Simple Folding is Really Hard
Hugo A. Akitaya∗ Erik D. Demaine† Jason S. Ku†
One of the most researched subsets of computa-tional origami studies flat foldings—folded states ofa polygonal paper that lie in a plane. If we unfoldsuch a folding, we obtain a flat-foldable crease pat-tern which is the planar straight-line graph formedby the creases. Each crease originates from oneof two types of fold: mountain (the paper foldsbackwards) or valley (the paper folds forwards). Acrease pattern is called assigned, if each of its creasesare labeled either mountain or valley, or unassigned,if no crease is labeled. The Flat-Foldabilityproblem asks whether a given crease pattern comesfrom some flat folding. This decision problem isknown to be NP-complete for both assigned andunassigned crease patterns [2].
A simple fold is an operation that transformsa flat folding into another by a rigid 180◦ rota-tion of a subset of the paper around an axis `.During the motion, the paper is not allowed totear or self-cross. This restriction is motivatedby practical sheet-metal bending, where a singlerobotic tool can fold the sheet material at once.The Simple-Foldability problem asks whethera given crease pattern can be folded by a se-quence of simple folds (unfolding is not allowed).Arkin et al. [1] introduced many models of sim-ple folds with respect to the number of layersfolded: they are one-layer (Fig. 1 (1), (5)), all-layers (Fig. 1 (1), (2), (3)), and some-layers (whichimposes no restriction). They prove that Simple-Foldability is weakly NP-complete for: one-layer (assigned), some-layers (assigned/unassigned)and all-layers (assigned/unassigned) if the paper isan orthogonal polygon and the creases are paper-aligned orthogonal (abbreviated ); and for some-layers (assigned) and all-layers (assigned) if the pa-per is square and the creases are paper alignedat multiple of 45◦ (abbreviated ). They alsoprovide a polynomial-time algorithm for Simple-Foldability with rectangular paper with paper-aligned orthogonal creases (abbreviated ). Arkinet al. pose as an open problem whether there exista pseudo-polynomial time algorithm for the modelsproven weakly NP-hard.
We settle this long standing open problem by
∗Tufts University, hugo.alves [email protected]†MIT, {edemaine,jasonku}@mit.edu
proving strong NP-completeness for all models withcrease patterns (assigned/unassigned), and for
some-layers and all-layers models with creasepatterns (assigned/unassigned). We reduce from 3-Partition, which is NP-complete [3]: can a set ofintegers A = {a1, . . . , an} be partitioned into n/3triples each with sum
∑A/(n/3) = t? Given an in-
stance of 3-Partition, we construct the creasepattern shown in Fig. 2, where ∞ = 10nt. The ver-tical creases force the long vertical uncreased stripof paper on the left of the construction to passthrough the right part. Collision is only avoidedby folding through horizontal creases that encodethe integers ai if and only if the 3-Partition in-stance has a solution. To prove the results forwe create a crease pattern that forces the squarepaper to be folded into a long rectangular strip andthen, using turn gadgets, into the orthogonal poly-gon shown in Fig. 2. We also point out an errorin the NP-hardness proof in [1](Theorem 7.1) whenthe crease pattern is unassigned.
If it is hard to decide simple-foldability, a natu-ral question arises: how close can we estimate thenumber of possible simple folds that can be per-formed? We define MaxFold, the natural op-timization version of the decision problem askingfor the maximum number of simple folds that canbe folded given a crease pattern. We show thatgiven a crease pattern admitting a maximum se-quence of m simple folds, approximating MaxFoldto within a factor of m1−ε for any constant ε > 0 isNP-complete in the some-layers and all-layers mod-els. To achieve such result, we transform the reduc-tion in Figure 2 into a gap-producing reduction byadding O(n1/ε) horizontal creases splitting the ex-isting vertical creases. The new creases can only be
(1)
1
(3)(2) (5) (6) (7)(4)2 2
0 10 2
2
Figure 1: Example folding steps demonstrating thedifferences between simple folding models. The axis` is a directed dotted line and the simple fold rotatesthe textured subset of the paper.
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folded if all the preexisting creases of the construc-tion are already folded.
Additionally, we propose three new simple foldsmodels, namely infinite-one-layer, infinite-some-layers and infinite all-layers that require that ex-actly one, at least one, or all layers are folded atthe intersection of ` and the flat folding, respec-tively. Examples are shown in Fig. 1 (1), Fig. 1(1), (3), (4), and Fig. 1 (1), (3), respectively.We prove strong NP-hardness for the infinite-one-layer and infinite-some-layers models by a reduc-tion from 3-Partition. Given an instance of 3-Partition, we construct the crease patternshown in Fig. 3, where δ = 3
2n . The uncreasedportion of the construction forces the creases la-beled ci, i ∈ {1, . . . , 2n3 } to be folded only whenthey are aligned with a crease labeled si after h1
and h2 are folded, or else the axis ` would intersectan uncreased region of the paper. Alignment is onlypossible by folding correct creases vj , j ∈ {1, . . . , n},whose positions encode integers in A, if and only ifthe 3-Partition instance has a solution.
Finally, we show a polynomial-time algorithm forcrease patterns in all infinite models based on the
algorithm in [1]. These results motivate why rectan-gular maps have orthogonal, not diagonal creases.
Acknowledgements. The authors started this work in the con-text of an open problem session associated with 6.890, an MITcourse on Algorithmic Lower Bounds taught by the second au-thor. Supported in part by NSF ODISSEI grant EFRI-1240383,NSF grants CCF-1138967 and CCF-1422311, and the Sciencewithout Borders scholarship program.
References
[1] E. M. Arkin, M. A. Bender, E. D. Demaine, M. L. Demaine,J. S. B. Mitchell, S. Sethia, and S. S. Skiena. When can youfold a map? Computational Geometry: Theory and Appli-cations, 29(1):23–46, September 2004. Preliminary resultspublished in the Proceedings of WADS 2001.
[2] M. Bern and B. Hayes. The complexity of flat origami.In Proc. 7th ACM-SIAM Symp. Discrete Algorithms,ACM/SIAM, pages 175–183, New York/Philadelphia, 1996.
[3] M. R. Garey and D. S. Johnson. Computers and Intractabil-ity: A Guide to the Theory of NP-Completeness. W. H.Freeman & Co., New York, NY, USA, 1979.
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Figure 2: Reduction from 3-Partition to Simple-Foldability for one-layer, some-layers, and all-layers. Mountains and valleys are drawn in red andblue respectively.
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h2
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n3 (2t+ 2δ)
2t2t+ 2δ2t+ 2δ t+ 1 a1 t+ 1 a2 t+ 1 a3 an
tn3 + n(t+ 1)
Figure 3: Reduction from 3-Partition to Simple-Foldability in the infinite-one-layer and infinite-some-layers models. Crease assignment is drawn in red and blue for mountain and valley respectively.
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