O problema da árvore geradora de comunicação ótima recebe um grafo com comprimentos não negativos nas arestas e um requerimento não negativo entre cada par de vértices; sendo o objetivo encontrar uma árvore geradora do grafo que minimize o custo de comunicação, que é a soma sobre cada par de vértice da distância entre eles na árvore vezes o requerimento entre eles. Este problema é NP-difícil, assim como vários casos particulares dele. Neste trabalho estudamos algumas variantes deste problema, introduzimos novos casos particulares que são também NP-difíceis e propomos esquemas de aproximação polinomial para alguns deles. / The optimum communication spanning tree problem receives a graph with non-negative lengths over the edges and non-negative requirements for each pair of nodes; being the objective to find a spanning tree of the graph that minimizes the communication cost, which is given by the sum, over each pair of nodes, of the distance, in the tree, between the nodes multiplied by the requirement between them. This problem and several of its particular cases are NP-hard. In this work we study some of the variants, also we introduce new NP-hard particular cases of the problem and propose polynomial approximation schemes for some of them.

O objetivo de projetos de reforma agrária é redistribuir terras de grandes latifúndios para terrenos menores, com destino à agricultura familiar. Um dos principais problemas do Instituto Nacional de Colonização e Reforma Agrária (INCRA) é subdividir uma parcela grande de terra em lotes menores que são balanceados com relação a certos atributos. Este problema é difícil por que precisa considerar diversas restrições legais e éticas. As soluções atuais são auxiliadas por computador, mas manuais, demoradas e suscetíveis a erros, tipicamente produzindo lotes retangulares de áreas similares mas que são injustos com relação a critérios como aptidão do solo ou acesso a recursos hidrográficos. Nesta dissertação, nós propomos um algoritmo genético para gerar subdivisões justas de forma automática. Nós apresentamos um algoritmo construtivo guloso randomizado baseado em locação-alocação para gerar soluções iniciais, assim como operadores de mutação e recombinação que consideram especificidades do problema. Experimentos com 5 instâncias reais e 25 instâncias geradas artificialmente confirmam a efetividade dos diferentes componentes do método proposto, e mostram que ele gera soluções mais balanceadas que as atualmente usadas na prática. / The goal of agrarian reform projects is the redistribution of farmland from large latifundia to smaller, often family farmers. One of the main problems the Brazilian National Institute of Colonization and Agrarian Reform (INCRA) has to solve is to subdivide a large parcel of land into smaller lots that are balanced with respect to certain attributes. This problem is difficult since it considers several constraints originating from legislation as well as ethical considerations. Current solutions are computer-assisted, but manual, time-consuming and error-prone, leading to rectangular lots of similar areas which are unfair with respect to soil aptitude and access to hydric resources. In this thesis, we propose a genetic algorithm to produce fair land subdivisions automatically. We present a greedy randomized constructive heuristic based on location-allocation to generate initial solutions, as well as mutation and recombination operators that consider specifics of the problem. Experiments on 5 real-world and 25 artificial instances confirm the effectiveness of the different components of our method, and show that it leads to fairer solutions than those currently applied in practice.

