A triangulation of a planar point set S is a maximal plane straight-line graph with vertex set S. In the minimum-weight triangulation (MWT) problem, we are looking for a triangulation of a given point set that minimizes the sum of the edge lengths. We prove that the decision version of this problem is NP-hard. We use a reduction from PLANAR-1-IN-3-SAT. The correct working of the gadgets is established with computer assistance, using dynamic programming on polygonal faces, as well as the beta-skeleton heuristic to certify that certain edges belong to the minimum-weight triangulation.
Two vertex-labelled polygons are \emph{compatible} if they have the same clockwise cyclic ordering of vertices. The definition extends to polygonal regions (polygons with holes) and to triangulations---for every face, the clockwise cyclic order of vertices on the boundary must be the same. It is known that every pair of compatible $n$-vertex polygonal regions can be extended to compatible triangulations by adding $O(n^2)$ Steiner points. Furthermore, $Ω(n^2)$ Steiner points are sometimes necessary, even for a pair of polygons. Compatible triangulations provide piecewise linear homeomorphisms a…
Given a set of $n$ points on a plane, in the Minimum Weight Triangulation problem, we wish to find a triangulation that minimizes the sum of Euclidean length of its edges. This incredibly challenging problem has been studied for more than four decades and has been only recently shown to be NP-Hard. In this paper we present a novel polynomial-time algorithm that computes a $14$-approximation of the minimum weight triangulation -- a constant that is significantly smaller than what has been previously known. In our algorithm, we use grids to partition the edges into levels where shorter edges app…
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.
Minimum-weight triangulation (MWT) is NP-hard. It has a polynomial-time constant-factor approximation algorithm, and a variety of effective polynomial- time heuristics that, for many instances, can find the exact MWT. Linear programs (LPs) for MWT are well-studied, but previously no connection was known between any LP and any approximation algorithm or heuristic for MWT. Here we show the first such connections: for an LP formulation due to Dantzig et al. (1985): (i) the integrality gap is bounded by a constant; (ii) given any instance, if the aforementioned heuristics find the MWT, then so doe…
We apply state-of-the-art computational geometry methods to the problem of reconstructing a time-varying sea surface from tide gauge records. Our work builds on a recent article by Nitzke et al.~(Computers \& Geosciences, 157:104920, 2021) who have suggested to learn a triangulation $D$ of a given set of tide gauge stations. The objective is to minimize the misfit of the piecewise linear surface induced by $D$ to a reference surface that has been acquired with satellite altimetry. The authors restricted their search to k-order Delaunay ($k$-OD) triangulations and used an integer linear program…
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. To gain more insight into flips of triangulations and to characterize "where edges go" when flipping from one triangulation to another, flips in edge-labeled triangulations have lately attracted considerable interest. In a recent breakthrough, Lubiw, Masárová, and Wagner (in Proc. 33rd Symp. of Computational Geometry, 2017) prove the so-called "Orbit Conjecture" for edge-labeled triangulations and ask for the complexity of the flip distance…
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 $T_2$.
We study the problem of finding a triangulation T of a planar point set S such as to minimize the expected distance between two points x and y chosen uniformly at random from S. By distance we mean the length of the shortest path between x and y along edges of T. The length of a path is the sum of the weights of its edges. Edge weights are assumed to be given as part of the problem for every pair of distinct points (x,y) in S^2. 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 th…
We show that Delaunay triangulations and compressed quadtrees are equivalent structures. More precisely, we give two algorithms: the first computes a compressed quadtree for a planar point set, given the Delaunay triangulation; the second finds the Delaunay triangulation, given a compressed quadtree. Both algorithms run in deterministic linear time on a pointer machine. Our work builds on and extends previous results by Krznaric and Levcopolous and Buchin and Mulzer. Our main tool for the second algorithm is the well-separated pair decomposition(WSPD), a structure that has been used previously…
We consider practical methods for the problem of finding a minimum-weight triangulation (MWT) of a planar point set, a classic problem of computational geometry with many applications. While Mulzer and Rote proved in 2006 that computing an MWT is NP-hard, Beirouti and Snoeyink showed in 1998 that computing provably optimal solutions for MWT instances of up to 80,000 uniformly distributed points is possible, making use of clever heuristics that are based on geometric insights. We show that these techniques can be refined and extended to instances of much bigger size and different type, based on…
University of Notre Dame（圣母大学）；Utrecht University（乌得勒支大学）；TU Eindhoven（埃因霍温理工大学）；KTH Royal Institute of Technology（皇家理工学院）；Vrije Universiteit Amsterdam（阿姆斯特丹自由大学）
Multiple-view triangulation by $\ell_{\infty}$ minimisation has become established in computer vision. State-of-the-art $\ell_{\infty}$ triangulation algorithms exploit the quasiconvexity of the cost function to derive iterative update rules that deliver the global minimum. Such algorithms, however, can be computationally costly for large problem instances that contain many image measurements, e.g., from web-based photo sharing sites or long-term video recordings. In this paper, we prove that $\ell_{\infty}$ triangulation admits a coreset approximation scheme, which seeks small representative …
We compute the number of triangulations of a convex $k$-gon each of whose sides is subdivided by $r-1$ points. We find explicit formulas and generating functions, and we determine the asymptotic behaviour of these numbers as $k$ and/or $r$ tend to infinity. We connect these results with the question of finding the planar set of points in general position that has the minimum possible number of triangulations - a well-known open problem from computational geometry.
