The inapproximability for NP-hard combinatorial optimization problems lies in the heart of theoretical computer science. A negative result can be either conditional, where the starting point is a complexity assumption, or unconditional, where the inapproximability holds for a restricted model of computation. Algorithms based on Linear Programming (LP) and Semidefinite Programming (SDP) relaxations are among the most prominent models of computation. The related and common measure of efficiency is the integrality gap, which sets the limitations of the associated algorithmic schemes. A number of systematic procedures, known as lift-and-project systems, have been proposed to improve the integrality gap of standard relaxations. These systems build strong hierarchies of either LP relaxations, such as the Lovasz-Schrijver (LS) and the Sherali-Adams (SA) systems, or SDP relaxations, such as the Lovasz-Schrijver SDP (LS+), the Sherali-Adams SDP (SA+) and the Lasserre (La) systems. In this thesis we prove integrality gap lower bounds for the aforementioned lift-and-project systems and for a number of combinatorial optimization problems, whose inapproximability is yet unresolved. Given that lift-and-project systems produce relaxations that have given the best algorithms known for a series of combinatorial problems, the lower bounds can be read as strong evidence of the inapproximability of the corresponding optimization problems. From the results found in the thesis we highlight the following: For every epsilon>0, the level-Omega(sqrt(log n/ log log n)) LS+ relaxation of the Vertex Cover polytope has integrality gap 2-epsilon. The integrality gap of the standard SDP for Vertex Cover remains 2-o(1) even if all hypermetric inequalities are added to the relaxation. The resulting relaxations are incomparable to the SDP relaxations derived by the LS+ system. Finally, the addition of all ell1 inequalities eliminates all solutions not in the integral hull. For every epsilon>0, the level-Omega(sqrt(log n/ log log n)) SA relaxation of Vertex Cover has integrality gap 2-epsilon. The integrality gap remains tight even for superconstant-level SA+ relaxations. We prove a tight lower bound for the number of tightenings that the SA system needs in order to prove the Pigeonhole Principle. We also prove sublinear and linear rank bounds for the La and SA systems respectively for the Tseitin tautology. Linear levels of the SA+ system treat highly unsatisfiable instances of fixed predicate-P constraint satisfaction problems over q-ary alphabets as fully satisfiable, when the satisfying assignments of the predicates P can be equipped with a balanced and pairwise independent distribution. We study the performance of the Lasserre system on the cut polytope. When the input is the complete graph on 2d+1 vertices, we show that the integrality gap is at least 1+1/(4d(d+1)) for the level-d SDP relaxation.

Made available in DSpace on 2014-12-17T15:48:07Z (GMT). No. of bitstreams: 1 CamilaNOT_DISSERT.pdf: 4321465 bytes, checksum: d103bd2da647997e8dfd0a8784c2060d (MD5) Previous issue date: 2013-02-01 / Nonogram is a logical puzzle whose associated decision problem is NP-complete. It has applications in pattern recognition problems and data compression, among others. The puzzle consists in determining an assignment of colors to pixels distributed in a N  M matrix that satisfies line and column constraints. A Nonogram is encoded by a vector whose elements specify the number of pixels in each row and column of a figure without specifying their coordinates. This work presents exact and heuristic approaches to solve Nonograms. The depth first search was one of the chosen exact approaches because it is a typical example of brute search algorithm that is easy to implement. Another implemented exact approach was based on the Las Vegas algorithm, so that we intend to investigate whether the randomness introduce by the Las Vegas-based algorithm would be an advantage over the depth first search. The Nonogram is also transformed into a Constraint Satisfaction Problem. Three heuristics approaches are proposed: a Tabu Search and two memetic algorithms. A new function to calculate the objective function is proposed. The approaches are applied on 234 instances, the size of the instances ranging from 5 x 5 to 100 x 100 size, and including logical and random Nonograms / O Nonograma ? um jogo l?gico cujo problema de decis?o associado ? NP-completo. Ele possui aplica??o em problemas de identifica??o de padr?es e de compacta??o de dados, dentre outros. O jogo consiste em determinar uma aloca??o de cores em pixels distribu?dos em uma matriz N  M atendendo restri??es em linhas e colunas. Um Nonograma ? codificado atrav?s de vetores cujos elementos especificam o n?mero de pixels existentes em cada coluna e linha de uma figura, sem especificar suas coordenadas. Este trabalho apresenta abordagens exatas e heur?sticas para solucionar o Nonograma. A Busca em Profundidade foi uma das abordagens exatas escolhida, por ser um exemplo t?pico de algoritmo de for?a bruta de f?cil implementa??o. Outra abordagem exata implementada foi baseada no algoritmo Las Vegas, atrav?s do qual se pretende investigar se a aleatoriedade introduzida pelo algoritmo Las Vegas traria algum benef?cio em rela??o ? Busca em Profundidade. O Nonograma tamb?m ? transformado em um Problema de Satisfa??o de Restri??es. Tr?s abordagens heur?sticas s?o propostas: uma Busca Tabu e dois algoritmos Mem?tico. Uma nova abordagem para o c?lculo da fun??o objetivo ? proposta neste trabalho. As abordagens s?o testadas em 234 casos de teste de tamanho entre 5 x 5 e 100 x 100, incluindo Nonogramas l?gicos e aleat?rios