Les Problèmes d’Optimisation Combinatoire (COP) sont largement utilisés pour modéliser et résoudre un grand nombre de problèmes industriels. La résolution de ces problèmes pose un véritable défi en raison de leur inhérente difficulté, la plupart étant NP-difficiles. En effet, les COP sont difficiles à résoudre par des méthodes exactes car la taille de l’espace de recherche à explorer croît de manière exponentielle par rapport à la taille du problème. Les méta-heuristiques sont souvent les méthodes les plus efficaces pour résoudre les problèmes les plus difficiles. Malheureusement, bien des problèmes réels restent hors de portée des meilleures méta-heuristiques. Le parallélisme permet d’améliorer les performances des méta-heuristiques. L’idée de base est d’avoir plusieurs instances d’une méta-heuristique explorant de manière simultanée l’espace de recherche pour accélérer la recherche de solution. Les meilleures techniques font communiquer ces instances pour augmenter la probabilité de trouver une solution. Cependant, la conception d’une méthode parallèle coopérative n’est pas une tâche aisée, et beaucoup de choix cruciaux concernant la communication doivent être résolus. Malheureusement, nous savons qu’il n’existe pas d’unique configuration permettant de résoudre efficacement tous les problèmes. Ceci explique que l’on trouve aujourd’hui des systèmes coopératifs efficaces mais conçus pour un problème spécifique ou bien des systèmes plus génériques mais dont les performances sont en général limitées. Dans cette thèse nous proposons un cadre général pour les méta-heuristiques parallèles coopératives (CPMH). Ce cadre prévoit plusieurs paramètres permettant de contrôler la coopération. CPMH organise les instances de méta-heuristiques en équipes ; chaque équipe vise à intensifier la recherche dans une région particulière de l’espace de recherche. Cela se fait grâce à des communications intra-équipes. Des communications inter-équipes permettent quant a` elles d’assurer la diversification de la recherche. CPMH offre à l’utilisateur la possibilité d’ajuster le compromis entre intensification et diversification. De plus, ce cadre supporte différentes méta-heuristiques et permet aussi l’hybridation de méta-heuristiques. Nous proposons également X10CPMH, une implémentation de CPMH, écrite en langage parallèle X10. Pour valider notre approche, nous abordons deux COP du monde industriel : des variantes difficiles du Problème de Stable Matching (SMP) et le Problème d’Affectation Quadratique (QAP). Nous proposons plusieurs méta-heuristiques originales en version séquentielle et parallèle, y compris un nouvelle méthode basée sur l’optimisation extrémale ainsi qu’un nouvel algorithme hybride en parallèle coopératif pour QAP. Ces algorithmes sont implémentés grâce à X10CPMH. L’évaluation expérimentale montre que les versions avec parallélisme coopératif offrent un très bon passage à l’échelle tout en fournissant des solutions de haute qualité. Sur les variantes difficiles de SMP, notre méthode coopérative offre des facteurs d’accélération super-linéaires. En ce qui concerne QAP, notre méthode hybride en parallèle coopératif fonctionne très bien sur les cas les plus difficiles et permet d’améliorer les meilleures solutions connues de plusieurs instances. / Combinatorial Optimization Problems (COP) are widely used to model and solve real-life problems in many different application domains. These problems represent a real challenge for the research community due to their inherent difficulty, as many of them are NP-hard. COPs are difficult to solve with exact methods due to the exponential growth of the problem’s search space with respect to the size of the problem. Metaheuristics are often the most efficient methods to make the hardest problems tractable. However, some hard and large real-life problems are still out of the scope of even the best metaheuristic algorithms. Parallelism is a straightforward way to improve metaheuristics performance. The basic idea is to perform concurrent explorations of the search space in order to speed up the search process. Currently, the most advanced techniques implement some communication mechanism to exchange information between metaheuristic instances in order to try and increase the probability to find a solution. However, designing an efficient cooperative parallel method is a very complex task, and many issues about communication must be solved. Furthermore, it is known that no unique cooperative configuration may efficiently tackle all problems. This is why there are currently efficient cooperative solutions dedicated to some specific problems or more general cooperative methods but with limited performances in practice. In this thesis we propose a general framework for Cooperative Parallel Metaheuristics (CPMH). This framework includes several parameters to control the cooperation. CPMH organizes the explorers into teams; each team aims at intensifying the search in a particular region of the search space and uses intra-team communication. In addition, inter-team communication is used to ensure search diversification. CPMH allows the user to tune the trade-off between intensification and diversification. However, our framework supports different metaheuristics and metaheuristics hybridization. We also provide X10CPMH, an implementation of our CPMH framework developed in the X10 parallel language. To assess the soundness of our approach we tackle two hard real-life COP: hard variants of the Stable Matching Problem (SMP) and the Quadratic Assignment Problem (QAP). For all problems we propose new sequential and parallel metaheuristics, including a new Extremal Optimization-based method and a new hybrid cooperative parallel algorithm for QAP. All algorithms are implemented thanks to X10CPMH. A complete experimental evaluation shows that the cooperative parallel versions of our methods scale very well, providing high-quality solutions within a limited timeout. On hard and large variants of SMP, our cooperative parallel method reaches super-linear speedups. Regarding QAP, the cooperative parallel hybrid algorithm performs very well on the hardest instances, and improves the best known solutions of several instances.

