Ce manuscrit aborde des problèmes d’ordonnancement et de transport avec une modélisation explicite du transport. De tels problèmes se modélisent communément sous forme de graphes qui sont évalués afin d’obtenir les dates de début des opérations.Les évaluations classiques des graphes sont effectuées au moyen d’algorithmes de plus long chemin permettant d’obtenir une solution semi-active, où toutes les dates des opérations sont au plus tôt. Néanmoins, ces évaluations permettent généralement de ne prendre en compte que des critères de temps ou de distance à minimiser. Les travaux présentés dans ce manuscrit proposent de tenir compte de critères de qualité de service dans la fonction objectif. Cette prise en considération nécessite de nouvelles fonctions d’évaluation du graphe afin d’obtenir des solutions non nécessairement semi-actives permettant de maximiser la qualité de service. En effet, une solution semi-active propose rarement une qualité de service optimale. Les critères de qualité de service adoptés portent sur les ordonnancements et sur le transport.Trois problèmes intégrés sont successivement traités. Le premier problème est un problème de Job-shop avec transport et qualité de service, appelé Job-shop Scheduling Problem with Routing (JSPR). Des pièces, définies par une succession d’opérations, sont à fabriquer sur différentes machines, et entre deux opérations, la pièce doit être transportée de machine en machine. Le critère de qualité de service dans ce problème est dépendant des délais entre, d’une part les différentes opérations sur les machines, et d’autre part entre les différentes opérations de transport. Les gammes opératoires et les opérations de transport sont dépendantes les unes des autres.Le second problème est un problème de Workforce Scheduling and Routing Problem (WSRP), assimilable à un problème de planification de visites à domicile par un ensemble d’employés, et où le transport est pris en compte. Pour ce problème, le critère de qualité de service dépend des dates de début des visites. Les tournées sont indépendantes les unes des autres.Le troisième problème est le Generalised Workforce Scheduling and Routing Problem (GWSRP), qui prend en compte des contraintes de coordination entre les employés. Les tournées de ces derniers sont dépendantes les unes des autres. Elles nécessitent d’être toutes considérées simultanément pour évaluer les dates des visites respectant les contraintes de coordination et maximisant la qualité de service.Pour chaque problème, une nouvelle fonction d’évaluation est proposée. Pour le JSPR, cette fonction est basée sur l’algorithme de (Cordeau and Laporte, 2003) qui est initialement prévu pour le Dial-A-Ride Problem, ainsi que sur l’insertion de time-lags dans le graphe disjonctif du JSPR. Cette évaluation est incluse dans une métaheuristique. Pour le WSRP, la fonction d’évaluation est basée sur un algorithme de calcul du plus court chemin avec un algorithme de type programmation dynamique à labels. Elle est généralisée pour être utilisée dans une génération de colonnes. Et enfin, pour le GWSRP, l’évaluation est effectuée par un modèle PPC qui combiné à une génération de colonnes définissent tous deux un schéma d’optimisation global. / This manuscript addresses scheduling and transport problems where the transport is explicitly taken into account. Such problems are commonly modelled by graphs that are evaluated to obtain the starting times of operations.Classic graph evaluations are performed using longer path algorithms to obtain a semi-active solution, where all operations are left shifted. Nevertheless, these evaluations generally allow only time or distance criteria to be taken into account. The work presented in this thesis propose to take the quality of service criteria into account in the objective function. These considerations require new graph evaluation functions in order to obtain non-semi-active solutions that maximise the quality of service. Indeed, a semi-active solution rarely offers maximum quality of service. Three integrated problems are successively addressed. The first problem is a Job-shop Scheduling Problem with transport and quality of service, referred to as Job-shop Scheduling Problem with Routing (JSPR). Jobs, defined by a succession of operations, are to be performed on different machines, and between two operations, the job must be transported from a machine to another machine. The quality of service criterion in this problem depends on the delay between, on the one hand, the different operations belonging to the same job, and on the other hand, between the different transport operations. Machine-operations and transport-operations are dependent.The second problem is a Workforce Scheduling and Routing Problem (WSRP), which is similar to a problem of planning home services by a set of employees, and where transport is taken into account. For this problem, the quality of service criterion depends on the starting times of the visits. The trips of employees are independent.The third problem is the Generalised Workforce Scheduling and Routing Problem (GWSRP), which takes coordination constraints between employees into account. The trips are dependent on each other. The evaluation function of the starting times must consider simultaneously all trips in order to respect all coordination constraints and to maximise the service quality.For each problem, a new evaluation function is proposed. For the JSPR, this function is based on the algorithm of (Cordeau and Laporte, 2003) which is introduced first for the Dial-A-Ride Problem. The evaluation function, for the JSPR, is based on the insertion of time-lags in the disjunctive graph. This evaluation is included in a metaheuristic. For the WSRP, the evaluation function is based on the dynamic labelling algorithm used for an Elementary Shortest Path Problem With Resource Constraints. This function is generalised in order to be included in a column generation scheme. Finally, for the GWSRP, the evaluation is performed by a PPC model combined with a generation of columns and both define an overall optimisation scheme.

