Minimizing the stabbing number of matchings, trees, and triangulations

The (axis-parallel) stabbing number of a given set of line segments is the maximum number of segments that can be intersected by any one (axis-parallel) line. This paper deals with finding perfect matchings, spanning trees, or triangulations of minimum stabbing number for a given set of points. The complexity of these problems has been a long-standing open question; in fact, it is one of the original 30 outstanding open problems in computational geometry on the list by Demaine, Mitchell, and O'Rourke. The answer we provide is negative for a number of minimum stabbing problems by showing them NP-hard by means of a general proof technique. It implies non-trivial lower bounds on the approximability. On the positive side we propose a cut-based integer programming formulation for minimizing the stabbing number of matchings and spanning trees. We obtain lower bounds (in polynomial time) from the corresponding linear programming relaxations, and show that an optimal fractional solution always contains an edge of at least constant weight. This result constitutes a crucial step towards a constant-factor approximation via an iterated rounding scheme. In computational experiments we demonstrate that our approach allows for actually solving problems with up to several hundred points optimally or near-optimally.Comment: 25 pages, 12 figures, Latex. To appear in "Discrete and Computational Geometry". Previous version (extended abstract) appears in SODA 2004, pp. 430-43

The minimum stabbing triangulation problem: IP models and computational evaluation

The minimum stabbing triangulation of a set of points in the plane (mstr) was previously investigated in the literature. The complexity of the mstr remains open and, to our knowledge, no exact algorithm was proposed and no computational results were reported earlier in the literature of the problem. This paper presents integer programming (ip) formulations for the mstr, that allow us to solve it exactly through ip branch-and-bound (b&b) algorithms. Moreover, one of these models is the basis for the development of a sophisticated Lagrangian heuristic for the problem. Computational tests were conducted with two instance classes comparing the performance of the latter algorithm against that of a standard (exact) b&b. The results reveal that the Lagrangian algorithm yielded solutions with minute, and often null, duality gaps for instances with several hundreds of points in small computation times

Integer programming approaches for minimum stabbing problems

The problem of finding structures with minimum stabbing number has received considerable attention from researchers. Particularly, [10] study the minimum stabbing number of perfect matchings (mspm), spanning trees (msst) and triangulations (mstr) associated to set of points in the plane. The complexity of the mstr remains open whilst the other two are known to be . This paper presents integer programming (ip) formulations for these three problems, that allowed us to solve them to optimality through ip branch-and-bound (b&b) or branch-and-cut (b&c) algorithms. Moreover, these models are the basis for the development of Lagrangian heuristics. Computational tests were conducted with instances taken from the literature where the performance of the Lagrangian heuristics were compared with that of the exact b&b and b&c algorithms. The results reveal that the Lagrangian heuristics yield solutions with minute, and often null, duality gaps for instances with several hundreds of points in small computation times. To our knowledge, this is the first computational study ever reported in which these three stabbing problems are considered and where provably optimal solutions are given. © 2014 EDP Sciences, ROADEF, SMAI.The problem of finding structures with minimum stabbing number has received considerable attention from researchers. Particularly, [10] study the minimum stabbing number of perfect matchings (mspm), spanning trees (msst) and triangulations (mstr) associat482211233CNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOFAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULO301732/2007-8; 473867/2010-9; 147619/2010-607/52015-

