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Esta dissertação trata do problema da k-árvore de custo mínimo (kMST): dados um grafo conexo G, um custo não-negativo c_e para cada aresta e e um número inteiro positivo k, encontrar uma árvore com k vértices que tenha custo mínimo. O kMST é um problema NP-difícil e portanto não se conhece um algoritmo polinomial para resolvê-lo. Nesta dissertação discutimos alguns casos em que é possível resolver o problema em tempo polinomial. Também são estudados algoritmos de aproximação para o kMST. Entre os algoritmos de aproximação estudados, apresentamos a 2-aproximação desenvolvida por Naveen Garg, que atualmente é o algoritmo com melhor fator de aproximação. / This dissertation studies the minimum cost k-tree problem (kMST): given a connected graph G, a nonnegative cost function c_e for each edge e and a positive integer k, find a minimum cost tree with k vertices. The kMST is an NP-hard problem, which implies that it is not known a polynomial algorithm to solve it. In this dissertation we discuss some cases that can be solved in polynomial time. We also study approximation algorithms for the kMST. Among the approximation algorithms we present the 2-approximation developed by Naveen Garg, which is currently the algorithm with the best approximation factor.

Die vorliegende Arbeit besteht aus zwei Teilen. Der erste Teil der Arbeit befasst sich mit unteren Schranken für approximative Steinerbaumalgorithmen. Ein Steinerbaum ist ein kürzester Teilgraph, der eine gegebene Teilmenge der Knoten eines Graphen spannt. Das Berechnen eines Steinerbaumes ist ein klassisches NP-schweres Problem, und es existieren mehrere Approximationsalgorithmen, wobei bei den meisten Algorithmen die Approximationsgüte nur durch untere und obere Schranken eingegrenzt werden kann. Für einige dieser Algorithmen werden in dieser Arbeit Instanzen vorgestellt, welche die unteren Schranken verbessern. Für den Relativen Greedy Algorithmus wird außerdem ein Verfahren vorgestellt, mit dem die Güte des Algorithmus eingeschränkt auf die Graphenklasse mit k Terminalen auf einen beliebigen Faktor genau bestimmt werden kann. Der zweite Teil der Arbeit widmet sich vollständig bipartiten Subgraphen mit gleicher Partitionsgrößse, sogenannten balancierten Bicliquen. Seit langem ist bekannt, dass in dichten bipartiten Graphen balancierte Bicliquen mit Omega(log(n)) Knoten existieren, aber es ist unbekannt, ob und wie diese in polynomieller Zeit konstruiert werden können. Der zweite Teil liefert dazu einen Beitrag, indem ein polynomieller Algorithmus vorgestellt wird, der in hinreichend großen dichten bipartiten Graphen eine balancierte Biclique mit Omega(sqrt(log(n))) Knoten konstruiert. / This thesis consists of two parts. The first part is concerned with lower bounds for approximating Steiner trees. The Steiner tree problem is to find a shortest subgraph that spans a given set of vertices in a graph and is a classical NP-hard problem. Several approximation algorithms exist, but for most algorithms only lower and upper bounds for the approximation ratio are known. For some of these algorithms we present instances which improve the lower bounds. When the problem is restricted to the class of graphs with k terminals, we also present a method which can be used to determine the approximation ratio of the Relative Greedy Algorithm with arbitrary precision. The second part is about balanced bicliques, i.e. complete bipartite subgraphs with equal partition sizes. It has been known for a long time that every dense bipartite graph contains a balanced biclique of size Omega(log(n)), but whether and how such a biclique can be constructed in polynomial time is still unknown. Our contribution to this problem is a polynomial time algorithm which constructs a balanced biclique of size Omega(sqrt(log(n))) in sufficiently large and dense bipartite graphs.

