Related papers: Computational Complexities of Folding
Origami metamaterials typically consist of folded sheets with periodic patterns, conferring them with remarkable mechanical properties. In the context of Continuum Mechanics, the majority of existing predictive methods are mechanism analogs…
Given a sheet of paper and a prescribed folding of its boundary, is there a way to fold the paper's interior without stretching so that the boundary lines up with the prescribed boundary folding? For polygonal boundaries nonexpansively…
Origami is an ancient art that continues to yield both artistic and scientific insights to this day. In 2012, Buhler, Butler, de Launey, and Graham extended these ideas even further by developing a mathematical construction inspired by…
Thick origami structures are considered here as assemblies of polygonal panels hinged to each other along their edges according to a corresponding origami crease pattern. The determination of the internal actions caused by external loads in…
We study a popular algorithm for fitting polynomial curves to scattered data based on the least squares with gradient weights. We show that sometimes this algorithm admits a substantial reduction of complexity, and, furthermore, find…
Traditional origami starts from flat surfaces, leading to crease patterns consisting of Euclidean vertices. However, Euclidean vertices are limited in their folding motions, are degenerate, and suffer from misfolding. Here we show how…
Ghomi proved that every convex polyhedron could be stretched via an affine transformation so that it has an edge-unfolding to a net [Gho14]. A net is a simple planar polygon; in particular, it does not self-overlap. One can view his result…
An "origami" (or flat structure) on a closed oriented surface, $S_g$, of genus $g \geq 2$ is obtained from a finite collection of unit Euclidean squares by gluing each right edge to a left one and each top edge to a bottom one. The main…
Edge-matching problems, also called edge matching puzzles, are abstractions of placement problems with neighborhood conditions. Pieces with colored edges have to be placed on a board such that adjacent edges have the same color. The problem…
As a confined thin sheet crumples, it spontaneously segments into flat facets delimited by a network of ridges. Despite the apparent disorder of this process, statistical properties of crumpled sheets exhibit striking reproducibility.…
We consider combinatorial problems that can be solved in polynomial time for graphs of bounded treewidth but where the order of the polynomial that bounds the running time is expected to depend on the treewidth bound. First we review some…
We study the theoretical and practical aspects of computing braids described by approximate descriptions of paths in the plane. Exact algorithms rely on the lexicographic ordering of the points in the plane, which is unstable under…
Given a graph G and an integer k, the objective of the $\Pi$-Contraction problem is to check whether there exists at most k edges in G such that contracting them in G results in a graph satisfying the property $\Pi$. We investigate the…
We analyze the computational complexity of several new variants of edge-matching puzzles. First we analyze inequality (instead of equality) constraints between adjacent tiles, proving the problem NP-complete for strict inequalities but…
Fold-and-cut theorem claims the possibility to cut out from a sheet a set of straight-line drawing using only one cut of scissors, without producing any other cut in the sheet and separating all the figures at the same time, just by folding…
In the longest plane spanning tree problem, we are given a finite planar point set $\mathcal{P}$, and our task is to find a plane (i.e., noncrossing) spanning tree for $\mathcal{P}$ with maximum total Euclidean edge length. Despite more…
We consider drawings of trees in which all edges incident to leaves can be extended to infinite rays without crossing, partitioning the plane into infinite convex polygons. Among all such drawings we seek the one maximizing the angular…
This paper addresses the problem of finding minimum forcing sets in origami. The origami material folds flat along straight lines called creases that can be labeled as mountains or valleys. A forcing set is a subset of creases that force…
We show that several problems of compacting orthogonal graph drawings to use the minimum number of rows, area, length of longest edge or total edge length cannot be approximated better than within a polynomial factor of optimal in…
Optimization problems consist of either maximizing or minimizing an objective function. Instead of looking for a maximum solution (resp. minimum solution), one can find a minimum maximal solution (resp. maximum minimal solution). Such…