<p>This thesis presents exact means to solve a family of NP-hard problems. Starting with the well-studied Exact Satisfiability problem (XSAT) parents, siblings and daughters are derived and studied, each with interesting practical and theoretical properties. While developing exact algorithms to solve the problems, we gain new insights into their structure and mutual similarities and differences.</p><p>Given a Boolean formula in CNF, the XSAT problem asks for an assignment to the variables such that each clause contains exactly one true literal. For this problem we present an <em>O</em>(1.1730ⁿ) time algorithm, where n is the number of variables. XSAT is a special case of the General Exact Satisfiability problem which asks for an assignment such that in each clause exactly i literals be true. For this problem we present an algorithm which runs in <em>O</em>(2^(1-<em>ε</em>⁾<em>n</em>) time, with 0 < <em>ε</em> < 1 for every fixed <em>i</em>; for <em>i</em>=2, 3 and 4 we have running times in <em>O</em>(1.4511ⁿ), <em>O</em>(1.6214ⁿ) and <em>O</em>(1.6848ⁿ) respectively.</p><p>For the counting problems we present an O(1.2190ⁿ) time algorithm which counts the number of models for an XSAT instance. We also present algorithms for #2SAT<em>w</em><em> </em>and #3SAT<em>w</em><em>,</em> two well studied Boolean problems. The algorithms have running times in O(1.2561ⁿ) and <em>O</em>(1.6737ⁿ) respectively.</p><p>Finally we study optimisation problems: As a variant of the Maximum Exact Satisfiability problem, consider the problem of finding an assignment exactly satisfying a maximum number of clauses while the rest are left with no true literal. This problem is reducible to #2SAT<em>w</em> without the addition of new variables and thus is solvable in time <em>O</em>(1.2561ⁿ). Another interesting optimisation problem is to find two XSAT models which differ in as many variables as possible. This problem is shown to be solvable in O(1.8348ⁿ) time.</p>

This thesis presents exact means to solve a family of NP-hard problems. Starting with the well-studied Exact Satisfiability problem (XSAT) parents, siblings and daughters are derived and studied, each with interesting practical and theoretical properties. While developing exact algorithms to solve the problems, we gain new insights into their structure and mutual similarities and differences. Given a Boolean formula in CNF, the XSAT problem asks for an assignment to the variables such that each clause contains exactly one true literal. For this problem we present an O(1.1730n) time algorithm, where n is the number of variables. XSAT is a special case of the General Exact Satisfiability problem which asks for an assignment such that in each clause exactly i literals be true. For this problem we present an algorithm which runs in O(2(1-ε) n) time, with 0 &lt; ε &lt; 1 for every fixed i; for i=2, 3 and 4 we have running times in O(1.4511n), O(1.6214n) and O(1.6848n) respectively. For the counting problems we present an O(1.2190n) time algorithm which counts the number of models for an XSAT instance. We also present algorithms for #2SATw and #3SATw, two well studied Boolean problems. The algorithms have running times in O(1.2561n) and O(1.6737n) respectively. Finally we study optimisation problems: As a variant of the Maximum Exact Satisfiability problem, consider the problem of finding an assignment exactly satisfying a maximum number of clauses while the rest are left with no true literal. This problem is reducible to #2SATw without the addition of new variables and thus is solvable in time O(1.2561n). Another interesting optimisation problem is to find two XSAT models which differ in as many variables as possible. This problem is shown to be solvable in O(1.8348n) time.

The topic of exact, exponential-time algorithms for NP-hard problems has received a lot of attention, particularly with the focus of producing algorithms with stronger theoretical guarantees, e.g. upper bounds on the running time on the form O(c^n) for some c. Better methods of analysis may have an impact not only on these bounds, but on the nature of the algorithms as well. The most classic method of analysis of the running time of DPLL-style ("branching" or "backtracking") recursive algorithms consists of counting the number of variables that the algorithm removes at every step. Notable improvements include Kullmann's work on complexity measures, and Eppstein's work on solving multivariate recurrences through quasiconvex analysis. Still, one limitation that remains in Eppstein's framework is that it is difficult to introduce (non-trivial) restrictions on the applicability of a possible recursion. We introduce two new kinds of complexity measures, representing two ways to add such restrictions on applicability to the analysis. In the first measure, the execution of the algorithm is viewed as moving between a finite set of states (such as the presence or absence of certain structures or properties), where the current state decides which branchings are applicable, and each branch of a branching contains information about the resultant state. In the second measure, it is instead the relative sizes of the modelled attributes (such as the average degree or other concepts of density) that controls the applicability of branchings. We adapt both measures to Eppstein's framework, and use these tools to provide algorithms with stronger bounds for a number of problems. The problems we treat are satisfiability for sparse formulae, exact 3-satisfiability, 3-hitting set, and counting models for 2- and 3-satisfiability formulae, and in every case the bound we prove is stronger than previously known bounds.

