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### Flip Distance Between Triangulations of a Simple Polygon is NP-Complete

Oswin Aichholzer, Wolfgang Mulzer, Alexander Pilz
2015 Discrete & Computational Geometry
We show that computing the flip distance between two triangulations of a simple polygon is NP-complete.  ...  Let T be a triangulation of a simple polygon. A flip in T is the operation of removing one diagonal of T and adding a different one such that the resulting graph is again a triangulation.  ...  Is there a PTAS for the flip distance between triangulations of a polygon? Even a constant-factor approximation would be interesting.  ...

### Flip Distance between Triangulations of a Simple Polygon is NP-Complete [chapter]

Oswin Aichholzer, Wolfgang Mulzer, Alexander Pilz
2013 Lecture Notes in Computer Science
We show that computing the flip distance between two triangulations of a simple polygon is NP-hard.  ...  Let T be a triangulation of a simple polygon. A flip in T is the operation of replacing one diagonal of T by a different one such that the resulting graph is again a triangulation.  ...  Is there a PTAS for the flip distance between triangulations of a polygon? Even a constant-factor approximation would be interesting.  ...

### Flip Distance Between Two Triangulations of a Point-Set is NP-complete [article]

Anna Lubiw, Vinayak Pathak
2012 arXiv   pre-print
We prove that two natural generalizations of the problem are NP-complete, namely computing the minimum number of flips between two triangulations of (1) a polygon with holes; (2) a set of points in the  ...  Given two triangulations of a convex polygon, computing the minimum number of flips required to transform one to the other is a long-standing open problem.  ...  The main result of our paper is that it is NP-complete to compute the flip distance between two triangulations of a a polygon with holes, or a set of points in the plane.  ...

### Flip distance between two triangulations of a point set is NP-complete

Anna Lubiw, Vinayak Pathak
2015 Computational geometry
We prove that two natural generalizations of the problem are NP-complete, namely computing the minimum number of flips between two triangulations of (1) a polygon with holes; (2) a set of points in the  ...  Given two triangulations of a convex polygon, computing the minimum number of flips required to transform one to the other is a long-standing open problem.  ...  The main result of our paper is that it is NP-complete to compute the flip distance between two triangulations of a polygon with holes, or of a set of points in the plane.  ...

### Computing the Flip Distance Between Triangulations

Iyad Kanj, Eric Sedgwick, Ge Xia
2017 Discrete & Computational Geometry
The Flip Distance problem asks if the flip distance between two given triangulations of P is k, for some given k ∈ N. It is a fundamental and a challenging problem.  ...  The flip distance between two triangulations of P is the minimum number of flips needed to transform one triangulation into the other.  ...   showed the problem to be N P-complete for triangulations of a simple polygon.  ...

### A Note on the Flip Distance Problem for Edge-Labeled Triangulations [article]

Alexander Pilz
2018 arXiv   pre-print
For both triangulations of point sets and simple polygons, it is known that determining the flip distance between two triangulations is an NP-hard problem.  ...  triangulations of simple polygons.  ...  Simple Polygons The reduction for the flip distance problem on simple polygons in  is from the strongly NP-complete Rectilinear Steiner Arborescence problem  .  ...

### Conceptual Framework for Finding Approximations to Minimum Weight Triangulation and Traveling Salesman Problem of Planar Point Sets

Marko Dodig, Milton Smith
2020 International Journal of Advanced Computer Science and Applications
We provide motivation for our research and introduce the fields of triangulation and polygonization of planar point sets as theoretical bases of our approach, namely, we present the Isoperimetric Inequality  ...  MWT is a classical problem of Computational Geometry with various applications, whereas TSP is perhaps the most researched problem in Combinatorial Optimization.  ...  distance between the corresponding points.  ...

### Minimum Average Distance Triangulations [article]

Laszlo Kozma
2012 arXiv   pre-print
In a different variant of the problem, the points are vertices of a simple polygon and we look for a triangulation of the interior of the polygon that is optimal in the same sense.  ...  We prove that a general formulation of the problem in which the weights are arbitrary positive numbers is strongly NP-complete.  ...  In contrast to both mwt and budgeted network design  , madt is interesting even for unit weights. In case of simple polygons, the problem is neither trivial, nor NP-hard.  ...