Let $\Poly$ be a simple polygon with $n$ vertices. The \emph{dual graph} $\triang^*$ of a triangulation~$\triang$ of~$\Poly$ is the graph whose vertices correspond to the bounded faces of $\triang$ and whose edges connect those faces of~$\triang$ that share an edge. We consider triangulations of~$\Poly$ that minimize or maximize the diameter of their dual graph. We show that both triangulations can be constructed in $O(n^3\log n)$ time using dynamic programming. If $\Poly$ is convex, we show that any minimizing triangulation has dual diameter exactly $2\cdot\lceil\log_2(n/3)\rceil$ or $2\cdot\…
The flip graph for a set $P$ of points in the plane has a vertex for every triangulation of $P$, and an edge when two triangulations differ by one flip that replaces one triangulation edge by another. The flip graph is known to have some connectivity properties: (1) the flip graph is connected; (2) connectivity still holds when restricted to triangulations containing some constrained edges between the points; (3) for $P$ in general position of size $n$, the flip graph is $\lceil \frac{n}{2} -2 \rceil$-connected, a recent result of Wagner and Welzl (SODA 2020). We introduce the study of connect…
We describe a randomized algorithm that, given a set $P$ of points in the plane, computes the best location to insert a new point $p$, such that the Delaunay triangulation of $P\cup\{p\}$ has the largest possible minimum angle. The expected running time of our algorithm is at most cubic, improving the roughly quartic time of the best previously known algorithm. It slows down to slightly super-cubic if we also specify a set of non-crossing segments with endpoints in $P$ and insist that the triangulation respect these segments, i.e., is the constrained Delaunay triangulation of the points and se…
The (axis-parallel) stabbing number of a given set of line segments is the maximum number of segments that can be intersected by any one (axis-parallel) line. This paper deals with finding perfect matchings, spanning trees, or triangulations of minimum stabbing number for a given set of points. The complexity of these problems has been a long-standing open question; in fact, it is one of the original 30 outstanding open problems in computational geometry on the list by Demaine, Mitchell, and O'Rourke. The answer we provide is negative for a number of minimum stabbing problems by showing them N…
Carleton University（卡尔顿大学）；Université Libre de Bruxelles（布鲁塞尔自由大学）；Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）
We study affine invariant 2D triangulation methods. That is, methods that produce the same triangulation for a point set $S$ for any (unknown) affine transformation of $S$. Our work is based on a method by Nielson [A characterization of an affine invariant triangulation. Geom. Mod, 191-210. Springer, 1993] that uses the inverse of the covariance matrix of $S$ to define an affine invariant norm, denoted $A_{S}$, and an affine invariant triangulation, denoted ${DT}_{A_{S}}[S]$. We revisit the $A_{S}$-norm from a geometric perspective, and show that ${DT}_{A_{S}}[S]$ can be seen as a standard Del…
Online routing in a planar embedded graph is central to a number of fields and has been studied extensively in the literature. For most planar graphs no $O(1)$-competitive online routing algorithm exists. A notable exception is the Delaunay triangulation for which Bose and Morin [Online routing in triangulations. SIAM Journal on Computing, 33(4):937-951, 2004] showed that there exists an online routing algorithm that is $O(1)$-competitive. However, a Delaunay triangulation can have $Ω(n)$ vertex degree and a total weight that is a linear factor greater than the weight of a minimum spanning tre…
Given a finite point set P in general position in the plane, a full triangulation is a maximal straight-line embedded plane graph on P. A partial triangulation is a full triangulation of some subset P' of P containing all extreme points in P. A bistellar flip on a partial triangulation flips an edge (an edge flip), removes a non-extreme point of degree 3, or adds a point in P \ P' as vertex of degree 3. The bistellar flip graph has all partial triangulations as vertices, and a pair of partial triangulations is adjacent if they can be obtained from one another by a bistellar flip. The edge flip…