The minimum cost bipartite matching problem is a well-studied optimization problem in computer science and operations research, with wide-ranging applications in fields such as machine learning, economics, transportation, logistics and biology. A special instance of this problem is the computation of the p-Wasserstein distance which we define next. Given a complete bipartite graph with two disjoint sets of n points in d-dimensional Euclidean space and an integer p ≥ 1, let the cost of an edge be the p-th power of the Euclidean distance between its endpoints. The objective of this problem is to find a minimum-cost matching in this complete bipartite graph. The Hungarian algorithm is a classical method that solves this problem in O(n^3) time. There are many algorithms that have a run time better than that of the Hungarian algorithm if the graphs have non-negative integer edge costs bounded by C. Since the input points have real-valued coordinates and the Euclidean distances can be irrational, such algorithms only return an approximate matching. Thus, the Hungarian algorithm remains the fastest known algorithm to compute an exact matching. In this thesis, we implement a new algorithm in the divide and conquer framework that computes the exact p-Wasserstein distance and has a run time asymptotically better than the Hungarian algorithm for stochastic point sets. Inspired by the techniques used in the algorithm, we also design an alternate version of the Hungarian algorithm that uses a grid- based approach. Our experimental analysis shows that both of our algorithms significantly outperform the classical Hungarian algorithm. / Master of Science / Suppose we have two sets of equal number of items and a list of compatible pairs of items, where a pair is considered compatible if its items belong to different sets. A perfect matching is a subset of compatible pairs where each item is paired with exactly one other item. When trying to find a perfect matching, there may be multiple options, and minimizing the cost of the perfect matching is often desired. This is referred to as the minimum cost bipartite matching problem, which is extensively studied due to its importance in algorithmic theory and operations research. A special instance of this problem is the calculation of the p- Wasserstein distance. It has many practical applications in fields such as machine learning, economics, transportation, logistics and biology. The Hungarian algorithm is the only known algorithm that can compute the exact p-Wasserstein distance. Therefore, our focus is to develop exact algorithms for this problem that perform better than the Hungarian algorithm and can scale efficiently to large datasets.

Virtualização permite o compartilhamento de uma rede física entre uma ou mais redes virtuais. O Problema de Mapeamento de Redes Virtuais é um dos principais desafios na virtualização de redes. Esse problema consiste em mapear uma rede virtual em uma rede física, respeitando restrições de capacidade. O presente trabalho mostra que encontrar uma solução factível para esse problema é NP-Difícil. Mesmo assim, muitas instâncias podem ser pode ser resolvidas na prática através da exploração de sua estrutura. Nós apresentamos um algoritmo de Branch & Price aplicado a instâncias de diferentes topologias e tamanhos. Os experimentos realizados sugerem que o algoritmo proposto é superior ao modelo de programação linear resolvido com CPLEX. / Virtualization allows one or more virtual networks to share physical infrastructures. The Virtual Network Embedding problem (VNEP) is one of the main challenges in the virtualization of physical networks. This problem consists in mapping a virtual network into a physical network while respecting capacity constraints. This work shows that finding a feasible solution for this problem is NP-Hard. However, many instances can be solved up to optimality in practice by exploiting the problem structure. We present a Branch & Price algorithm applied to instances of different topologies and sizes. The experimental results suggest that the proposed algorithm is superior to the Integer Linear Programming model solved by CPLEX.

