Bounds on the maximum multiplicity of some common geometric graphs

We obtain new lower and upper bounds for the maximum multiplicity of some weighted and, respectively, non-weighted common geometric graphs drawn on n points in the plane in general position (with no three points collinear): perfect matchings, spanning trees, spanning cycles (tours), and triangulations. (i) We present a new lower bound construction for the maximum number of triangulations a set of n points in general position can have. In particular, we show that a generalized double chain formed by two almost convex chains admits {\Omega}(8.65^n) different triangulations. This improves the bound {\Omega}(8.48^n) achieved by the double zig-zag chain configuration studied by Aichholzer et al. (ii) We present a new lower bound of {\Omega}(12.00^n) for the number of non-crossing spanning trees of the double chain composed of two convex chains. The previous bound, {\Omega}(10.42^n), stood unchanged for more than 10 years. (iii) Using a recent upper bound of 30^n for the number of triangulations, due to Sharir and Sheffer, we show that n points in the plane in general position admit at most O(68.62^n) non-crossing spanning cycles. (iv) We derive lower bounds for the number of maximum and minimum weighted geometric graphs (matchings, spanning trees, and tours). We show that the number of shortest non-crossing tours can be exponential in n. Likewise, we show that both the number of longest non-crossing tours and the number of longest non-crossing perfect matchings can be exponential in n. Moreover, we show that there are sets of n points in convex position with an exponential number of longest non-crossing spanning trees. For points in convex position we obtain tight bounds for the number of longest and shortest tours. We give a combinatorial characterization of the longest tours, which leads to an O(nlog n) time algorithm for computing them

Approximating Longest Spanning Tree with Neighborhoods

We study the following maximization problem in the Euclidean plane: Given a collection of neighborhoods (polygonal regions) in the plane, the goal is to select a point in each neighborhood so that the longest spanning tree on selected points has maximum length. It is not known whether or not this problem is NP-hard. We present an approximation algorithm with ratio 0.548 for this problem. This improves the previous best known ratio of 0.511. The presented algorithm takes linear time after computing a diameter. Even though our algorithm itself is fairly simple, its analysis is rather involved. In some part we deal with a minimization problem with multiple variables. We use a sequence of geometric transformations to reduce the number of variables and simplify the analysis

Planar Bichromatic Bottleneck Spanning Trees

Given a set P of n red and blue points in the plane, a planar bichromatic spanning tree of P is a geometric spanning tree of P, such that each edge connects between a red and a blue point, and no two edges intersect. In the bottleneck planar bichromatic spanning tree problem, the goal is to find a planar bichromatic spanning tree T, such that the length of the longest edge in T is minimized. In this paper, we show that this problem is NP-hard for points in general position. Our main contribution is a polynomial-time (8?2)-approximation algorithm, by showing that any bichromatic spanning tree of bottleneck ? can be converted to a planar bichromatic spanning tree of bottleneck at most 8?2 ?

Non-Crossing Hamiltonian Paths and Cycles in Output-Polynomial Time

We show that, for planar point sets, the number of non-crossing Hamiltonian paths is polynomially bounded in the number of non-crossing paths, and the number of non-crossing Hamiltonian cycles (polygonalizations) is polynomially bounded in the number of surrounding cycles. As a consequence, we can list the non-crossing Hamiltonian paths or the polygonalizations, in time polynomial in the output size, by filtering the output of simple backtracking algorithms for non-crossing paths or surrounding cycles respectively. To prove these results we relate the numbers of non-crossing structures to two easily-computed parameters of the point set: the minimum number of points whose removal results in a collinear set, and the number of points interior to the convex hull. These relations also lead to polynomial-time approximation algorithms for the numbers of structures of all four types, accurate to within a constant factor of the logarithm of these numbers

Acoplamientos óptimos de caminos de longitud dos

96 páginas. Maestría en Optimización.Let P be a set of 3k points in the Euclidean plane. A 3-matching is a partition of P into k subsets of 3 points each, called triplets. The cost of each triplet fa; b; cg is given by minfjabj + jbcj; jbcj + jcaj; jcaj + jabjg, and the cost of the 3-matching is the sum of the costs of its triplets. The Euclidean 3-matching problem consists on finding a minimum cost 3-matching of P under the Euclidean metric. In the usual formulation of the Euclidean 3- matching problem we need to find a minimum cost 3-matching of P. This problem has several applications, especially in the insertion of components on a printed circuit board. Johnsson, Magyar, and Nevalainen introduced two integer programming formulations for this problem, and proved that its decision version is NP-complete if each triplet has an arbitrary positive cost (i.e., not necessarily Euclidean). The problem remains NP-complete even if the points of P correspond to vertices of a unit distance graph (a metric cost function). In this work, we prove that the linear programming relaxations of these two models are equivalent. Then we introduce three new integer programming models that use fewer variables than those from Johnsson, Magyar, and Nevalainen. We also compare the linear programming relaxations of the models. Besides the minimization problem, we are also interested in a similar maximization problem: finding a maximum cost non-crossing Euclidean 3-matching of P, where non-crossing means that no two segments intersect in a common interior point. Both problems, minimum cost and maximum cost non-crossing, are challenging, and we believe that both are NP-hard. Exact solutions to both problems can be attained through integer programming; however, in order to obtain good solutions in feasible times, we fix our attention to heuristics. We present three heuristics specially designed for our problems and compare their solutions and execution times against solving the exact models

Algorithmic and Combinatorial Results in Selection and Computational Geometry

This dissertation investigates two sets of algorithmic and combinatorial problems. Thefirst part focuses on the selection problem under the pairwise comparison model. For the classic “median of medians” scheme, contrary to the popular belief that smaller group sizes cause superlinear behavior, several new linear time algorithms that utilize small groups are introduced. Then the exact number of comparisons needed for an optimal selection algorithm is studied. In particular, the implications of a long standing conjecture known as Yao’s hypothesis are explored. For the multiparty model, we designed low communication complexity protocols for selecting an exact or an approximate median of data that is distributed among multiple players. In the second part, three computational geometry problems are studied. For the longestspanning tree with neighborhoods, approximation algorithms are provided. For the stretch factor of polygonal chains, upper bounds are proved and almost matching lower bound constructions in \mathbb{R}^2 and higher dimensions are developed. For the piercing number τ and independence number ν of a family of axis-parallel rectangles in the plane, a lower bound construction for ν = 4 that matches Wegner’s conjecture is analyzed. The previous matching construction for ν = 3, due to Wegner himself, dates back to 1968