<p>In this thesis a method for solving the Generalized Assignment Problem (GAP) is described. It is based on a reformulation of the original problem into a Set Partitioning Problem (SPP), in which the columns represent partial solutions to the original problem. For solving this problem, column generation, with systematic overgeneration of columns, is used. Conditions that guarantee that an optimal solution to a restricted SPP is optimal also in the original problem are given. In order to satisfy these conditions, not only columns with the most negative Lagrangian reduced costs need to be generated, but also others; this observation leads to the use of overgeneration of columns.</p><p>The Generalized Assignment Problem has shown to be NP-hard and therefore efficient algorithms are needed, especially for large problems. The application of the proposed method decomposes GAP into several knapsack problems via Lagrangian relaxation, and enumerates solutions to each of these problems. The solutions obtained from the knapsack problems form a Set Partitioning Problem, which consists of combining one solution from each knapsack problem to obtain a solution to the original problem. The algorithm has been tested on problems with 10 agents and 60 jobs. This leads to 10 knapsack problems, each with 60 variables.</p>

In this thesis a method for solving the Generalized Assignment Problem (GAP) is described. It is based on a reformulation of the original problem into a Set Partitioning Problem (SPP), in which the columns represent partial solutions to the original problem. For solving this problem, column generation, with systematic overgeneration of columns, is used. Conditions that guarantee that an optimal solution to a restricted SPP is optimal also in the original problem are given. In order to satisfy these conditions, not only columns with the most negative Lagrangian reduced costs need to be generated, but also others; this observation leads to the use of overgeneration of columns. The Generalized Assignment Problem has shown to be NP-hard and therefore efficient algorithms are needed, especially for large problems. The application of the proposed method decomposes GAP into several knapsack problems via Lagrangian relaxation, and enumerates solutions to each of these problems. The solutions obtained from the knapsack problems form a Set Partitioning Problem, which consists of combining one solution from each knapsack problem to obtain a solution to the original problem. The algorithm has been tested on problems with 10 agents and 60 jobs. This leads to 10 knapsack problems, each with 60 variables.

Dans cette thèse nous proposons de nouveaux algorithmes d'analyse vidéo. La première contribution de cette thèse concerne le domaine de la segmentation de vidéos avec pour objectif d'obtenir une segmentation dense et spatio-temporellement cohérente. Nous proposons de combiner les aspects spatiaux et temporels d'une vidéo en une seule notion, celle de Fibre. Une fibre est un ensemble de trajectoires qui sont spatialement connectées par un maillage. Les fibres sont construites en évaluant simultanément les aspects spatiaux et temporels. Par rapport a l’état de l'art une segmentation de vidéo a base de fibres présente comme avantages d’accéder naturellement au voisinage grâce au maillage et aux correspondances temporelles pour la plupart des pixels de la vidéo. De plus, cette segmentation à base de fibres a une complexité quasi linéaire par rapport au nombre de pixels. La deuxième contribution de cette thèse concerne le suivi d'objets multiples. Nous proposons une approche de suivi qui utilise des caractéristiques des points suivis, la cinématique des objets suivis et l'apparence globale des détections. L'unification de toutes ces caractéristiques est effectuée avec un champ conditionnel aléatoire. Ensuite ce modèle est optimisé en combinant les techniques de passage de message et une variante de processus ICM (Iterated Conditional Modes) pour inférer les trajectoires d'objet. Une troisième contribution mineure consiste dans le développement d'un descripteur pour la mise en correspondance d'apparences de personne. Toutes les approches proposées obtiennent des résultats compétitifs ou meilleurs (qualitativement et quantitativement) que l’état de l'art sur des base de données. / In this thesis we propose novel algorithms for video analysis. The first contribution of this thesis is in the domain of video segmentation wherein the objective is to obtain a dense and coherent spatio-temporal segmentation. We propose joining both spatial and temporal aspects of a video into a single notion Fiber. A fiber is a set of trajectories which are spatially connected by a mesh. Fibers are built by jointly assessing spatial and temporal aspects of the video. Compared to the state-of-the-art, a fiber based video segmentation presents advantages such as a natural spatio-temporal neighborhood accessor by a mesh, and temporal correspondences for most pixels in the video. Furthermore, this fiber-based segmentation is of quasi-linear complexity w.r.t. the number of pixels. The second contribution is in the realm of multiple object tracking. We proposed a tracking approach which utilizes cues from point tracks, kinematics of moving objects and global appearance of detections. Unification of all these cues is performed on a Conditional Random Field. Subsequently this model is optimized by a combination of message passing and an Iterated Conditional Modes (ICM) variant to infer object-trajectories. A third, minor, contribution relates to the development of suitable feature descriptor for appearance matching of persons. All of our proposed approaches achieve competitive and better results (both qualitatively and quantitatively) than state-of-the-art on open source datasets.