Virtualização permite o compartilhamento de uma rede física entre uma ou mais redes virtuais. O Problema de Mapeamento de Redes Virtuais é um dos principais desafios na virtualização de redes. Esse problema consiste em mapear uma rede virtual em uma rede física, respeitando restrições de capacidade. O presente trabalho mostra que encontrar uma solução factível para esse problema é NP-Difícil. Mesmo assim, muitas instâncias podem ser pode ser resolvidas na prática através da exploração de sua estrutura. Nós apresentamos um algoritmo de Branch & Price aplicado a instâncias de diferentes topologias e tamanhos. Os experimentos realizados sugerem que o algoritmo proposto é superior ao modelo de programação linear resolvido com CPLEX. / Virtualization allows one or more virtual networks to share physical infrastructures. The Virtual Network Embedding problem (VNEP) is one of the main challenges in the virtualization of physical networks. This problem consists in mapping a virtual network into a physical network while respecting capacity constraints. This work shows that finding a feasible solution for this problem is NP-Hard. However, many instances can be solved up to optimality in practice by exploiting the problem structure. We present a Branch & Price algorithm applied to instances of different topologies and sizes. The experimental results suggest that the proposed algorithm is superior to the Integer Linear Programming model solved by CPLEX.

Virtualização permite o compartilhamento de uma rede física entre uma ou mais redes virtuais. O Problema de Mapeamento de Redes Virtuais é um dos principais desafios na virtualização de redes. Esse problema consiste em mapear uma rede virtual em uma rede física, respeitando restrições de capacidade. O presente trabalho mostra que encontrar uma solução factível para esse problema é NP-Difícil. Mesmo assim, muitas instâncias podem ser pode ser resolvidas na prática através da exploração de sua estrutura. Nós apresentamos um algoritmo de Branch & Price aplicado a instâncias de diferentes topologias e tamanhos. Os experimentos realizados sugerem que o algoritmo proposto é superior ao modelo de programação linear resolvido com CPLEX. / Virtualization allows one or more virtual networks to share physical infrastructures. The Virtual Network Embedding problem (VNEP) is one of the main challenges in the virtualization of physical networks. This problem consists in mapping a virtual network into a physical network while respecting capacity constraints. This work shows that finding a feasible solution for this problem is NP-Hard. However, many instances can be solved up to optimality in practice by exploiting the problem structure. We present a Branch & Price algorithm applied to instances of different topologies and sizes. The experimental results suggest that the proposed algorithm is superior to the Integer Linear Programming model solved by CPLEX.

Cette thèse présente des algorithmes de résolution de quatre problèmes de clique : clique de poids maximum (MVWCP), s-plex maximum (MsPlex), clique maximum équilibrée dans un graphe biparti (MBBP) et clique partition (CPP). Les trois premiers problèmes sont des généralisations ou relaxations du problème de la clique maximum, tandis que le dernier est un problème de couverture. Ces problèmes, ayant de nombreuses applications pratiques, sont NP-difficiles, rendant leur résolution ardue dans le cas général. Nous présentons ici des algorithmes de recherche locale, principalement basés sur la recherche tabou, permettant de traiter efficacement ces problèmes ; chacun de ces algorithmes emploie des composants originaux et spécifiquement adaptés aux problèmes traités, comme de nouveaux opérateurs ou mécanismes perturbatifs. Nous y intégrons également des stratégies telles que la réduction de graphe ou la propagation afin de traiter des réseaux de plus grande taille. Des expérimentations basées sur des jeux d’instances nombreux et variés permettent de montrer la compétitivité de nos algorithmes en comparaison avec les autres stratégies existantes. / This thesis considers four clique problems: the maximum vertex weight clique problem (MVWCP), the maximum s-plex problem (MsPlex), the maximum balanced biclique problem (MBBP) and the clique partitioning problem (CPP). The first three are generalization and relaxation of the classic maximum clique problem (MCP), while the last problem belongs to a clique grouping problem. These combinatorial problems have numerous practical applications. Given that they all belong to the NP-Hard family, it is computationally difficult to solve them in the general case. For this reason, this thesis is devoted to develop effective algorithms to tackle these challenging problems. Specifically, we propose two restart tabu search algorithms based on a generalized PUSH operator for MVWCP, a frequency driven local search algorithms for MsPlex, a graph reduction based tabu search as well as effective exact branch and bound algorithms for MBBP and lastly, a three phase local search algorithm for CPP. In addition to the design of efficient move operators for local search algorithms, we also integrate components like graph reduction or upper bound propagation in order to deal deal with very large real-life networks. The experimental tests on a wide range of instances show that our algorithms compete favorably with the main state-of-the-art algorithms.

