Formulations and algorithms for the maximum area poligonization problem

Orientador: Fábio Luiz UsbertiDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: O problema do caixeiro viajante euclidiano (Traveling Salesman Problem - TSP) do ponto de vista geométrico tem por objetivo encontrar um polígono simples sobre um dado conjunto de vértices cujo perímetro é mínimo. É possível derivar o problema de modo que o objetivo seja encontrar um polígono simples cuja área interna seja máxima: tal problema é conhecido como Poligonização de Área Máxima (Maximum Area Polygonization - MAXAP). O MAXAP é um problema de otimização combinatória NP-difícil com aplicações práticas em segmentos como reconhecimento de padrões, reconstrução de imagens, clusterização e robótica. Este trabalho propõe novas metodologias de solução e formulações matemáticas para o MAXAP, visando a implementação de algoritmos para metodologias exata, aproximada e heurística, bem como um estudo computacional para avaliar o desempenho das metodologias para o conjunto de instâncias desenvolvido. São propostos neste trabalho dois modelos matemáticos de programação linear inteira, duas heurísticas construtivas, uma metaheurística GRASP e uma matheuristic aplicada sobre um dos modelos matemáticos. Experimentos computacionais foram executados para comparar as metodologias propostas entre si e com um algoritmo 1/2-aproximado da literatura. Análises comparativas de desempenho foram realizadas sobre os resultados obtidos mostrando que o GRASP obteve o melhor desempenho no critério de qualidade das soluções. As heurísticas construtivas propostas por sua vez obtiveram um desempenho superior sobre o algoritmo aproximado. Finalmente, os modelos matemáticos propostos mostram a dificuldade de resolver de maneira exata o MAXAP, encontrando soluções ótimas em uma hora somente para as instâncias de 10 pontos, em um conjunto de instâncias de até 100 pontosAbstract: The Traveling Salesman Problem (TSP) from a geometric point of view aims to find a simple polygon with minimum perimeter. It is possible to derive the problem so that the objective is to find a simple polygon whose enclosed area is maximum, such problem is known as Maximum Area Polygonization (MAXAP). The MAXAP is an NP-hard combinatorial optimization problem with practical applications in segments such as pattern recognition, image reconstruction, clustering and robotics. This work proposes new solution methodologies and mathematical formulations for MAXAP, aiming the implementation of algorithms for exact, approximate and heuristic solutions, as well as a computational study to evaluate the performance of the methodologies for a benchmark set of instances. Two mathematical models based on integer linear programming are proposed. In addition, two constructive heuristics, a GRASP metaheuristic, and a matheuristic are proposed for the solution of larger instances. Computacional experiments were conducted to compare the proposed methodologies among themselves and a 1/2-approximation algorithm from literature. Comparative perfomance analysis were made on the results showing that the GRASP outperformed all other approaches with respect to solution quality. The constructive heuristics, on the other hand, outperformed the literature 1/2-approximation algorithm. Finally, the proposed mathematical models have shown the hardness of exact solution for the MAXAP, finding optimal solutions in one hour only for the 10-vertices instances in a set of instances with 10, 25, 50 and 100 verticesMestradoCiência da ComputaçãoMestre em Ciência da ComputaçãoCAPE

Outside-Obstacle Representations with All Vertices on the Outer Face

An obstacle representation of a graph $G$ consists of a set of polygonal obstacles and a drawing of $G$ as a visibility graph with respect to the obstacles: vertices are mapped to points and edges to straight-line segments such that each edge avoids all obstacles whereas each non-edge intersects at least one obstacle. Obstacle representations have been investigated quite intensely over the last few years. Here we focus on outside-obstacle representations (OORs) that use only one obstacle in the outer face of the drawing. It is known that every outerplanar graph admits such a representation [Alpert, Koch, Laison; DCG 2010]. We strengthen this result by showing that every (partial) 2-tree has an OOR. We also consider restricted versions of OORs where the vertices of the graph lie on a convex polygon or a regular polygon. We characterize when the complement of a tree and when a complete graph minus a simple cycle admits a convex OOR. We construct regular OORs for all (partial) outerpaths, cactus graphs, and grids.Comment: Appears in the Proceedings of the 30th International Symposium on Graph Drawing and Network Visualization (GD 2022

On indecomposable polyhedra and the number of interior Steiner points

The existence of 3d {\it indecomposable polyhedra}, that is, the interior of every such polyhedron cannot be decomposed into a set of tetrahedra whose vertices are all of the given polyhedron, is well-known. While the geometry and combinatorial structure of such polyhedra are much less studied. In this article, we first investigate the geometry of some well-known examples, the so-called {\it Sch\"on\-hardt polyhedron}~\cite{Schonhardt1928} and the Bagemihl's generalization of it~\cite{Bagemihl48-decomp-polyhedra}, which will be called {\it Bagemihl polyhedra}. We provide a construction of an interior point, so-called {\it Steiner point}, which can be used to tetrahedralize the Sch\"on\-hardt and the Bagemihl polyhedra. We then provide a construction of a larger class of three-dimensional indecomposable polyhedra which often appear in grid generation problems. We show that such polyhedra have the same combinatorial structure as the Sch\"onhardt and Bagemihl polyhedra, but they may need more than one interior Steiner point to be tetrahedralized. Given such a polyhedron with $n \ge 6$ vertices, we show that it can be tetrahedralized by adding at most $\left\lceil \frac{n - 5}{2}\right\rceil$ interior Steiner points. %, is sufficient to decompose it. We also show that this number is optimal in the worst case

Stringy Canonical Forms

Canonical forms of positive geometries play an important role in revealing hidden structures of scattering amplitudes, from amplituhedra to associahedra. In this paper, we introduce "stringy canonical forms", which provide a natural definition and extension of canonical forms for general polytopes, deformed by a parameter $\alpha'$. They are defined by real or complex integrals regulated with polynomials with exponents, and are meromorphic functions of the exponents, sharing various properties of string amplitudes. As $\alpha' \to 0$, they reduce to the usual canonical form of a polytope given by the Minkowski sum of the Newton polytopes of the regulating polynomials, or equivalently the volume of the dual of this polytope, naturally determined by tropical functions. At finite $\alpha'$, they have simple poles corresponding to the facets of the polytope, with the residue on the pole given by the stringy canonical form of the facet. There is the remarkable connection between the $\alpha' \to 0$ limit of tree-level string amplitudes, and scattering equations that appear when studying the $\alpha' \to \infty$ limit. We show that there is a simple conceptual understanding of this phenomenon for any stringy canonical form: the saddle-point equations provide a diffeomorphism from the integration domain to the interior of the polytope, and thus the canonical form can be obtained as a pushforward via summing over saddle points. When the stringy canonical form is applied to the ABHY associahedron in kinematic space, it produces the usual Koba-Nielsen string integral, giving a direct path from particle to string amplitudes without an a priori reference to the string worldsheet. We also discuss a number of other examples, including stringy canonical forms for finite-type cluster algebras (with type A for string amplitudes), and other natural integrals over the positive Grassmannian.Comment: 61 pages, 12 figures; v3, match the JHEP versio