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.
@article{arxiv.1411.6371,
title = {Folding a Paper Strip to Minimize Thickness},
author = {Erik D. Demaine and David Eppstein and Adam Hesterberg and Hiro Ito and Anna Lubiw and Ryuhei Uehara and Yushi Uno},
journal= {arXiv preprint arXiv:1411.6371},
year = {2016}
}