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

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

Simusoccer App: business plan

National regulations introduced in Portugal in 2015 impacted the online gambling market (betting real money), closing sports betting websites and, consequently blocking players from online betting. The research aims to investigate the potential of the launch of a mobile app (SimuSoccer) fully dedicated to recreational gambling (not betting real money) on football results, not violating 2015’s law. The methodology adopted qualitative and quantitative measures, through structured questionnaires, based on 151 respondents. The research explores if there is a market of consumers driven solely by the pleasure of playing in a fan-loyalty relation with player’s favorite leagues and clubs, instead of betting real money. The key conclusions suggest a window of opportunity to launch SimuSoccer as a viable risk-free game app - following the freemium business model - while taking advantage of users’ [apparent] preference for interface’s intuitiveness, football exclusivity, and fanloyalty- gaming approach

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

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

On the ratio between maximum weight perfect matchings and maximum weight matchings in grids

Given a graph G that admits a perfect matching, we investigate the parameter η(G) (originally motivated by computer graphics applications) which is defined as follows. Among all nonnegative edge weight assignments, η(G) is the minimum ratio between (i) the maximum weight of a perfect matching and (ii) the maximum weight of a general matching. In this paper, we determine the exact value of η for all rectangular grids, all bipartite cylindrical grids, and all bipartite toroidal grids. We introduce several new techniques to this endeavor

Short Flip Sequences to Untangle Segments in the Plane

A (multi)set of segments in the plane may form a TSP tour, a matching, a tree, or any multigraph. If two segments cross, then we can reduce the total length with the following flip operation. We remove a pair of crossing segments, and insert a pair of non-crossing segments, while keeping the same vertex degrees. The goal of this paper is to devise efficient strategies to flip the segments in order to obtain crossing-free segments after a small number of flips. Linear and near-linear bounds on the number of flips were only known for segments with endpoints in convex position. We generalize these results, proving linear and near-linear bounds for cases with endpoints that are not in convex position. Our results are proved in a general setting that applies to multiple problems, using multigraphs and the distinction between removal and insertion choices when performing a flip.Comment: 19 pages, 10 figure

Identidade e autodeterminação informacional no novo Regulamento Geral de Proteção de Dados: a inevitável privatização dos deveres estaduais de proteção

The problematic issue of personal data protection appears to be an unavoidable reality of today’s risk society, in which “technological time” is not compatible with “legal time”, imposing on the administrative function the observance of a set of rules and regulation duties regarding the collection, storage, processing and use of personal data that proves to be fundamental to the effective protection of the subjective legal positions of individuals. Throughout this article, it will be sought to analyze the autonomous fundamental nature of the right to data protection in the context of national and comparative law, in order to identify its own dogmatic shape. It will be equally important to critically appraise the solutions that are adopted by the European legislator in the current revision of the EU regulatory framework, which has a decisive role in shaping the national legal system on data protection.A problemática da proteção de dados pessoais afigura-se como uma realidade incontornável da sociedade de risco hodierna, na qual o «tempo tecnológico» não se compadece com o «tempo jurídico», impondo- se à função administrativa a observância de um conjunto de regras e deveres de regulação jurídica das atividades de recolha, armazenamento, tratamento e utilização de dados pessoais que se revelam fundamentais para a tutela efetiva das posições jurídicas subjetivas dos particulares. Ao longo do presente artigo, procurar-se-á analisar da jusfundamentalidade autónoma do direito à proteção de dados no contexto do direito nacional e do direito comparado, visando delimitar o seu recorte dogmático próprio. Caberá, de igual modo, versar, de forma crítica, sobre as soluções adotadas pelo legislador europeu, na atual revisão do quadro normativo da UE, que assumem um papel determinante na conformação do quadro da legislação nacional sobre proteção de dados

A África do Sul e o Sistema-Mundo: da guerra dos Bôeres à globalização

A África do Sul é provavelmente o país mais representativo da lógica de funcionamento global do sistema-mundo capitalista. Aí se encontram alguns dos mais elevados índices de desigualdade social e de criminalidade violenta, bem como, em coexistência precária, alguns dos maiores índices de desenvolvimento tecnológico assim como de exclusão social, de que é exemplo uma das mais graves incidências de SIDA, com cerca de 20% da população total infectada com o vírus HIV. No mesmo contexto, ali coexistem parcelas nacionais do “centro”, da “semiperiferia”, da “periferia” e da “arena externa”. O autor parte da premissa do carácter paradigmático, ou exemplar, da evolução histórica da África do Sul (em particular ao longo do século XX) como ilustração representativa da evolução do sistema capitalista, na sua necessária vertente expansionista, de âmbito planetário e a partir das suas origens europeias. Este livro procura, numa perspectiva pouco ortodoxa, explicar o trajecto histórico da comunidade nacional Africânder e o modo como os seus dirigentes mais cosmopolitas, aproveitando uma determinada “janela de oportunidade” proporcionada pela evolução da conjuntura política internacional, prepararam a transição para a democracia não-racial e a possibilidade da criação de uma nação “arco-íris”, na feliz expressão do Arcebispo Desmond Tutu