Complexity of Simple Folding of Mixed Orthogonal Crease Patterns

Discrete and Computational Geometry, Graphs, and Games

Abstract

Continuing results from JCDCGGG 2016 and 2017, we solve several new cases of the simple foldability 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 orthogonal one-layer simple folds. By contrast, we show strong NP-completeness for mixed orthogonal 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.