Funções submodulares aparecem naturalmente em diversas áreas, tais como probabilidade, geometria e otimização combinatória. Pode-se dizer que o papel desempenhado por essas funções em otimização discreta é similar ao desempenhado por convexidade em otimização contínua. Com efeito, muitos problemas em otimização combinatória podem ser formulados como um problema de minimizar uma função submodular sobre um conjunto apropriado. Além disso, submodularidade está presente em vários teoremas ou problemas combinatórios e freqüentemente desempenha um papel essencial em uma demonstração ou na eficiência de um algoritmo. Nesta dissertação, estudamos aspectos estruturais e algorítmicos de funções submodulares, com ênfase nos recentes avanços em algoritmos combinatórios para minimização dessas funções. Descrevemos com detalhes os primeiros algoritmos combinatórios e fortemente polinomiais para esse propósito, devidos a Schrijver e Iwata, Fleischer e Fujishige, além de algumas outras extensões. Aplicações de submodularidade em otimização combinatória também estão presentes neste trabalho. / Submodular functions arise naturally in various fields, including probability, geometry and combinatorial optimization. The role assumed by these functions in discrete optimization is similar to that played by convexity in continuous optimization. Indeed, we can state many problems in combinatorial optimization as a problem of minimizing a submodular function over an appropriate set. Moreover, submodularity appears in many combinatorial theorems or problems and frequently plays an essencial role in a proof or an algorithm. In this dissertation, we study structural and algorithmic aspects of submodular functions. In particular, we focus on the recent advances in combinatorial algorithms for submodular function minimization. We describe in detail the first combinatorial strongly polynomial-time algorithms for this purpose, due to Schrijver and Iwata, Fleischer, and Fujishige, as well as some extensions. Some applications of submodularity in combinatorial optimization are also included in this work.

nÃo hà / Este trabalho foi motivado pela necessidade da exploraÃÃo do potencial do paralelismo distribuÃdo em aplicaÃÃes em OtimizaÃÃo CombinatÃria. Para tanto, propomos uma interface de programaÃÃo distribuÃda, na qual prezamos dois requisitos principais: eficiÃncia e reuso. O primeiro advÃm da necessidade de aplicaÃÃes de CAD exigirem mÃximo desempenho possÃvel. Assim sendo, especificamos esta interface como uma extensÃo da biblioteca MPI, a qual à assumida como eficiente para aplicaÃÃes distribuÃdas. O requisito reuso deve tornar compatÃveis duas caracterÃsticas importantes: assincronismo e operaÃÃes coletivas. O assincronismo deve estar presente na interface, uma vez que as aplicaÃÃes em OtimizaÃÃo CombinatÃria, em sua maioria, possuem uma natureza assÃncrona. OperaÃÃes coletivas sÃo funcionalidades que devem estar disponÃveis na interface, de modo que possam ser utilizadas por aplicaÃÃes em suas execuÃÃes. Tendo em vista atender o requisito reuso, baseamos esta interface nos Modelos de ComputaÃÃo DistribuÃda Dirigidos por Eventos e por Pulsos, pois os mesmos sÃo assÃncronos e permitem a incorporaÃÃo de operaÃÃes coletivas. Implementamos parcialmente a inteface definida neste trabalho. Tendo em vista validar uso desta inteface por aplicaÃÃes em OtimizaÃÃo CombinatÃria, selecionamos duas aplicaÃÃes e as implementamos utilizando a interface. SÃo elas a tÃcnica Branch-and-Bound e o Problema do Conjunto Independente MÃximo (CIM). Fornecemos tambÃm alguns resultados experimentais. / This work was motivated by the need of exploiting the potential of distributed paralelism in combinatorial optimization applications. propose a distributed programming interface, To achieve this goal, we in which we cherish two main requirements: eciency and reuse. The rst stems from the need of HPC (High applications require maximum possible performance. Performance Computing) Therefore, we specify our interface as an extension of the MPI library, which is assumed to be ecient for distributed applications. The reuse requirement must make compatible two important features: asynchronism and collective operations. Asynchronism must be present at our interface, once most of combinatorial optimization applications have an asynchronous nature. Collective operations are features that should be available in the interface, so that they can be used by applications in their execution. In order reach the reuse requirement, we based this interface on the Event- and Pulse-driven Models of Distributed Computing, once they are asynchronous and allow the incorporation of collective operations. We implemented partially the interface dened in this work. In order to validate the use of the inteface by combinatorial optimization applications, we selected two applications and implemented them using our interface. They are the Branch-and-Bound technique and the Maximum Stable Set Problem (MSSP). We also provide some experimental results.

