On the Combinatorial Complexity of Approximating Polytopes

Approximating convex bodies succinctly by convex polytopes is a fundamental problem in discrete geometry. A convex body $K$ of diameter $\mathrm{diam}(K)$ is given in Euclidean $d$-dimensional space, where $d$ is a constant. Given an error parameter $\varepsilon > 0$, the objective is to determine a polytope of minimum combinatorial complexity whose Hausdorff distance from $K$ is at most $\varepsilon \cdot \mathrm{diam}(K)$. By combinatorial complexity we mean the total number of faces of all dimensions of the polytope. A well-known result by Dudley implies that $O(1/\varepsilon^{(d-1)/2})$ facets suffice, and a dual result by Bronshteyn and Ivanov similarly bounds the number of vertices, but neither result bounds the total combinatorial complexity. We show that there exists an approximating polytope whose total combinatorial complexity is $\tilde{O}(1/\varepsilon^{(d-1)/2})$, where $\tilde{O}$ conceals a polylogarithmic factor in $1/\varepsilon$. This is a significant improvement upon the best known bound, which is roughly $O(1/\varepsilon^{d-2})$. Our result is based on a novel combination of both old and new ideas. First, we employ Macbeath regions, a classical structure from the theory of convexity. The construction of our approximating polytope employs a new stratified placement of these regions. Second, in order to analyze the combinatorial complexity of the approximating polytope, we present a tight analysis of a width-based variant of B\'{a}r\'{a}ny and Larman's economical cap covering. Finally, we use a deterministic adaptation of the witness-collector technique (developed recently by Devillers et al.) in the context of our stratified construction.Comment: In Proceedings of the 32nd International Symposium Computational Geometry (SoCG 2016) and accepted to SoCG 2016 special issue of Discrete and Computational Geometr

Shadoks Approach to Convex Covering

We describe the heuristics used by the Shadoks team in the CG:SHOP 2023 Challenge. The Challenge consists of 206 instances, each being a polygon with holes. The goal is to cover each instance polygon with a small number of convex polygons. Our general strategy is the following. We find a big collection of large (often maximal) convex polygons inside the instance polygon and then solve several set cover problems to find a small subset of the collection that covers the whole polygon.Comment: SoCG CG:SHOP 2023 Challeng

The Cost of Perfection for Matchings in Graphs

Perfect matchings and maximum weight matchings are two fundamental combinatorial structures. We consider the ratio between the maximum weight of a perfect matching and the maximum weight of a general matching. Motivated by the computer graphics application in triangle meshes, where we seek to convert a triangulation into a quadrangulation by merging pairs of adjacent triangles, we focus mainly on bridgeless cubic graphs. First, we characterize graphs that attain the extreme ratios. Second, we present a lower bound for all bridgeless cubic graphs. Third, we present upper bounds for subclasses of bridgeless cubic graphs, most of which are shown to be tight. Additionally, we present tight bounds for the class of regular bipartite graphs

Efficient Algorithms for Battleship

We consider an algorithmic problem inspired by the Battleship game. In the variant of the problem that we investigate, there is a unique ship of shape $S \subset Z^2$ which has been translated in the lattice $Z^2$. We assume that a player has already hit the ship with a first shot and the goal is to sink the ship using as few shots as possible, that is, by minimizing the number of missed shots. While the player knows the shape $S$, which position of $S$ has been hit is not known. Given a shape $S$ of $n$ lattice points, the minimum number of misses that can be achieved in the worst case by any algorithm is called the Battleship complexity of the shape $S$ and denoted $c(S)$. We prove three bounds on $c(S)$, each considering a different class of shapes. First, we have $c(S) \leq n-1$ for arbitrary shapes and the bound is tight for parallelogram-free shapes. Second, we provide an algorithm that shows that $c(S) = O(\log n)$ if $S$ is an HV-convex polyomino. Third, we provide an algorithm that shows that $c(S) = O(\log \log n)$ if $S$ is a digital convex set. This last result is obtained through a novel discrete version of the Blaschke-Lebesgue inequality relating the area and the width of any convex body.Comment: Conference version at 10th International Conference on Fun with Algorithms (FUN 2020

10 passos essenciais para a inseminação artificial em caprinos e ovinos.

bitstream/item/47281/1/FD-Inseminacao-artificial.pdf1. reimpr

First records of 13 echinoderm species on the southwest coast of Portugal

En el ámbito de un análisis de caracterización del área marina adyacente al Parque Natural do Sudoeste Alentejano y Costa Vicentina (PNSACV) se ha estudiado material biológico diverso proveniente de campañas del programa SEPLAT (SEdimentos de la PLATaforma), del que es responsable el Instituto Hidrográfico de la Marina Portuguesa. Las muestras se obtuvieron en el área de plataforma continental adyacente al PNSACV, entre Sines y el cabo de San Vicente, a profundidades comprendidas entre los 15 y los 387 metros. Los equinodermos constituyeron, en esas campañas, un grupo bien representativo de la fauna marina presente en esta área de la plataforma continental. Se recogieron 60 especies de equinodermos distribuidos en las clases siguientes: 2 Crinoidea, 2 Asteroidea, 32 Ophiuroidea, 12 Echinoidea y 12 Holothurioidea. Trece de las 60 especies tienen aquí su primera cita para la costa portuguesa.As part of a characterisation study of the marine area contiguous to the Sudoeste Alentejano e Costa Vicentina Natural Park (PNSACV), biological material proceeding from cruises of the SEPLAT (PLATform SEdiments) programme (Portuguese Navy's Hydrographic Institute) has been studied. The sampling took place on the continental shelf contiguous to the PNSACV, between Sines and Cape St Vicente, at water depths ranging from 15-387 m. During these cruises, equinoderms were found to be a well-represented faunal group. Sixty echinoderm species were identified, distributed as follows, by classes: 2 crinoids, 2 asteroids, 32 ophiuroids, 12 equinoids and 12 holothuroids. Thirteen of the 60 species are first records on the Portuguese coast.Instituto Español de Oceanografí