This Thesis is focused on the arranging of the melody, which is accompanied by jazz chords. It deals with creating a more harmonious voices using Block Voicing method. Distribution to target notes and passing notes is made using techniques of constraint programming (CSP). Passing notes are reharmonized by dominant seventh chord or by parallel chord. Using CSP a bass part is also created. To solve CSP is used Gecode library. The harmonious voices are arranged by Four Part Close Voicing. The application result is a tool for the music arranger.

First, basic aspects of interval analysis, roles of intervals and their applications are addressed. Then, various classes of interval matrices are described and their relations are depicted. This material forms a prelude to the unifying theme of the rest of the work - solving interval linear systems. Several methods for enclosing the solution set of square and overdetermined interval linear systems are covered and compared. For square systems the new shaving method is introduced, for overdetermined systems the new subsquares approach is introduced. Detecting unsolvability and solvability of such systems is discussed and several polynomial conditions are compared. Two strongest condi- tions are proved to be equivalent under certain assumption. Solving of interval linear systems is used to approach other problems in the rest of the work. Computing enclosures of determinants of interval matrices is addressed. NP- hardness of both relative and absolute approximation is proved. New method based on solving square interval linear systems and Cramer's rule is designed. Various classes of matrices with polynomially computable bounds on determinant are characterized. Solving of interval linear systems is also used to compute the least squares linear and nonlinear interval regression. It is then applied to real...

The modal satisfiability problem has to date been solved using either a specifically designed algorithm, or by translating the modal logic formula into a different class of problem, such as a first-order logic, a propositional satisfiability problem or a constraint satisfaction problem. These approaches and the solvers developed to support them are surveyed and a synthesis thereof is presented. The translation of a modal K formula into a constraint satisfaction problem, as developed by Brand et al. [18], is further enhanced. The modal formula, which must be in conjunctive normal form, is translated into layered propositional formulae. Each of these layers is translated into a constraint satisfaction problem and solved using the constraint solver ECLiPSe. I extend this translation to deal with reflexive and transitive accessibility relations, thereby providing for the modal logics KT and S4. Two of the difficulties that arise when these accessibility relations are added are that the resultant formula increases considerably in complexity, and that it is no longer in conjunctive normal form (CNF). I eliminate the need for the conversion of the formula to CNF and deal instead with formulae that are in negation normal form (NNF). I apply a number of enhancements to the formula at each modal layer before it is translated into a constraint satisfaction problem. These include extensive simplification, the assignment of a single value to propositional variables that occur only positively or only negatively, and caching the status of the formula at each node of the search tree. All of these significantly prune the search space. The final results I achieve compare favorably with those obtained by other solvers. / Computing / M.Sc. (Computer Science)