### The dual diameter of triangulations

Matias Korman, Stefan Langerman, Wolfgang Mulzer, Alexander Pilz, Maria Saumell, Birgit Vogtenhuber
2018 Computational geometry
Let \$\Poly\$ be a simple polygon with \$n\$ vertices.  ...  In contrast, we give examples of general simple polygons where every triangulation that maximizes the number of ears has dual diameter that is quadratic in the minimum possible value.  ...  Part of this work has been done while A.P. was recipient of a DOC-fellowship of the Austrian Academy of Sciences at the Institute for Software Technology, Graz University of Technology, Austria.  ...

### Flip Distances Between Graph Orientations

Oswin Aichholzer, Jean Cardinal, Tony Huynh, Kolja Knauer, Torsten Mütze, Raphael Steiner, Birgit Vogtenhuber
2020 Algorithmica
We prove that deciding whether the flip distance between two α -orientations of a planar graph G is at most two is NP-complete.  ...  Skeletons of associahedra, for instance, are the graphs induced by quadrilateral flips in triangulations of a convex polygon.  ...  To view a copy of this licence, visit http://creat iveco mmons .org/licen ses/by/4.0/.  ...

### Computing Distances between Evolutionary Trees [chapter]

Bhaskar DasGupta, Xin He, Tao Jiang, Ming Li, John Tromp, Lusheng Wang, Louxin Zhang
1998 Handbook of Combinatorial Optimization
Theorem 1 Computing the nni distance {between two labeled or unlabeled trees) ·is NP-complete.  ...  A diagonal-flip is an operation that transforms one triangulation of a conve.x polygon into another as showed in Figure 18 . A diagonal inside the polygon is removed, creating a quadrilateral.  ...

### The Polygon Burning Problem [article]

William Evans, Rebecca Lin
2021 arXiv   pre-print
We prove that PB is NP-hard when k is arbitrary.  ...  Lastly, we define and characterize a new type of polygon, the sliceable polygon. A sliceable polygon is a convex polygon that contains no Voronoi vertex from the Voronoi diagram of its vertices.  ...  If P is without holes, then it is a simple polygon.  ...

### Flip distances between graph orientations [article]

Oswin Aichholzer and Jean Cardinal and Tony Huynh and Kolja Knauer and Torsten Mütze and Raphael Steiner and Birgit Vogtenhuber
2019 arXiv   pre-print
We prove that deciding whether the flip distance between two α-orientations of a planar graph G is at most two is -complete.  ...  Skeletons of associahedra, for instance, are the graphs induced by quadrilateral flips in triangulations of a convex polygon.  ...  We thank the organizers and participants of this workshop for the stimulating atmosphere.  ...

### Approximating uniform triangular meshes in polygons

Franz Aurenhammer, Naoki Katoh, Hiromichi Kojima, Makoto Ohsaki, Yinfeng Xu
2002 Theoretical Computer Science
The minimum (non-zero) distance between two point sets X and Y is defined as l(X, Y ) = min{l(x, y) | x ∈ X, y ∈ Y, x = y}. When X is a singleton set {x} we simply write l(X, Y ) as l(x, Y ).  ...  Note that the solution we construct is neccessarily a triangulation of constant vertex degree.  ...  In addition, the fifth author was also supported by NSF of China (No. 19731001) and the Japan Society for the Promotion of Science of Japan. We gratefully acknowledge all these supports.  ...

### Approximating Uniform Triangular Meshes in Polygons [chapter]

Franz Aurenhammer, Naoki Katoh, Hiromichi Kojima, Makoto Ohsaki, Yinfeng Xu
2000 Lecture Notes in Computer Science
The minimum (non-zero) distance between two point sets X and Y is defined as l(X, Y ) = min{l(x, y) | x ∈ X, y ∈ Y, x = y}. When X is a singleton set {x} we simply write l(X, Y ) as l(x, Y ).  ...  Note that the solution we construct is neccessarily a triangulation of constant vertex degree.  ...  In addition, the fifth author was also supported by NSF of China (No. 19731001) and the Japan Society for the Promotion of Science of Japan. We gratefully acknowledge all these supports.  ...
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