Inspired by a planar partitioning problem involving multiple unbounded chambers, this thesis investigates using classical techniques what can be said of the existence, uniqueness, and regularity of minimizers in a certain free-endpoint isoperimetric problem. In two cases, a full existence-uniqueness-regularity result is proved using a convexity technique inspired by work of Talenti. The problem studied here can be interpreted physically as the identification of the equilibrium shape of a sessile liquid drop in half-space (in the absence of gravity). This is a well-studied variational problem whose full resolution requires the use of geometric measure theory, in particular the theory of sets of finite perimeter. A crash course on the theory required for the modern statement of the equilibrium shape theorem is presented in an appendix. / Thesis / Master of Science (MSc)

Nesta tese apresentamos a abordagem da Mecânica Estatística para o clássico problema de otimização denominado problema da partição numérica (PPN), que é definido como: Dada uma seqüência de N números reais positivos {a1, a2, a3,....aN}, o problema consiste em particioná-los em dois conjuntos complementares, A e Ac, tais que o valor absoluto da diferença da soma dos ais nos dois conjuntos seja minimizada. No caso em que os aj\'s são variáveis aleatórias estatisticamente independentes distribuídas uniformemente no intervalo unitário, este problema NP-completo equivale ao problema de encontrar o estado fundamental de um modelo de Ising antiferromagnético aleatório de alcance infinito. Conseqüentemente, a análise probabilística do PPN pode ser realizada com as ferramentas da Mecânica Estatística de sistemas desordenados. Neste trabalho empregamos a aproximação recozida (annealed) para derivar uma expressão analítica para o limitante inferior do valor médio da diferença para partições tanto com vínculo de cardinalidade quanto sem vínculo para grandes valores de N. Além disso, calculamos analiticamente a fração de estados metaestáveis, isto é, estados que possuem a menor energia mediante todos os vizinhos (estados que diferem pela troca de um único spin). Concluímos a análise da abordagem direta, cujas instâncias . / In this thesis we present a statistical mechanics approach to a classical optimization problem called the number partitioning problem (NPP), which is stated as follows. Given a sequence of N positive real numbers , the number partitioning problem consists of partitioning them into two sets A and its complementary set Ac such that the absolute value of the difference of the sums of aj over the two sets is minimized. In each case in which the aj\'s are statistically independent random variables uniformly distributed in the unit interval, this NP-complete problem is equivalent to the problem of finding the ground state of an infinite range, random antiferromagnetic Ising model. Hence the probabilistic analysis of the NPP can be carried out within the framework of the standard statistical mechanics of disordered systems. In this vein we employ the annealed approximation to derive analytical lower bounds to the average value of the difference for the best-constrained and unconstrained partitions in the large N limit. Furthermore, we calculate analytically the fraction of metastable states, i.e. states that are stable against all single spin flips. We conclude the analysis of the so-called direct approach, in which the instances {ai} are fixed and the partitions are variable, with the analytical study of the linear programming relaxation of this NP-complete integer programming. In the second part of this thesis we propose and explore an inverse approach to the NPP, in which the optimal partitions are fixed and the instances are variable. Specifically, using the replica framework we study analytically the instance space of the number partitioning problem. We show that, regardless of the distribution of the instance entries, there is an upper bound &#945cN to the number of perfect random partitions (i.e. partitions for which that difference is zero). In particular, in the case where the two sets have the same cardinality (balanced partitions) we find &#945c =1/2. Moreover, in the case of unbalanced partitions, we show that perfect random partitions exist only if the difference between the cardinalities of the two sets scales like m N-1/2}.

Nesta tese apresentamos a abordagem da Mecânica Estatística para o clássico problema de otimização denominado problema da partição numérica (PPN), que é definido como: Dada uma seqüência de N números reais positivos {a1, a2, a3,....aN}, o problema consiste em particioná-los em dois conjuntos complementares, A e Ac, tais que o valor absoluto da diferença da soma dos ais nos dois conjuntos seja minimizada. No caso em que os aj\'s são variáveis aleatórias estatisticamente independentes distribuídas uniformemente no intervalo unitário, este problema NP-completo equivale ao problema de encontrar o estado fundamental de um modelo de Ising antiferromagnético aleatório de alcance infinito. Conseqüentemente, a análise probabilística do PPN pode ser realizada com as ferramentas da Mecânica Estatística de sistemas desordenados. Neste trabalho empregamos a aproximação recozida (annealed) para derivar uma expressão analítica para o limitante inferior do valor médio da diferença para partições tanto com vínculo de cardinalidade quanto sem vínculo para grandes valores de N. Além disso, calculamos analiticamente a fração de estados metaestáveis, isto é, estados que possuem a menor energia mediante todos os vizinhos (estados que diferem pela troca de um único spin). Concluímos a análise da abordagem direta, cujas instâncias . / In this thesis we present a statistical mechanics approach to a classical optimization problem called the number partitioning problem (NPP), which is stated as follows. Given a sequence of N positive real numbers , the number partitioning problem consists of partitioning them into two sets A and its complementary set Ac such that the absolute value of the difference of the sums of aj over the two sets is minimized. In each case in which the aj\'s are statistically independent random variables uniformly distributed in the unit interval, this NP-complete problem is equivalent to the problem of finding the ground state of an infinite range, random antiferromagnetic Ising model. Hence the probabilistic analysis of the NPP can be carried out within the framework of the standard statistical mechanics of disordered systems. In this vein we employ the annealed approximation to derive analytical lower bounds to the average value of the difference for the best-constrained and unconstrained partitions in the large N limit. Furthermore, we calculate analytically the fraction of metastable states, i.e. states that are stable against all single spin flips. We conclude the analysis of the so-called direct approach, in which the instances {ai} are fixed and the partitions are variable, with the analytical study of the linear programming relaxation of this NP-complete integer programming. In the second part of this thesis we propose and explore an inverse approach to the NPP, in which the optimal partitions are fixed and the instances are variable. Specifically, using the replica framework we study analytically the instance space of the number partitioning problem. We show that, regardless of the distribution of the instance entries, there is an upper bound &#945cN to the number of perfect random partitions (i.e. partitions for which that difference is zero). In particular, in the case where the two sets have the same cardinality (balanced partitions) we find &#945c =1/2. Moreover, in the case of unbalanced partitions, we show that perfect random partitions exist only if the difference between the cardinalities of the two sets scales like m N-1/2}.