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

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

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

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Perspectivas de estudos ecologicos sobre o Pantanal.

Este trabalho originou-se das discussoes realizadas no "Curso de Ecologia de Areas Alagaveis", ministrado pelo Dr. Wolfgang J. Junk, do Max-Planck Institute fur limnologie - Alemanha e pela Dra Carolina Joana da Silva do Departamento de Biologia da Universidade Federal de Mato Grosso (Cuiaba), em agosto de 1991, no Centro de Pesquisa Agropecuaria do Pantanal - EMBRAPA/CPAP. Sua leitura leva ao entendimento de que, aprofundando-se os conhecimentos e reunindo-se as informacoes, e possivel formar uma base tecnico-cientifica solida para que acoes de manejo possam ser encontradas na regiao dentro dos principios do desenvolvimento sustentavel. Reeditado de forma ampliada quatro anos depois da realizacao daquele evento, este documento foi elaborado em duas partes interdependentes. Na primeira parte sao apresentadas informacoes sobre a regiao - algumas bastantes recentes, com enfase no Pantanal, como forma de atualizar e subsidiar recomendacoes a pesquisa ecologica, que constituem a segunda parte deste trabalho. Algumas dessas questoes ja fazem parte dos objetivos das pesquisas que vem sendo conduzidas pelas diferentes instituicoes que atuam na regiao. Outras necessitam ser incluidas ou implementadas. A organizacao desse curso visou inicialmente promover a troca e a atualizacao de conhecimentos teoricos e metodologicos em Limnologia para profissionais atuantes na regiao da Bacia Hidrografica do Alto Paraguai, em particular na planicie pantaneira/chaquenha, pertencente ao Brasil, Bolivia e Paraguai.bitstream/item/37750/1/DOC18.pd