Approximations on minimum weight pseudo-triangulation problem using ant colony optimization

In this work, we consider the Minimum Weight Pseudo-Triangulation (MWPT) problem of a given set of n points in the plane. Globally optimal pseudo-triangulations with respect to the weight, as optimization criteria, are difficult to be found by deterministic methods, since no polynomial algorithm is known. We show how the Ant Colony Optimization (ACO) metaheuristic can be used to find high quality pseudo-triangulations of minimum weight. We present the experimental and statistical study based on our own set of instances since no reference to benchmarks for these problems were found in the literature. Throughout the experimental evaluation, we appraise the ACO metaheuristic performance for MWPT problem

Approximations on minimum weight pseudo-triangulations using ant colony optimization metaheuristic

Globally optimal pseudo-triangulations are di cult to be found by deterministic methods as, for most type of criteria, no polynomial algorithm is known. In this work, we consider the Minimum Weight Pseudo-Triangulation (MWPT) problem of a given set of n points in the plane. This paper shows how the Ant Colony Optimization (ACO) metaheuristic can be used to nd optimal pseudo-triangulations of minimum weight. For the experimental study presented here we have created a set of instances for MWPT since no reference to benchmarks for these problems were found in the literature. We assess through the experimental evaluation the applicability of the ACO metaheuristic for MWPT.Presentado en el X Workshop Agentes y Sistemas InteligentesRed de Universidades con Carreras en Informática (RedUNCI

Approximations on minimum weight pseudo-triangulations using ant colony optimization metaheuristic

Globally optimal pseudo-triangulations are di cult to be found by deterministic methods as, for most type of criteria, no polynomial algorithm is known. In this work, we consider the Minimum Weight Pseudo-Triangulation (MWPT) problem of a given set of n points in the plane. This paper shows how the Ant Colony Optimization (ACO) metaheuristic can be used to nd optimal pseudo-triangulations of minimum weight. For the experimental study presented here we have created a set of instances for MWPT since no reference to benchmarks for these problems were found in the literature. We assess through the experimental evaluation the applicability of the ACO metaheuristic for MWPT.Presentado en el X Workshop Agentes y Sistemas InteligentesRed de Universidades con Carreras en Informática (RedUNCI

Metaheuristic approaches for MWT and MWPT problems

It is known that the Minimum Weight Triangulation problem is NP-hard. Also the complexity of Minimum Weight Pseudo-Triangulation problem is unknown, suspecting that it is also a NP-hard problem. Therefore we focused on the development of approximate algorithms to find high quality triangulations and pseudo-triangulations of minimum weight. In this work we propose the use of two metaheuristics to solve these problems: Ant Colony Optimization (ACO) and Simulated Annealing (SA). For the experimental study we have created a set of instances for MWT and MWPT problems since no reference to benchmarks for these problems were found in the literature. Through the experimental evaluation, we assess the applicability of the ACO and SA metaheuristics for MWT and MWPT problems. These results are compared with those obtained from the application of deterministic algorithms for the same problems (Delaunay Triangulation for MWT and a Greedy algorithm respectively for MWT and MWPT).Presentado en el XII Workshop Agentes y Sistemas Inteligentes (WASI)Red de Universidades con Carreras en Informática (RedUNCI

Approximations on minimum weight pseudo-triangulations using ant colony optimization metaheuristic

Globally optimal pseudo-triangulations are di cult to be found by deterministic methods as, for most type of criteria, no polynomial algorithm is known. In this work, we consider the Minimum Weight Pseudo-Triangulation (MWPT) problem of a given set of n points in the plane. This paper shows how the Ant Colony Optimization (ACO) metaheuristic can be used to nd optimal pseudo-triangulations of minimum weight. For the experimental study presented here we have created a set of instances for MWPT since no reference to benchmarks for these problems were found in the literature. We assess through the experimental evaluation the applicability of the ACO metaheuristic for MWPT.Presentado en el X Workshop Agentes y Sistemas InteligentesRed de Universidades con Carreras en Informática (RedUNCI

Maximizing Maximal Angles for Plane Straight-Line Graphs

Let $G=(S, E)$ be a plane straight-line graph on a finite point set $S\subset\R^2$ in general position. The incident angles of a vertex $p \in S$ of $G$ are the angles between any two edges of $G$ that appear consecutively in the circular order of the edges incident to $p$. A plane straight-line graph is called $\phi$-open if each vertex has an incident angle of size at least $\phi$. In this paper we study the following type of question: What is the maximum angle $\phi$ such that for any finite set $S\subset\R^2$ of points in general position we can find a graph from a certain class of graphs on $S$ that is $\phi$-open? In particular, we consider the classes of triangulations, spanning trees, and paths on $S$ and give tight bounds in most cases.Comment: 15 pages, 14 figures. Apart of minor corrections, some proofs that were omitted in the previous version are now include

Geodesic-Preserving Polygon Simplification

Polygons are a paramount data structure in computational geometry. While the complexity of many algorithms on simple polygons or polygons with holes depends on the size of the input polygon, the intrinsic complexity of the problems these algorithms solve is often related to the reflex vertices of the polygon. In this paper, we give an easy-to-describe linear-time method to replace an input polygon $\mathcal{P}$ by a polygon $\mathcal{P}'$ such that (1) $\mathcal{P}'$ contains $\mathcal{P}$, (2) $\mathcal{P}'$ has its reflex vertices at the same positions as $\mathcal{P}$, and (3) the number of vertices of $\mathcal{P}'$ is linear in the number of reflex vertices. Since the solutions of numerous problems on polygons (including shortest paths, geodesic hulls, separating point sets, and Voronoi diagrams) are equivalent for both $\mathcal{P}$ and $\mathcal{P}'$, our algorithm can be used as a preprocessing step for several algorithms and makes their running time dependent on the number of reflex vertices rather than on the size of $\mathcal{P}$

Triangulating the Real Projective Plane

We consider the problem of computing a triangulation of the real projective plane P2, given a finite point set S={p1, p2,..., pn} as input. We prove that a triangulation of P2 always exists if at least six points in S are in general position, i.e., no three of them are collinear. We also design an algorithm for triangulating P2 if this necessary condition holds. As far as we know, this is the first computational result on the real projective plane