[pt] Uma maneira de localizar uma informação em uma base de dados grande e caótica como a Internet é utilizar um índice hierárquico que respeita alguma maneira de categorizar os dados. Exemplos desta hierarquia são os serviços de diretório, comuns em sites de busca. Porém, esta abordagem pode apresentar algumas desvantagens, como a necessidade de percorrer muitas páginas até chegar em uma informação muito acessada. Uma maneira de tratar este problema é o uso de hotlinks, hyperlinks adicionais que servem como atalho em uma busca. Estudamos algoritmos eficientes para atribuir hotlinks em um diretório web, de modo a reduzir o número máximo ou o número médio de acessos em uma busca. Fornecemos para o problema de minimização do número máximo de acessos um algoritmo (14/3)-aproximado e um algoritmo polinomial exato baseado em programação dinâmica. Por outro lado, para o problema de minimizar o número médio de acessos, adaptamos o algoritmo exato do problema anterior. Entretanto, este algoritmo adaptado é polinomial apenas para sites representados por árvores com altura O(log n). Por isso, introduzimos um parâmetro que permite ao usuário reduzir o tempo de execução em detrimento da qualidade da solução. Para este problema de minimizar o número médio de acessos, realizamos também experimentos comparando nosso algoritmo, um modelo em programação inteira, e alguns algoritmos propostos por outros autores. Introduzimos modificações práticas que melhoraram a performance do nosso algoritmo. / [en] An approach to search an information in a large and chaotic data base like the Internet is to use a hierarquical index regarding some categorization of the data. As an example, we have the web directories, usually found in search engines. However, this approach may have problems, as the need of visiting too many web pages to find a very accessed information. A way to address this problem is the use of hotlinks, which are hyperlinks added to the web site and used as shortcuts in a search. We studied efficient algorithms to assign hotlinks in web directories, in such a way to minimize the maximum or the average number of accesses to find an information. For the problem of minimizing the maximum number of accesses, we provide an (14/3)-approximation algorithm and an exact polinomial time algorithm based on dynamic programming. On the other hand, for the problem of minimizing the expected number of accesses, we adapted the previous exact algorithm. However, this adapted algorithm is polinomial only for web sites represented by trees with height O(log n). So, we introduce a parameter that allows the user to reduce the execution time under the cost of reducing the solution quality. For this problem of minimizing the expected number of accesses, we also made experiments comparing our algorithm, an integer programming model, and some algorithms proposed by other authors. We introduce pratical changes that improved the performance of our algorithm.

O tema central deste trabalho é o estudo de problemas sobre caminhos mais longos em grafos, de pontos de vista tanto estrutural como algorítmico. A primeira parte tem como foco o estudo de problemas motivados pela seguinte questão levantada por T. Gallai em 1966: é verdade que em todo grafo conexo existe um vértice comum a todos os seus caminhos mais longos? Hoje, já se conhecem diversos grafos conexos cuja intersecção de todos os seus caminhos mais longos é vazia. Entretanto, existem classes de grafos para as quais a resposta à pergunta de Gallai é afirmativa. Nessa linha, apresentamos alguns resultados da literatura e duas novas classes que obtivemos: os grafos exoplanares e as 2-árvores. Motivado por esse problema, nos anos 80, T. Zamfirescu formulou a seguinte pergunta que permanece em aberto: é verdade que em todo grafo conexo existe um vértice comum a quaisquer três de seus caminhos mais longos? Apresentamos, além de alguns resultados conhecidos, uma prova de que a resposta é afirmativa para grafos em que todo bloco não trivial é hamiltoniano. Notamos que esse último resultado e o acima mencionado para grafos exoplanares generalizam um teorema de M. Axenovich (2009) que afirma que quaisquer três caminhos mais longos em um grafo exoplanar têm um vértice em comum. Finalmente, mencionamos alguns outros resultados da literatura relacionados com o tema. Na segunda parte, investigamos o problema de encontrar um caminho mais longo em um grafo. Este problema é NP-difícil para grafos arbitrários. Isto motiva investigações em duas linhas a respeito da busca de tais caminhos. Pode-se procurar classes especiais de grafos para as quais existem algoritmos polinomiais, ou pode-se abrir mão da busca de um caminho mais longo, e projetar um algoritmo eficiente que encontra um caminho cujo comprimento esteja próximo do comprimento de um mais longo. Nesse trabalho estudamos ambas as abordagens e apresentamos alguns resultados da literatura. / The central theme of this thesis is the study of problems related to longest paths in graphs, both from a structural and an algorithmic point of view. The first part focuses on the study of problems motivated by the following question raised by T. Gallai in 1966: is it true that every connected graph has a vertex common to all its longest paths? Today, many connected graphs in which all longest paths have empty intersection are known. However, there are classes of graphs for which Gallais question has a positive answer. In this direction, we present some results from the literature, as well as two new classes we obtained: outerplanar graphs and 2-trees. Motivated by this problem, T. Zamfirescu, in the 80s, proposed the following question which remains open: is it true that every connected graph has a vertex common to any three of its longest paths? We present, in addition to some known results, a proof that the answer to this question is positive for graphs in which all non-trivial blocks are Hamiltonian. We note that this result and the one mentioned above for outerplanar graphs generalize a theorem of M. Axenovich (2009) that states that any three longest paths in an outerplanar graph have a common vertex. Finally, we mention some other related results from the literature. In the second part, we investigate the problem of finding a longest path in a graph. This problem is NP-hard for arbitrary graphs. This motivates investigations in two directions with respect to the search for such paths. We can look for special classes of graphs for which the problem is polynomially solvable, or we can relinquish the search for a longest path and design an efficient algorithm that finds a path whose length is close to that of a longest path. In this thesis we study both approaches and present some results from the literature.

