221 research outputs found
Flip Distance Between 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. The flip distance between two triangulations is
the smallest number of flips required to transform one triangulation into the
other. For the special case of convex polygons, the problem of determining the
shortest flip distance between two triangulations is equivalent to determining
the rotation distance between two binary trees, a central problem which is
still open after over 25 years of intensive study. We show that computing the
flip distance between two triangulations of a simple polygon is NP-complete.
This complements a recent result that shows APX-hardness of determining the
flip distance between two triangulations of a planar point set.Comment: Accepted versio
Flip Distance Between Triangulations of a Planar Point Set is APX-Hard
In this work we consider triangulations of point sets in the Euclidean plane,
i.e., maximal straight-line crossing-free graphs on a finite set of points.
Given a triangulation of a point set, an edge flip is the operation of removing
one edge and adding another one, such that the resulting graph is again a
triangulation. Flips are a major way of locally transforming triangular meshes.
We show that, given a point set in the Euclidean plane and two
triangulations and of , it is an APX-hard problem to minimize
the number of edge flips to transform to .Comment: A previous version only showed NP-completeness of the corresponding
decision problem. The current version is the one of the accepted manuscrip
Flip Distance Between Two Triangulations of a Point-Set is NP-complete
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.
It is not known whether the problem is in P or NP-complete. 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 plane
IST Austria Thesis
This thesis considers two examples of reconfiguration problems: flipping edges in edge-labelled triangulations of planar point sets and swapping labelled tokens placed on vertices of a graph. In both cases the studied structures – all the triangulations of a given point set or all token placements on a given graph – can be thought of as vertices of the so-called reconfiguration graph, in which two vertices are adjacent if the corresponding structures differ by a single elementary operation – by a flip of a diagonal in a triangulation or by a swap of tokens on adjacent vertices, respectively. We study the reconfiguration of one instance of a structure into another via (shortest) paths in the reconfiguration graph.
For triangulations of point sets in which each edge has a unique label and a flip transfers the label from the removed edge to the new edge, we prove a polynomial-time testable condition, called the Orbit Theorem, that characterizes when two triangulations of the same point set lie in the same connected component of the reconfiguration graph. The condition was first conjectured by Bose, Lubiw, Pathak and Verdonschot. We additionally provide a polynomial time algorithm that computes a reconfiguring flip sequence, if it exists. Our proof of the Orbit Theorem uses topological properties of a certain high-dimensional cell complex that has the usual reconfiguration graph as its 1-skeleton.
In the context of token swapping on a tree graph, we make partial progress on the problem of finding shortest reconfiguration sequences. We disprove the so-called Happy Leaf Conjecture and demonstrate the importance of swapping tokens that are already placed at the correct vertices. We also prove that a generalization of the problem to weighted coloured token swapping is NP-hard on trees but solvable in polynomial time on paths and stars
An Time FPT Algorithm for Convex Flip Distance
Let be a convex polygon in the plane, and let be a triangulation of
. An edge in is called a diagonal if it is shared by two triangles
in . A flip of a diagonal is the operation of removing and adding
the opposite diagonal of the resulting quadrilateral to obtain a new
triangulation of from . The flip distance between two triangulations of
is the minimum number of flips needed to transform one triangulation into
the other. The Convex Flip Distance problem asks if the flip distance between
two given triangulations of is at most , for some given parameter .
We present an FPT algorithm for the Convex Flip Distance problem that runs in
time and uses polynomial space, where is the number of flips.
This algorithm significantly improves the previous best FPT algorithms for the
problem
The diameter of type D associahedra and the non-leaving-face property
Generalized associahedra were introduced by S. Fomin and A. Zelevinsky in
connection to finite type cluster algebras. Following recent work of L. Pournin
in types and , this paper focuses on geodesic properties of generalized
associahedra. We prove that the graph diameter of the -dimensional
associahedron of type is precisely for all greater than .
Furthermore, we show that all type associahedra have the non-leaving-face
property, that is, any geodesic connecting two vertices in the graph of the
polytope stays in the minimal face containing both. This property was already
proven by D. Sleator, R. Tarjan and W. Thurston for associahedra of type .
In contrast, we present relevant examples related to the associahedron that do
not always satisfy this property.Comment: 18 pages, 14 figures. Version 3: improved presentation,
simplification of Section 4.1. Final versio
A proof of the orbit conjecture for flipping edge-labelled triangulations
Given a triangulation of a point set in the plane, a flip deletes an edge e whose removal leaves a convex quadrilateral, and replaces e by the opposite diagonal of the quadrilateral. It is well known that any triangulation of a point set can be reconfigured to any other triangulation by some sequence of flips. We explore this question in the setting where each edge of a triangulation has a label, and a flip transfers the label of the removed edge to the new edge. It is not true that every labelled triangulation of a point set can be reconfigured to every other labelled triangulation via a sequence of flips, but we characterize when this is possible. There is an obvious necessary condition: for each label l, if edge e has label l in the first triangulation and edge f has label l in the second triangulation, then there must be some sequence of flips that moves label l from e to f, ignoring all other labels. Bose, Lubiw, Pathak and Verdonschot formulated the Orbit Conjecture, which states that this necessary condition is also sufficient, i.e. that all labels can be simultaneously mapped to their destination if and only if each label individually can be mapped to its destination. We prove this conjecture. Furthermore, we give a polynomial-time algorithm (with (8) being a crude bound on the run-time) to find a sequence of flips to reconfigure one labelled triangulation to another, if such a sequence exists, and we prove an upper bound of (7) on the length of the flip sequence. Our proof uses the topological result that the sets of pairwise non-crossing edges on a planar point set form a simplicial complex that is homeomorphic to a high-dimensional ball (this follows from a result of Orden and Santos; we give a different proof based on a shelling argument). The dual cell complex of this simplicial ball, called the flip complex, has the usual flip graph as its 1-skeleton. We use properties of the 2-skeleton of the flip complex to prove the Orbit Conjecture
- …