A partir da reestruturação dos serviços públicos de energia elétrica, foi criada uma série de novas ferramentas regulatórias, simulando e/ou criando um ambiente competitivo, para que as empresas busquem continuamente a evolução de seus indicadores e custos. Com a edição da Resolução nº 024, de 27 de janeiro de 2000, a Agência Nacional de Energia Elétrica (ANEEL) atualizou a regulamentação dos aspectos relativos à continuidade do fornecimento de energia elétrica. As metas de continuidade são definidas através do cluster ao qual cada conjunto de consumidores está vinculado. Os conjuntos são agrupados pelas suas características físicas: área, km de rede primária, número de consumidores, potência de transformadores instalada e consumo médio do conjunto. Um dos pontos focais desta resolução é a possibilidade de uma concessionária agrupar unidades consumidoras, considerando as características técnicas específicas de seu sistema elétrico. Desta forma, o agente regulador permite que as concessionárias modifiquem seus conjuntos de consumidores, desde que fiquem evidenciadas vantagens técnicas, econômicas e sociais da nova proposta em relação ao critério vigente de agrupamento. Visando aperfeiçoar a utilização dos recursos, direcionando as ações para modicidade tarifária e considerando a capacidade de prover condições de atendimento homogêneo, este trabalho busca combinar os consumidores de uma concessionária em conjuntos que minimizem o risco de multa e a necessidade de investimentos nas redes. Este é um problema semelhante ao de redistribuição de eleitores nos distritos de votação nos EUA, conhecido como Political Districting. Para resolver o problema de explosão combinatória resultante das possíveis combinações de áreas e minimizar as multas, o modelo proposto neste trabalho utiliza técnicas de computação evolutiva. A metodologia é ilustrada alterando os 419 conjuntos iniciais de uma concessionária por meio de um algoritmo genético (AG) e um algoritmo imunológico (AI) que otimiza o resultado proposto, minimizando o risco de multas pelo não cumprimento das metas de continuidade. / From the restructuring of the Public Electric Power Sector, new regulatory tools were devised to simulate and create a competitive environment for companies to continuously seek targets for their indicators and costs. With the issue of Resolution nº 024 of January 27, 2000, the National Agency of Electric Energy (ANEEL) updated the rules in dealing with electricity supply continuity. The goals related to the continuity of service are defined through the cluster in which each set of consumers is bound. Consumers are grouped by their physical characteristics: area, length (km) of primary network, the number of consumers, power transformers installed capacity and average consumption. ANEEL allows the utilities to modify their sets of consumers, whenever the technical advantages, economic and social implications of the new proposal in relation to the current criterion of grouping become evident. Considering the possibility of avoiding unnecessary investments in networks, burdening the distribution tariff, this paper attempts to combine the consumers of a utility in sets that minimize the risk of penalties and network investments. This problem is similar to the redistribution in voting districts in the U.S., known as Political Districting. In order to solve the combinatorial explosion problem resulting from the possible combinations of areas and minimization of penalties, the model proposed in this paper uses evolutionary computation techniques. The case study alters the initial 419 sets of consumers of a utility through a genetic algorithm and an artificial immune algorithm, which were proposed to optimize the outcome, minimizing the risk of penalties in not meeting the goals related to continuity of service.

In the present thesis we provide compact extended formulations for a wide range of polytopes associated with the constraint satisfaction problem (CSP), monadic second order logic (MSO) on graphs, and extensions of MSO, when the given instances have bounded treewidth. We show that our extended formulations have additional useful properties, and we uncover connections between MSO and CSP. We conclude that a combination of the MSO logic, CSP and geometry provides an extensible framework for the design of compact extended formulations and parameterized algorithms for graphs of bounded treewidth. Putting our framework to use, we settle the parameterized complexity landscape for various extensions of MSO when parameterized by two important graph width parameters, namely treewidth and neighborhood diversity. We discover that the (non)linearity of the MSO extension determines the difference between fixedparameter tractability and intractability when parameterized by neighborhood diversity. Finally, we study shifted combinatorial optimization, a new nonlinear optimization framework generalizing standard combinatorial optimization, and provide initial findings from the perspective of parameterized complexity

Remote sensors and unmanned aerial vehicles (UAVs) have the potential to dramatically improve infrastructure health monitoring in terms of accuracy of the information and frequency of data collection. UAV automation has made significant progress but that automation is also creating vast amounts of data that needs to be processed into actionable information. A key aspect of this work is the optimization (not just automation) of data collection from UAVs for targeted planning of mission objectives. This work investigates the use of camera planning for Structure from Motion for 3D modeling of infrastructure. Included in this thesis is a novel multi-scale view-planning algorithm for autonomous targeted inspection. The method presented reduced the number of photos needed and therefore reduced the processing time while maintaining desired accuracies across the test site. A second focus in this work investigates various set covering problem algorithms to use for selecting the optimal camera set. The trade-offs between solve time and quality of results are explored. The Carousel Greedy algorithm is found to be the best method for solving the problem due to its relatively fast solve speeds and the high quality of the solutions found. Finally, physical flight tests are used to demonstrate the quality of the method for determining coverage. Each of the set covering problem algorithms are used to create a camera set that achieves 95% coverage. The models from the different camera sets are comparable despite having a large amount of variability in the camera sets chosen. While this study focuses on multi-scale view planning for optical sensors, the methods could be extended to other remote sensors, such as aerial LiDAR.

