (so that a "flat folding" is indeed close to flat), and minimizing the total amount of paper required to execute the folding (where "thicker" creases consume more paper). We prove both problems strongly NP-complete even for 1D folding. On the other hand, we prove the first problem fixed-parameter tractable in 1D with respect to the number of layers.