Made available in DSpace on 2014-12-17T15:47:03Z (GMT). No. of bitstreams: 1 SilviaMDMM_TESE.pdf: 3010194 bytes, checksum: 43610ec3f0a30c2e5ef7fb5c0b2dc5b0 (MD5) Previous issue date: 2013-12-16 / Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior / The Quadratic Minimum Spanning Tree Problem (QMST) is a version of the Minimum Spanning Tree Problem in which, besides the traditional linear costs, there is a quadratic structure of costs. This quadratic structure models interaction effects between pairs of edges. Linear and quadratic costs are added up to constitute the total cost of the spanning tree, which must be minimized. When these interactions are restricted to adjacent edges, the problem is named Adjacent Only Quadratic Minimum Spanning Tree (AQMST). AQMST and QMST are NP-hard problems that model several problems of transport and distribution networks design. In general, AQMST arises as a more suitable model for real problems. Although, in literature, linear and quadratic costs are added, in real applications, they may be conflicting. In this case, it may be interesting to consider these costs separately. In this sense, Multiobjective Optimization provides a more realistic model for QMST and AQMST. A review of the state-of-the-art, so far, was not able to find papers regarding these problems under a biobjective point of view. Thus, the objective of this Thesis is the development of exact and heuristic algorithms for the Biobjective Adjacent Only Quadratic Spanning Tree Problem (bi-AQST). In order to do so, as theoretical foundation, other NP-hard problems directly related to bi-AQST are discussed: the QMST and AQMST problems. Bracktracking and branch-and-bound exact algorithms are proposed to the target problem of this investigation. The heuristic algorithms developed are: Pareto Local Search, Tabu Search with ejection chain, Transgenetic Algorithm, NSGA-II and a hybridization of the two last-mentioned proposals called NSTA. The proposed algorithms are compared to each other through performance analysis regarding computational experiments with instances adapted from the QMST literature. With regard to exact algorithms, the analysis considers, in particular, the execution time. In case of the heuristic algorithms, besides execution time, the quality of the generated approximation sets is evaluated. Quality indicators are used to assess such information. Appropriate statistical tools are used to measure the performance of exact and heuristic algorithms. Considering the set of instances adopted as well as the criteria of execution time and quality of the generated approximation set, the experiments showed that the Tabu Search with ejection chain approach obtained the best results and the transgenetic algorithm ranked second. The PLS algorithm obtained good quality solutions, but at a very high computational time compared to the other (meta)heuristics, getting the third place. NSTA and NSGA-II algorithms got the last positions / O problema da ?rvore Geradora M?nima Quadr?tica (AGMQ) ? uma vers?o do problema da ?rvore Geradora M?nima na qual se considera, al?m dos custos lineares tradicionais, uma estrutura de custos quadr?tica. Tal estrutura quadr?tica modela efeitos de intera??o entre pares de arestas. Os custos lineares e quadr?ticos s?o somados para compor o custo total da ?rvore geradora, que deve ser minimizado. Quando as intera??es s?o restritas ?s arestas adjacentes, o problema ? denominado ?rvore Geradora M?nima Quadr?tica em Adjac?ncia de Arestas (AGMQA). A AGMQA e a AGMQ s?o problemas NP-dif?ceis que modelam diversos problemas de projeto de redes de transporte e distribui??o. Em geral, a AGMQA emerge como um modelo mais apropriado para a modelagem de problemas reais. Embora, na literatura, os custos lineares e quadr?ticos sejam somados, em aplica??es reais, os custos linear e quadr?tico podem ser conflitantes. Neste caso, seria mais interessante considerar os custos separadamente. Neste sentido, a Otimiza??o Multiobjetivo prov? uma modelagem mais realista para os problemas da AGMQ e da AGMQA. Uma revis?o do estado da arte, at? o momento, n?o foi capaz de encontrar qualquer trabalho que investigue esses problemas sob um ponto de vista biobjetivo. O objetivo desta Tese ?, pois, o desenvolvimento de algoritmos exatos e heur?sticos para o Problema Biobjetivo da ?rvore Geradora Quadr?tica em Adjac?ncia de Arestas (AGQA-bi). Para tanto, como fundamenta??o te?rica, discutem-se outros problemas NP-dif?ceis diretamente relacionados ? AGQA-bi, a saber: AGMQ e AGMQA. Algoritmos exatos backtracking e branch-and-bound s?o propostos para o problema-alvo desta investiga??o. Os algoritmos heur?sticos desenvolvidos s?o: busca local Pareto Local Search, Busca Tabu com ejection chain, Algoritmo Transgen?tico, NSGA-II e uma hibridiza??o das duas ?ltimas propostas mencionadas denominada NSTA. Os algoritmos propostos s?o comparados entre si por meio da an?lise de seus desempenhos em experimentos computacionais com casos de teste adaptados da literatura da AGMQ. No que se refere aos algoritmos exatos, a an?lise considera, em especial, o tempo de execu??o. No caso dos algoritmos heur?sticos, al?m do tempo de execu??o, a qualidade do conjunto de aproxima??o gerado ? avaliada. Indicadores de qualidade s?o empregados para aferir tal informa??o. Ferramentas estat?sticas apropriadas s?o usadas na an?lise de desempenho dos algoritmos exatos e heur?sticos. Para o conjunto de inst?ncias utilizado e considerando os crit?rios de qualidade dos conjuntos de aproxima??o gerados e tempo de execu??o dos algoritmos, os experimentos mostraram que o algoritmo de Busca Tabu com ejection chain obteve melhores resultados e que o algoritmo transgen?tico ficou em segundo lugar. A busca local PLS obteve solu??es de qualidade, mas a um tempo computacional muito alto se comparado ?s outras (meta)heur?sticas. Nesse sentido, ocupa a terceira coloca??o. Por fim, ficaram os algoritmos NSTA e NSGAII