Let $P$ be a collection of $n$ points moving along pseudo-algebraic trajectories in the plane. One of the hardest open problems in combinatorial and computational geometry is to obtain a nearly quadratic upper bound, or at least a subcubic bound, on the maximum number of discrete changes that the Delaunay triangulation $\DT(P)$ of $P$ experiences during the motion of the points of $P$. In this paper we obtain an upper bound of $O(n^{2+\eps})$, for any $\eps>0$, under the assumptions that (i) any four points can be co-circular at most twice, and (ii) either no triple of points can be collinear …
In the Minimum Installation Path problem, we are given a graph $G$ with edge weights $w(.)$ and two vertices $s,t$ of $G$. We want to assign a non-negative power $p(v)$ to each vertex $v$ of $G$ so that the edges $uv$ such that $p(u)+p(v)$ is at least $w(uv)$ contain some $s$-$t$-path, and minimize the sum of assigned powers. In the Minimum Barrier Shrinkage problem, we are given a family of disks in the plane and two points $x$ and $y$ lying outside the disks. The task is to shrink the disks, each one possibly by a different amount, so that we can draw an $x$-$y$ curve that is disjoint from t…
Let $P$ be a set of $n$ points in the plane. A crossing-free structure on $P$ is a plane graph with vertex set $P$. Examples of crossing-free structures include triangulations of $P$, spanning cycles of $P$, also known as polygonalizations of $P$, among others. In this paper we develop a general technique for computing the number of crossing-free structures of an input set $P$. We apply the technique to obtain algorithms for computing the number of triangulations, matchings, and spanning cycles of $P$. The running time of our algorithms is upper bounded by $n^{O(k)}$, where $k$ is the number o…
A data structure is presented for point location in connected planar subdivisions when the distribution of queries is known in advance. The data structure has an expected query time that is within a constant factor of optimal. More specifically, an algorithm is presented that preprocesses a connected planar subdivision G of size n and a query distribution D to produce a point location data structure for G. The expected number of point-line comparisons performed by this data structure, when the queries are distributed according to D, is H + O(H^{2/3}+1) where H=H(G,D) is a lower bound on the ex…
Given a plane forest $F = (V, E)$ of $|V| = n$ points, we find the minimum set $S \subseteq E$ of edges such that the edge-constrained minimum spanning tree over the set $V$ of vertices and the set $S$ of constraints contains $F$. We present an $O(n \log n )$-time algorithm that solves this problem. We generalize this to other proximity graphs in the constraint setting, such as the relative neighbourhood graph, Gabriel graph, $β$-skeleton and Delaunay triangulation. We present an algorithm that identifies the minimum set $S\subseteq E$ of edges of a given plane graph $I=(V,E)$ such that $I \su…
We use here the results on the influence graph by Boissonnat et al. to adapt them for particular cases where additional information is available. In some cases, it is possible to improve the expected randomized complexity of algorithms from O(n log n) to O(n log star n). This technique applies in the following applications: triangulation of a simple polygon, skeleton of a simple polygon, Delaunay triangulation of points knowing the EMST (euclidean minimum spanning tree).
Randomized HyperSteiner: A Stochastic Delaunay Triangulation Heuristic for the Hyperbolic Steiner Minimal Tree
School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology（KTH皇家理工学院电气工程与计算机科学学院）；Amsterdam Machine Learning Lab (AMLab), University of Amsterdam（阿姆斯特丹大学阿姆斯特丹机器学习实验室）；School of Engineering Sciences, KTH Royal Institute of Technology（KTH皇家理工学院工程科学学院）；Leiden Institute of Advanced Computer Science, Universiteit Leiden（莱顿大学莱顿高级计算机科学研究所）
Placing a minimum number of guards on a given watchman route in a polygonal domain is called the {\em minimum vision points problem}. We prove that finding the minimum number of vision points on a shortest watchman route in a simple polygon is APX-Hard. We then extend the proof to the class of rectilinear polygons having at most three dent orientations.
