Paper by Erik D. Demaine

Reference:

Josh Brunner, Erik D. Demaine, Della Hendrickson, Victor Luo, and Andy Tockman, “Complexity of Simple Folding Orthogonal Crease Patterns”, in Abstracts from the 23rd Thailand-Japan Conference on Discrete and Computational Geometry, Graphs, and Games (TJCDCGGG 2021), September 3–5, 2021, pages 26–27.

Abstract:

Continuing results from JCDCGGG 2016 and 2017, we solve several new cases of the simple folding problem — deciding which crease patterns can be folded flat by a sequence of (some model of) simple folds. We give new efficient algorithms for mixed crease patterns, where some creases are assigned mountain/valley while others are unassigned, for all 1D cases and for 2D rectangular paper with one-layer simple folds. By contrast, we show strong NP-completeness for mixed crease patterns on 2D rectangular paper with some-layers simple folds, complementing a previous result for all-layers simple folds. We also prove strong NP-completeness for finite simple folds (no matter the number of layers) of unassigned orthogonal crease patterns on arbitrary paper, complementing a previous result for assigned crease patterns, and contrasting with a previous positive result for infinite all-layers simple folds. In total, we obtain a characterization of polynomial vs. NP-hard for all cases — finite/infinite one/some/all-layers simple folds of assigned/unassigned/mixed orthogonal crease patterns on 1D/rectangular/arbitrary paper — except the unsolved case of infinite all-layers simple folds of assigned orthogonal crease patterns on arbitrary paper.

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MixedSimpleFolds_TJM (Complexity of Simple Folding of Mixed Orthogonal Crease Patterns)