Made available in DSpace on 2014-12-17T15:48:01Z (GMT). No. of bitstreams: 1 PatriciaMLLD_DISSERT.pdf: 2062279 bytes, checksum: edf20f81d921e118846850abb8ec8a1d (MD5) Previous issue date: 2012-03-05 / The Multiobjective Spanning Tree Problem is NP-hard and models applications in several areas. This research presents an experimental analysis of different strategies used in the literature to develop exact algorithms to solve the problem. Initially, the algorithms are classified according to the approaches used to solve the problem. Features of two or more approaches can be found in some of those algorithms. The approaches investigated here are: the two-stage method, branch-and-bound, k-best and the preference-based approach. The main contribution of this research lies in the fact that no research was presented to date reporting a systematic experimental analysis of exact algorithms for the Multiobjective Spanning Tree Problem. Therefore, this work can be a basis for other research that deal with the same problem. The computational experiments compare the performance of algorithms regarding processing time, efficiency based on the number of objectives and number of solutions found in a controlled time interval. The analysis of the algorithms was performed for known instances of the problem, as well as instances obtained from a generator commonly used in the literature / O Problema da ?rvore Geradora Multiobjetivo ? NP-?rduo e modela aplica??es em diversas ?reas. Esta pesquisa apresenta uma an?lise experimental de diferentes estrat?gias utilizadas na literatura para desenvolver algoritmos exatos para resolver o problema. Inicialmente, os algoritmos s?o classificados de acordo com as abordagens utilizadas para resolver o problema. Caracter?sticas de duas ou mais abordagens podem ser encontradas em alguns desses algoritmos. As abordagens aqui investigadas s?o: o m?todo duas fases, branch-and-bound, k-best e a abordagem baseada em prefer?ncia. A principal contribui??o deste trabalho est? no fato de que nenhuma pesquisa desenvolvida at? o momento relata uma an?lise sistem?tica experimental de algoritmos exatos para o problema da ?rvore Geradora Multiobjetivo. Portanto, este trabalho pode ser uma base para outras pesquisas que lidam com o mesmo problema. Os experimentos computacionais comparam o desempenho de algoritmos em rela??o ao tempo de processamento, ? efici?ncia com base no n?mero de objetivos e no n?mero de solu??es encontradas em um intervalo de tempo controlado. A an?lise dos algoritmos foi realizada para inst?ncias conhecidas do problema, bem como para inst?ncias obtidas a partir de um gerador bastante utilizado na literatura