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

Flip Paths Between Lattice Triangulations

Diagonal flip paths between triangulations have been studied in the combinatorial setting for nearly a century. One application of flip paths to Euclidean distance geometry and Moebius geometry is a recent, simple, constructive proof by Connelly and Gortler of the Koebe-Andreev-Thurston circle packing theorem that relies on the existence of a flip path between any two triangulation graphs. More generally, length and other structural quantities on minimum (length) flip paths are metrics on the space of triangulations. In the geometric setting, finding a minimum flip path between two triangulations is NP-complete. However, for two lattice triangulations, used to model electron spin systems, Eppstein and Caputo et al. gave algorithms running in $O\left(n^2\right)$ time, where $n$ is the number of points in the point-set. Their algorithms apply to constrained flip paths that ensure a set of \emph{constraint} edges are present in every triangulation along the path. We reformulate the problem and provide an algorithm that runs in $O\left(n^{\frac{3}{2}}\right)$ time. In fact, for a large, natural class of inputs, the bound is tight, i.e., our algorithm runs in time linear in the length of this output flip path. Our results rely on structural elucidation of minimum flip paths. Specifically, for any two lattice triangulations, we use Farey sequences to construct a partially-ordered sets of flips, called a minimum flip \emph{plan}, whose linear-orderings are minimum flip paths between them. To prove this, we characterize a minimum flip plan that starts from an equilateral lattice triangulation - i.e., a lattice triangulation whose edges are all unit-length - and \emph{forces a point-pair to become an edge}. To the best of our knowledge, our results are the first to exploit Farey sequences for elucidating the structure of flip paths between lattice triangulations.Comment: 24 pages (33 with appendices), 8 figure

Reconfiguring Triangulations

The results in this thesis lie at the confluence of triangulations and reconfiguration. We make the observation that certain solved and unsolved problems about triangulations can be cast as reconfiguration problems. We then solve some reconfiguration problems that provide us new insights about triangulations. Following are the main contributions of this thesis: 1. We show that computing the flip distance between two triangulations of a point set is NP-complete. A flip is an operation that changes one triangulation into another by replacing one diagonal of a convex quadrilateral by the other diagonal. The flip distance, then, is the smallest number of flips needed to transform one triangulation into another. For the special case when the points are in convex position, the problem of computing the flip distance is a long-standing open problem. 2. Inspired by the problem of computing the flip distance, we start an investigation into computing shortest reconfiguration paths in reconfiguration graphs. We consider the reconfiguration graph of satisfying assignments of Boolean formulas where there is a node for each satisfying assignment of a formula and an edge whenever one assignment can be changed to another by changing the value of exactly one variable from 0 to 1 or from 1 to 0. We show that computing the shortest path between two satisfying assignments in the reconfiguration graph is either in P, NP-complete, or PSPACE-complete depending on the class the Boolean formula lies in. 3. We initiate the study of labelled reconfiguration. For the case of triangulations, we assign a unique label to each edge of the triangulation and a flip of an edge from e to e' assigns the same label to e' as e. We show that adding labels may make the reconfiguration graph disconnected. We also show that the worst-case reconfiguration distance changes when we assign labels. We show tight bounds on the worst case reconfiguration distance for edge-labelled triangulations of a convex polygon and of a spiral polygon, and edge-labelled spanning trees of a graph. We generalize the result on spanning trees to labelled bases of a matroid and show non-trivial upper bounds on the reconfiguration distance

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 $S$ in the Euclidean plane and two triangulations $T_1$ and $T_2$ of $S$, it is an APX-hard problem to minimize the number of edge flips to transform $T_1$ to $T_2$.Comment: A previous version only showed NP-completeness of the corresponding decision problem. The current version is the one of the accepted manuscrip

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 $A$ and $B$, this paper focuses on geodesic properties of generalized associahedra. We prove that the graph diameter of the $n$-dimensional associahedron of type $D$ is precisely $2n-2$ for all $n$ greater than $1$. Furthermore, we show that all type $BCD$ 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 $A$. 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