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| Author(s): |
Aléx Fernando Brandt
Total Authors: 1
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| Document type: | Master's Dissertation |
| Press: | Campinas, SP. |
| Institution: | Universidade Estadual de Campinas (UNICAMP). Instituto de Computação |
| Defense date: | 2014-12-17 |
| Examining board members: |
Cid Carvalho de Souza;
Eduardo Candido Xavier;
Sebastián Alberto Urrutia
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| Advisor: | Pedro Jussieu de Rezende; Cid Carvalho de Souza |
| Abstract | |
Let P be a set of points in the plane. The geometric graph of P, G(P) =(P, E), is the complete weighted graph whose vertices correspond to the points of P and in which the cost of an edge {i, j} is given by the Euclidean distance between the points i and j. Initially, consider a general problem where one wants to build a network with good connection quality joining a set of sites represented by points in the plane. In many applications of this kind, the problem can be modeled with the aid of a geometric graph. This occurs when, for example, for a pair of points, the quality measure is defined as the ratio of the length of the shortest path that connects them in the designed network and their Euclidean distance. This ratio determines the dilation of that pair of points in the network. On the other hand, the dilation of the built network itself is given by the maximum dilation over all pair of points. In this dissertation we present exact methods for solving problems dedicated to finding a spanning tree or a planar triangulation of minimum dilation, named, respectively, the Minimum Dilation Spanning Tree Problem (MDSTP) and Minimum Dilation Triangulation Problem (MDTP). The described methods are composed primarily by the formulation, downsizing and resolution of mixed integer linear programs. The downsizing of these mathematical models is done by exploiting the geometry of the problems by means of routines that determine the need of the presence or the absence of certain elements of the formulation in solutions with dilation less than or equal to the primal bound supplied by a heuristic. Applying these routines in a pre-processing phase allows a significant reduction of the model size leading to the acceleration of its resolution time. With the combination of these techniques, for the first time, proven optimal solutions of instances with up to 20 points for the MDSTP and up to 70 points to the MDTP were obtained. The problems and their formulations, as well as ways of reducing and solving them, are presented in detail. Furthermore, analyzes of computational performance not only of the exact methods, but also of algorithms proposed by other authors are made (AU) | |
| FAPESP's process: | 12/17965-6 - Dual bounds and exact algorithms for minimum dilation problems in geometric graphs |
| Grantee: | Alex Fernando Brandt |
| Support Opportunities: | Scholarships in Brazil - Master |