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

2015
*
Discrete & Computational Geometry
*

We show that computing the

doi:10.1007/s00454-015-9709-7
fatcat:pzvdd6hrtbganezxx4ctl2ei7i
*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. ...##
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Flip Distance between Triangulations of a Simple Polygon is NP-Complete
[chapter]

2013
*
Lecture Notes in Computer Science
*

We show that computing the

doi:10.1007/978-3-642-40450-4_2
fatcat:rth2ogl3d5a6rfwsj22oxdi7mq
*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]

2012
*
arXiv
*
pre-print

We prove that two natural generalizations

arXiv:1205.2425v1
fatcat:5p7webflj5emfilfnxbc3xai44
*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. ...##
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Flip distance between two triangulations of a point set is NP-complete

2015
*
Computational geometry
*

We prove that two natural generalizations

doi:10.1016/j.comgeo.2014.11.001
fatcat:ebnw4dsrhfehvluhxqdzxqsnne
*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

2017
*
Discrete & Computational Geometry
*

The

doi:10.1007/s00454-017-9867-x
fatcat:hkjet6cuyrcz5ltmnn2i6jjwie
*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. ... [1] 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]

2018
*
arXiv
*
pre-print

For both

arXiv:1808.03126v1
fatcat:hnhowugk5vdi7gqpaxlsh7adcu
*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 [2]*is*from the strongly*NP*-*complete*Rectilinear Steiner Arborescence problem [14] . ...##
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Conceptual Framework for Finding Approximations to Minimum Weight Triangulation and Traveling Salesman Problem of Planar Point Sets

2020
*
International Journal of Advanced Computer Science and Applications
*

We provide motivation for our research and introduce the fields

doi:10.14569/ijacsa.2020.0110403
fatcat:kswkleldebbmzmwzchflbwdosi
*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. ...##
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Minimum Average Distance Triangulations
[article]

2012
*
arXiv
*
pre-print

In

arXiv:1112.1828v3
fatcat:kqckfoxjtre4dd43vx3thwywo4
*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 [1] , 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

2018
*
Computational geometry
*

Let $\Poly$ be

doi:10.1016/j.comgeo.2017.06.008
fatcat:vfckb57yjfag5ij3pi5rnafnai
*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

2020
*
Algorithmica
*

We prove that deciding whether the

doi:10.1007/s00453-020-00751-1
pmid:33583986
pmcid:PMC7846516
fatcat:5sdsxovs65goredrceeaqnvhrq
*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/. ...##
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Computing Distances between Evolutionary Trees
[chapter]

1998
*
Handbook of Combinatorial Optimization
*

Theorem 1 Computing the nni

doi:10.1007/978-1-4613-0303-9_11
fatcat:2o3tooun75abddi4qhrmeztqwy
*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. ...##
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The Polygon Burning Problem
[article]

2021
*
arXiv
*
pre-print

We prove that PB

arXiv:2111.09054v1
fatcat:3cd4gm25uvbw5ma2gsnqjcn5fe
*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*. ...##
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Flip distances between graph orientations
[article]

2019
*
arXiv
*
pre-print

We prove that deciding whether the

arXiv:1902.06103v1
fatcat:ruaqfglsvrcnpf5u7tnx567hke
*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. ...##
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Approximating uniform triangular meshes in polygons

2002
*
Theoretical Computer Science
*

The minimum (non-zero)

doi:10.1016/s0304-3975(01)00407-8
fatcat:ipiptzdw7ndy7lsagwlwfx6w6e
*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]

2000
*
Lecture Notes in Computer Science
*

The minimum (non-zero)

doi:10.1007/3-540-44968-x_3
fatcat:tf2jbitxbvhqxmrmybky2znt5m
*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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