Paper by Erik D. Demaine

Reference:: Esther M. Arkin, Michael A. Bender, Erik D. Demaine, Martin L. Demaine, Joseph S. B. Mitchell, Saurabh Sethia, and Steven S. Skiena, “When Can You Fold a Map?”, in Proceedings of the 7th Workshop on Algorithms and Data Structures (WADS 2001), Lecture Notes in Computer Science, volume 2125, Providence, Rhode Island, August 8–10, 2001, pages 401–413.
Abstract:: We explore the following problem: given a collection of creases on a piece of paper, each assigned a folding direction of mountain or valley, is there a flat folding by a sequence of simple folds? There are several models of simple folds; the simplest one-layer simple fold rotates a portion of paper about a crease in the paper by ±180°. We first consider the analogous questions in one dimension lower—bending a segment into a flat object—which lead to interesting problems on strings. We develop efficient algorithms for the recognition of simply foldable 1-D crease patterns, and reconstruction of a sequence of simple folds. Indeed, we prove that a 1-D crease pattern is flat-foldable by any means precisely if it is by a sequence of one-layer simple folds.
Next we explore simple foldability in two dimensions, and find a surprising contrast: “map” folding and variants are polynomial, but slight generalizations are NP-complete. Specifically, we develop a linear-time algorithm for deciding foldability of an orthogonal crease pattern on a rectangular piece of paper, and prove that it is (weakly) NP-complete to decide foldability of (1) an orthogonal crease pattern on a orthogonal piece of paper, (2) a crease pattern of axis-parallel and diagonal (45-degree) creases on a square piece of paper, and (3) crease patterns without a mountain/valley assignment.
Comments:: This paper is also available from the electronic LNCS volume as http://link.springer.de/link/service/series/0558/papers/2125/21250401.pdf.
Updates:: Ivars Peterson wrote an article describing these results, “Proof clarifies a map-folding problem”, Science News 158(26-27):406, December 23-30, 2002.
Helen Pearson also wrote an article describing these results, “Origami solves road map riddle”, Nature Science Update, February 18, 2002.
Copyright:: The paper is \copyright Springer-Verlag.
Length:: The paper is 12 pages.
Availability:: The paper is available in PostScript (145k) and gzipped PostScript (47k).; See information on file formats.; [Google Scholar search]
Related papers:: MapFolding_CGW2014 (Simple Folding is Strongly NP-Complete); MapFolding (When Can You Fold a Map?); CGW2000 (When Can You Fold a Map?)