Esta dissertação trata do problema da k-árvore de custo mínimo (kMST): dados um grafo conexo G, um custo não-negativo c_e para cada aresta e e um número inteiro positivo k, encontrar uma árvore com k vértices que tenha custo mínimo. O kMST é um problema NP-difícil e portanto não se conhece um algoritmo polinomial para resolvê-lo. Nesta dissertação discutimos alguns casos em que é possível resolver o problema em tempo polinomial. Também são estudados algoritmos de aproximação para o kMST. Entre os algoritmos de aproximação estudados, apresentamos a 2-aproximação desenvolvida por Naveen Garg, que atualmente é o algoritmo com melhor fator de aproximação. / This dissertation studies the minimum cost k-tree problem (kMST): given a connected graph G, a nonnegative cost function c_e for each edge e and a positive integer k, find a minimum cost tree with k vertices. The kMST is an NP-hard problem, which implies that it is not known a polynomial algorithm to solve it. In this dissertation we discuss some cases that can be solved in polynomial time. We also study approximation algorithms for the kMST. Among the approximation algorithms we present the 2-approximation developed by Naveen Garg, which is currently the algorithm with the best approximation factor.

Este trabalho tem por objetivo estudar os problemas computacionais que surgem ao se relacionar grafos com superfícies bidimensionais, dando especial atenção aos problemas do número de cruzamentos mínimo no plano (CROSSING NUMBER) e a problemas relacionados ao desenho de grafos em livros. Apresentamos uma redução do problema MULTICUT para CROSSING NUMBER, além de um resultado de complexidade em grafos de comparabilidade baseado em um resultado conhecido para desenhos em livros. / The objective of this text is to study computational problems that emerge from the relation between graphs and bidimensional surfaces, giving special attention to the crossing number problem and graph drawings on books. We present a reduction from MULTICUT to CROSSING NUMBER, in addition to a complexity result on comparability graphs based on a known result about drawings on books.