Remote sensors and unmanned aerial vehicles (UAVs) have the potential to dramatically improve infrastructure health monitoring in terms of accuracy of the information and frequency of data collection. UAV automation has made significant progress but that automation is also creating vast amounts of data that needs to be processed into actionable information. A key aspect of this work is the optimization (not just automation) of data collection from UAVs for targeted planning of mission objectives. This work investigates the use of camera planning for Structure from Motion for 3D modeling of infrastructure. Included in this thesis is a novel multi-scale view-planning algorithm for autonomous targeted inspection. The method presented reduced the number of photos needed and therefore reduced the processing time while maintaining desired accuracies across the test site. A second focus in this work investigates various set covering problem algorithms to use for selecting the optimal camera set. The trade-offs between solve time and quality of results are explored. The Carousel Greedy algorithm is found to be the best method for solving the problem due to its relatively fast solve speeds and the high quality of the solutions found. Finally, physical flight tests are used to demonstrate the quality of the method for determining coverage. Each of the set covering problem algorithms are used to create a camera set that achieves 95% coverage. The models from the different camera sets are comparable despite having a large amount of variability in the camera sets chosen. While this study focuses on multi-scale view planning for optical sensors, the methods could be extended to other remote sensors, such as aerial LiDAR.

Graph theory is fascinating branch of math. Leonhard Euler introduced the concept of Graph Theory in his paper about the seven bridges of Konigsberg published in 1736. In a nutshell, graph theory is the study of pair-wise relationships between objects. Each object is represented using a vertex and in case of a relationship between a pair of vertices, they will be connected using an edge. In this dissertation, graph theory is used to study several important combinatorial optimization problems. In chapter 2, we study the multi-dimensional assignment problem using their underlying hypergraphs. It will be shown how the MAP can be represented by a k-partite graph and how any solution to MAP is associated to a k-clique in the respective k-partite graph. Two heuristics are proposed to solve the MAP and computational studies are performed to compare the performance of the proposed methods with exact solutions. On the heels of chapter 3, a new branch-and-bound method is proposed to solve the problem of finding all k-cliques in a k-partite graph. The new method utilizes bitsets as the data-structure to represent graph data. A new pruning method is introduced in BitCLQ, and CPU instructions are used to improve the performance of the branch-and-bound method. BitCLQ gains up to 300% speed up over existing methods. In chapter 4, two new heuristic to solve decomposable cost MAP's are proposed. The proposed heuristic are based on the partitioning of the underlying graph representing the MAP. In the first heuristic method, MAP's are partitioned into several smaller MAP's with the same dimensiality and smaller cardinality. The solution to the original MAP is constructed incrementally, by using the solutions obtained from each of the smaller MAP's. The second heuristic works in the same fashion. But instead of partitioning the graph along the cardinality, graphs are divided into smaller graphs with the same cardinality but smaller dimensionality. The result of each heuristic is then compared with a well-known existing heuristic. An important problem in graph theory is the maximum clique problem (MCQ). A clique in a graph is defined as a complete subgraph. MCQ problem entails finding the size of the largest clique contained in a graph. General branch-and-bound methods to solve MCQ use graph coloring to find an upper bound on the size of the maximum clique. Recently, a new MAX-SAT based branch-and-bound method for MCQ is proposed that improves the quality of the upper bound obtained from graph coloring. In chapter 5, a branch and bound algorithm is proposed for the maximum clique problem. The proposed method uses bitsets as the data-structure. The result of the computational studies to compare the proposed method with existing methods for MCQ is provided. Chapter 6 contains an application of a graph theory in solving a risk management problem. A new mixed-integer mathematical model to formulate a risk-based network is provided. It will be shown that an optimal solution of the model is a maximal clique in the underlying graph representing the network. The model is then solved using a graph-theoretic approach and the results are compared to CPLEX.