We consider the Minimum Convex Partition problem: Given a set P of n points in the plane, draw a plane graph G on P, with positive minimum degree, such that G partitions the convex hull of P into a minimum number of convex faces. We show that Minimum Convex Partition is NP-hard, and we give several approximation algorithms, from an O(log OPT)-approximation running in O(n^8)-time, where OPT denotes the minimum number of convex faces needed, to an O(sqrt(n) log n)-approximation algorithm running in O(n^2)-time. We say that a point set is k-directed if the (straight) lines containing at least thr…
The minimum feature size of a crossing-free straight line drawing is the minimum distance between a vertex and a non-incident edge. This quantity measures the resolution needed to display a figure or the tool size needed to mill the figure. The spread is the ratio of the diameter to the minimum feature size. While many algorithms (particularly in meshing) depend on the spread of the input, none explicitly consider finding a mesh whose spread is similar to the input. When a polygon is partitioned into smaller regions, such as triangles or quadrangles, the degradation is the ratio of original to…
In a geometric network G = (S, E), the graph distance between two vertices u, v in S is the length of the shortest path in G connecting u to v. The dilation of G is the maximum factor by which the graph distance of a pair of vertices differs from their Euclidean distance. We show that given a set S of n points with integer coordinates in the plane and a rational dilation delta > 1, it is NP-hard to determine whether a spanning tree of S with dilation at most delta exists.
Given a set P of points in the plane, an Euclidean t-spanner for P is a geometric graph that preserves the Euclidean distances between every pair of points in P up to a constant factor t. The weight of a geometric graph refers to the total length of its edges. In this paper we show that the problem of deciding whether there exists an Euclidean t-spanner, for a given set of points in the plane, of weight at most w is NP-hard for every real constant t > 1, both whether planarity of the t-spanner is required or not.
A greedily routable region (GRR) is a closed subset of $\mathbb R^2$, in which each destination point can be reached from each starting point by choosing the direction with maximum reduction of the distance to the destination in each point of the path. Recently, Tan and Kermarrec proposed a geographic routing protocol for dense wireless sensor networks based on decomposing the network area into a small number of interior-disjoint GRRs. They showed that minimum decomposition is NP-hard for polygons with holes. We consider minimum GRR decomposition for plane straight-line drawings of graphs. Her…
Turaev Viro invariants are amongst the most powerful tools to distinguish 3-manifolds: They are implemented in mathematical software, and allow practical computations. The invariants can be computed purely combinatorially by enumerating colourings on the edges of a triangulation T. These edge colourings can be interpreted as embeddings of surfaces in T. We give a characterisation of how these embedded surfaces intersect with the tetrahedra of T. This is done by characterising isotopy classes of simple closed loops in the 3-punctured disk. As a direct result we obtain a new system of coordinate…
Institute for Advanced Studies in Basic Sciences (IASBS)（基础科学高级研究院）
In the minimum constraint removal ($MCR$), there is no feasible path to move from the starting point towards the goal and, the minimum constraints should be removed in order to find a collision-free path. It has been proved that $MCR$ problem is $NP-hard$ when constraints have arbitrary shapes or even they are in shape of convex polygons. However, it has a simple linear solution when constraints are lines and the problem is open for other cases yet. In this paper, using a reduction from Subset Sum problem, in three steps, we show that the problem is NP-hard for both weighted and unweighted lin…
Let $P$ be an orthogonal polygon. Consider a sliding camera that travels back and forth along an orthogonal line segment $s\in P$ as its \emph{trajectory}. The camera can see a point $p\in P$ if there exists a point $q\in s$ such that $pq$ is a line segment normal to $s$ that is completely inside $P$. In the \emph{minimum-cardinality sliding cameras problem}, the objective is to find a set $S$ of sliding cameras of minimum cardinality to guard $P$ (i.e., every point in $P$ can be seen by some sliding camera) while in the \emph{minimum-length sliding cameras problem} the goal is to find such a …
We consider the Minimum Coverage Kernel problem: given a set $B$ of $d$-dimensional boxes, find a subset of $B$ of minimum size covering the same region as $B$. This problem is $\mathsf{NP}$-hard, but as for many $\mathsf{NP}$-hard problems on graphs, the problem becomes solvable in polynomial time under restrictions on the graph induced by $B$. We consider various classes of graphs, show that Minimum Coverage Kernel remains $\mathsf{NP}$-hard even for severely restricted instances, and provide two polynomial time approximation algorithms for this problem.
We study two optimization problems on simplicial complexes with homology over $\mathbb{Z}_2$, the minimum bounded chain problem: given a $d$-dimensional complex $\mathcal{K}$ embedded in $\mathbb{R}^{d+1}$ and a null-homologous $(d-1)$-cycle $C$ in $\mathcal{K}$, find the minimum $d$-chain with boundary $C$, and the minimum homologous chain problem: given a $(d+1)$-manifold $\mathcal{M}$ and a $d$-chain $D$ in $\mathcal{M}$, find the minimum $d$-chain homologous to $D$. We show strong hardness results for both problems even for small values of $d$; $d = 2$ for the former problem, and $d=1$ for…