In this dissertation, we examine specific network connectivity problems, and achieve improved approximation algorithm and integrality gap results for them. We introduce an important new, highly useful and applicable, network connectivity problem - the Vital Core Connectivity Problem (VCC). Despite its many practical uses, this problem has not been previously studied. We present the first constant factor approximation algorithm for VCC, and provide an upper bound on the integrality gap of its linear programming relaxation. We also introduce a new, useful, extension of the minimum spanning tree problem, called the Extended Minimum Spanning Tree Problem (EMST), that is based on a special case of VCC; and provide both a polynomial-time algorithm and a complete linear description for it. Furthermore, we show how to generalize this new problem to handle numerous disjoint vital cores, providing the first complete linear description of, and polynomial-time algorithm for, the generalized problem. We examine the Survivable Network Design Problem (SNDP) with multiple copies of edges allowed in the solution (multi-SNDP), and present a new approximation algorithm for which the approximation guarantee is better than that of the current best known for certain cases of multi-SNDP. With our method, we also obtain improved bounds on the integrality gap of the linear programming relaxation of the problem. Furthermore, we show the application of these results to variations of SNDP. We investigate cases where the optimal values of multi-SNDP and SNDP are equal; and we present an improvement on the previously best known integrality gap bound and approximation guarantee for the special case of SNDP with metric costs and low vertex connectivity requirements, as well as for the similar special case of the Vertex Connected Survivable Network Design Problem (VC-SNDP). The quality of the results that one can obtain for a given network design problem often depends on its integer linear programming formulation, and, in particular, on its linear programming relaxation. In this connection, we investigate formulations for the Steiner Tree Problem (ST). We propose two new formulations for ST, and investigate their strength in terms of their associated integrality gaps.

In this dissertation, the problem of planning the motion of m Unmanned Vehicles (UVs) (or simply vehicles) through n points in a plane is considered. A motion plan for a vehicle is given by the sequence of points and the corresponding angles at which each point must be visited by the vehicle. We require that each vehicle return to the same initial location(depot) at the same heading after visiting the points. The objective of the motion planning problem is to choose at most q(≤ m) UVs and find their motion plans so that all the points are visited and the total cost of the tours of the chosen vehicles is a minimum amongst all the possible choices of vehicles and their tours. This problem is a generalization of the wellknown Traveling Salesman Problem (TSP) in many ways: (1) each UV takes the role of salesman (2) motion constraints of the UVs play an important role in determining the cost of travel between any two locations; in fact, the cost of the travel between any two locations depends on direction of travel along with the heading at the origin and destination, and (3) there is an additional combinatorial complexity stemming from the need to partition the points to be visited by each UV and the set of UVs that must be employed by the mission. In this dissertation, a sub-optimal, two-step approach to motion planning is presented to solve this problem:(1) the combinatorial problem of choosing the vehicles and their associated tours is based on Euclidean distances between points and (2) once the sequence of points to be visited is specified, the heading at each point is determined based on a Dynamic Programming scheme. The solution to the first step is based on a generalization of Held-Karp’s method. We modify the Lagrangian heuristics for finding a close sub-optimal solution. In the later chapters of the dissertation, we relax the assumption that all vehicles are homogenous. The motivation of heterogenous variant of Multi-depot, Multiple Traveling Salesmen Problem (MDMTSP) derives form applications involving Unmanned Aerial Vehicles (UAVs) or ground robots requiring multiple vehicles with different capabilities to visit a set of locations.

The Steiner Tree problem is one of the most popular graph problems and has many application areas. The rectilinear version of this problem, introduced by Hanan, has taken special attention since it addresses a fundamental matter in &ldquo / Physical Design&rdquo / phase of the Very Large Scale Integrated (VLSI) Computer Aided Design (CAD) process. The Rectilinear Steiner Tree Problem asks for a minimum length tree that interconnects a given set of points by only horizontal and vertical line segments, enabling the use of extra points. There are various exact algorithms. However the problem is NP-complete hence approximation algorithms have to be used especially for large instances. In this thesis work, first a survey on heuristics for the Rectilinear Steiner Tree Problem is conducted and then two recently developed successful algorithms, BGA by Kahng et. al. and RST by Zhou have been studied and investigated deeply. Both algorithms have subproblems, most of which have individual backgrounds in literature. After an analysis of BGA and RST, the thesis proposes a modification on RST, which leads to a faster and non-recursive version. The efficiency of the modified algorithm has been validated by computational tests using both random and VLSI benchmark instances. A partially parallelized version of the modified algorithm is also proposed for distributed computing environments. It is implemented using MPI (message passing interface) middleware and the results of comparative tests conducted on a cluster of workstations are presented. The proposed distributed algorithm has also proved to be promising especially for large problem instances.