The modal satisfiability problem has to date been solved using either a specifically designed algorithm, or by translating the modal logic formula into a different class of problem, such as a first-order logic, a propositional satisfiability problem or a constraint satisfaction problem. These approaches and the solvers developed to support them are surveyed and a synthesis thereof is presented. The translation of a modal K formula into a constraint satisfaction problem, as developed by Brand et al. [18], is further enhanced. The modal formula, which must be in conjunctive normal form, is translated into layered propositional formulae. Each of these layers is translated into a constraint satisfaction problem and solved using the constraint solver ECLiPSe. I extend this translation to deal with reflexive and transitive accessibility relations, thereby providing for the modal logics KT and S4. Two of the difficulties that arise when these accessibility relations are added are that the resultant formula increases considerably in complexity, and that it is no longer in conjunctive normal form (CNF). I eliminate the need for the conversion of the formula to CNF and deal instead with formulae that are in negation normal form (NNF). I apply a number of enhancements to the formula at each modal layer before it is translated into a constraint satisfaction problem. These include extensive simplification, the assignment of a single value to propositional variables that occur only positively or only negatively, and caching the status of the formula at each node of the search tree. All of these significantly prune the search space. The final results I achieve compare favorably with those obtained by other solvers. / Computing / M.Sc. (Computer Science)

Le test à base de modèles des systèmes réactifs est le processus de vérifier si un système sous test (SUT) est conforme à sa spécification. Il consiste à gérer à la fois la génération des données de test et le calcul de verdicts en utilisant des modèles. Nous spécifions le comportement des systèmes réactifs à l'aide des systèmes de transitions symboliques temporisées à entrée-sortie (TIOSTS). Quand les TIOSTSs sont utilisés pour tester des systèmes avec une interface centralisée, l'utilisateur peut ordonner complètement les événements (i.e., les entrées envoyées au système et les sorties produites). Les interactions entre le testeur et le SUT consistent en des séquences d'entrées et de sortie nommées traces, pouvant être séparées par des durées dans le cadre du test temporisé, pour former ce que l'on appelle des traces temporisées. Les systèmes distribués sont des collections de composants locaux communiquant entre eux et interagissant avec leur environnement via des interfaces physiquement distribuées. Différents événements survenant à ces différentes interfaces ne peuvent plus être ordonnés. Cette thèse concerne le test de conformité des systèmes distribués où un testeur est placé à chaque interface localisée et peut observer ce qui se passe à cette interface. Nous supposons qu'il n'y a pas d’horloge commune mais seulement des horloges locales pour chaque interface. La sémantique de tels systèmes est définie comme des tuples de traces temporisées. Nous considérons une approche du test dans le contexte de la relation de conformité distribuée dtioco. La conformité globale peut être testée dans une architecture de test en utilisant des testeurs locaux sans communication entre eux. Nous proposons un algorithme pour vérifier la communication pour un tuple de traces temporisées en formulant le problème de message-passing en un problème de satisfaction de contraintes (CSP). Nous avons mis en œuvre le calcul des verdicts de test en orchestrant à la fois les algorithmes du test centralisé off-line de chacun des composants et la vérification des communications par le biais d'un solveur de contraintes. Nous avons validé notre approche sur un cas étude de taille significative. / Model-based testing of reactive systems is the process of checking if a System Under Test (SUT) conforms to its model. It consists of handling both test data generation and verdict computation by using models. We specify the behaviour of reactive systems using Timed Input Output Symbolic Transition Systems (TIOSTS) that are timed automata enriched with symbolic mechanisms to handle data. When TIOSTSs are used to test systems with a centralized interface, the user may completely order events occurring at this interface (i.e., inputs sent to the system and outputs produced from it). Interactions between the tester and the SUT are sequences of inputs and outputs named traces, separated by delays in the timed framework, to form so-called timed traces. Distributed systems are collections of communicating local components which interact with their environment at physically distributed interfaces. Interacting with such a distributed system requires exchanging values with it by means of several interfaces in the same testing process. Different events occurring at different interfaces cannot be ordered any more. This thesis focuses on conformance testing for distributed systems where a separate tester is placed at each localized interface and may only observe what happens at this interface. We assume that there is no global clock but only local clocks for each localized interface. The semantics of such systems can be seen as tuples of timed traces. We consider a framework for distributed testing from TIOSTS along with corresponding test hypotheses and a distributed conformance relation called dtioco. Global conformance can be tested in a distributed testing architecture using only local testers without any communication between them. We propose an algorithm to check communication policy for a tuple of timed traces by formulating the verification of message passing in terms of Constraint Satisfaction Problem (CSP). Hence, we were able to implement the computation of test verdicts by orchestrating both localised off-line testing algorithms and the verification of constraints defined by message passing that can be supported by a constraint solver. Lastly, we validated our approach on a real case study of a telecommunications distributed system.