Encontrando estruturas geométricas com número de trespasse mínimo

Orientador: Cid Carvalho de SouzaTese (doutorado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: Problemas de trespasse têm sido investigados há tempos em Geometria Computacional pois aplicações para eles são encontradas em uma grande variedade de áreas. Em geral, a entrada é formada por dois conjuntos de objetos geométricos: o conjunto, finito ou infinito, L de trespassadores e o conjunto O. Uma solução viável é um subconjunto O' de O satisfazendo uma certa propriedade estrutural $\pi$. Dado O', o número de trespasse de l em L é a quantidade de elementos de O' intersectados por l. O número de trespasse de O' relativo a L é o número de trespasse máximo dentre qualquer l em L. O objetivo do problema é achar um subconjunto de O satisfazendo a propriedade $\pi$ com o menor número de trespasse possível relativo a L. Esta tese traz contribuições tanto teóricas quanto experimentais para alguns problemas de trespasse. Em [19, 20], Fekete, Lübbecke e Meijer resolveram o problema aberto a respeito da complexidade de encontrar uma árvore geradora com número de trespasse mínimo. Eles também mostraram que achar um emparelhamento perfeito com número de trespasse mínimo é NP-difícil. Modelos de programação inteira para os problemas foram apresentados. Porém, muito poucos experimentos computacionais foram realizados. Nesta tese, estudamos modelos de programação inteira para encontrar emparelhamentos perfeitos, árvores geradoras e triangulação com número de trespasse mínimo. Com base nestas formulações, apresentamos algoritmos exatos e heurísticas Lagrangianas para resolvê-los. Estes algoritmos mostraram que as heurísticas Lagrangianas proveem boas soluções, frequentemente ótimas, em um breve tempo computacional. De todos os dez problemas e variantes discutidos em [20], para apenas três deles a complexidade não foi provada: Triangulação com Número de Trespasse Mínimo, com trespassadores paralelos aos eixos e gerais, e Triangulação com Número de Cruzamento Mínimo, caso geral. Nesta tese, provamos que estes três problemas são NP-difíceis. Outro problema de trespasse mínimo é apresentado em [2] e também estudado em [16]. Este problema pede por uma partição retangular com número de trespasse mínimo em um polígono retilinear. Embora a complexidade do problema ainda seja desconhecida, em [2] um algoritmo de 3-aproximação é apresentado. Em [16] um modelo de programação inteira é dado e uma 2-aproximação reivindicada. Nesta tese, fortalecemos a formulação introduzida em [16]. Também propomos um modelo alternativo e comparamos os dois teórica e computacionalmente. Além disso, mostramos que o algoritmo proposto em [16] não provê uma 2-aproximação para o problemaAbstract: Stabbing problems have long being investigated in Computational Geometry since applications for them are found in a great variety of areas. In general, the input is formed by two sets of geometrical objects: the finite or infinite set L of stabbers and the set O. A feasible solution for the problem is a subset O' of O satisfying a given structural property $\pi$. Given O', the stabbing number of l in L is the total of elements of O' that are intersected by l. The stabbing number of L relative to O' is the maximum stabbing number of all its elements. The goal of the problem is to find a subset of O satisfying property $\pi$ and having the smallest possible stabbing number. This thesis brings both theoretical and experimental contributions to the investigation of some stabbing problems. The works of Fekete, Lübbecke and Meijer [19, 20] solved the open problem relative to the complexity of finding a spanning tree with minimum stabbing number. They also showed that finding a perfect matching with minimum stabbing number is NP-hard. Integer programming formulations for the problems were also presented. However, very few computational experiments were performed. In this thesis we study integer programming formulations for the problems of finding perfect matchings, spanning trees and triangulations with minimum stabbing number. Based on these formulations we present exact algorithms and Lagrangian heuristics to solve the problems. These algorithms show that the Lagrangian heuristics yield solutions with good quality, often optimal, in short time span. Of all the ten problems and variants discussed in [20], for only three of them the complexity was not proved: The Minimum Stabbing Triangulation, axis-parallel and general cases, and The Minimum Crossing Triangulation, general case. In this thesis we prove that the three problems are NP-hard. Another problem of finding a structure with minimum stabbing number is presented in [2] and also studied in [16]. This problem asks for a rectangular partition with minimum stabbing number in a rectilinear polygon. Although the complexity of the problem is still unkown, in [2] a 3-approximation algorithm is presented. In [16] an integer programming formulation is given and a 2-approximation is claimed. In this thesis we strengthen the formulation introduced in [16]. We also propose an alternative model and compare the formulations both theoretically and computationally. Furthermore, we show that the algorithm proposed in [16] can not provide a 2-approximation for the problemDoutoradoCiência da ComputaçãoDoutor em Ciência da Computação147619/2010-6CNP