Scheduling allocates scarce resources to activities such that certain constraints are satisfied and specific objectives are optimized. The activities to be executed are com- monly known or determined a priori in the planning stage. To improve the flexibility of scheduling systems, the concept of optional activities was invented. Optional activities are those activities whose presence in the resulting schedule is to be decided. Rather than determining which activities need to be executed and scheduling them in two consecu- tive phases, flexibility and efficiency can be improved significantly when both activity selection and time allocation are integrated within the same solver. Such an approach was implemented in a few Constraint Programming solvers and manifested great perfor- mance on multiple scheduling problems. In this thesis, we apply the concept of optional activities to scheduling problems that do not seem to involve optional activities, such as the production scheduling problem with sequence-dependent non-overlapping setups, but also on problems beyond the scheduling domain, such as the multi-agent path finding problem and its extension with weighted and capacitated arcs. 1

Les problèmes de séquencement et d'ordonnancement forment une famille de problèmes combinatoires qui implique la programmation dans le temps d'un ensemble d'opérations soumises à des contraintes de capacités et de ressources. Nous contribuons dans cette thèse à la résolution de ces problèmes dans un contexte de satisfaction de contraintes et d'optimisation combinatoire. Nos propositions concernent trois aspects différents: comment choisir le prochain nœud à explorer (recherche)? comment réduire efficacement l'espace de recherche (propagation)? et que peut-on apprendre des échecs rencontrés lors de la recherche (apprentissage)? Nos contributions commencent par une étude approfondie des heuristiques de branchement pour le problème de séquencement de chaîne d'assemblage de voitures. Cette évaluation montre d'abord les paramètres clés de ce qui constitue une bonne heuristique pour ce problème. De plus, elle montre que la stratégie de branchement est aussi importante que la méthode de propagation. Deuxièmement, nous étudions les mécanismes de propagation pour une classe de contraintes de séquencement à travers la conception de plusieurs algorithmes de filtrage. En particulier, nous proposons un algorithme de filtrage complet pour un type de contrainte de séquence avec une complexité temporelle optimale dans le pire cas. Troisièmement, nous investiguons l'impact de l'apprentissage de clauses pour résoudre le problème de séquencement de véhicules à travers une nouvelle stratégie d'explication réduite pour le nouveau filtrage. L'évaluation expérimentale montre l'importance de l'apprentissage de clauses pour ce problème. Ensuite, nous proposons une alternative pour la génération retardée de variables booléennes pour encoder les domaines. Finalement, nous revisitons les algorithmes d'analyse de conflits pour résoudre les problèmes d'ordonnancement disjonctifs. En particulier, nous introduisons une nouvelle procédure d'analyse de conflits dédiée pour cette famille de problèmes. La nouvelle méthode diffère des algorithmes traditionnels par l'apprentissage de clauses portant uniquement sur les variables booléennes de disjonctions. Enfin, nous présentons les résultats d'une large étude expérimentale qui démontre l'impact de ces nouveaux mécanismes d'apprentissage. En particulier, la nouvelle méthode d'analyse de conflits a permis de découvrir de nouvelle bornes inférieures pour des instances largement étudiées de la littérature / Sequencing and scheduling involve the organization in time of operations subject to capacity and resource constraints. We propose in this dissertation several improvements to the constraint satisfaction and combinatorial optimization methods for solving these problems. These contributions concern three different aspects: how to choose the next node to explore (search)? how much, and how efficiently, one can reduce the search space (propagation)? and what can be learnt from previous failures (learning)? Our contributions start with an empirical study of search heuristics for the well known car-sequencing problem. This evaluation characterizes the key aspects of a good heuristic and shows that the search strategy is as important as the propagation aspect in this problem. Second, we carefully investigate the propagation aspect in a class of sequencing problems. In particular, we propose an algorithm for filtering a type of sequence constraints which worst case time complexity is lower than the best known upper bound, and indeed optimal. Third, we investigate the impact of clause learning for solving the car-sequencing problem. In particular, we propose reduced explanations for the new filtering. The experimental evaluation shows compelling evidence supporting the importance of clause learning for solving efficiently this problem. Next, we revisit the general approach of lazy generation for the Boolean variables encoding the domains. Our proposition avoids a classical redundancy issue without computational overhead. Finally, we investigate conflict analysis algorithms for solving disjunctive scheduling problems. In particular, we introduce a novel learning procedure tailored to this family of problems. The new conflict analysis differs from conventional methods by learning clauses whose size are not function of the scheduling horizon. Our comprehensive experimental study with traditional academic benchmarks demonstrates the impact of the novel learning scheme that we propose. In particular, we find new lower bounds for a well known scheduling benchmark

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NP-hard problems of higher-dimensional orthogonal packing are considered. We look closer at their logical structure and show that they can be decomposed into problems of a smaller dimension with a special contiguous structure. This decomposition influences the modeling of the packing process, which results in three new solution approaches. Keeping this decomposition in mind, we model the smaller-dimensional problems in a single position-indexed formulation with non-overlapping inequalities serving as binding constraints. Thus, we come up with a new integer linear programming model, which we subject to polyhedral analysis. Furthermore, we establish general non-overlapping and density inequalities and prove under appropriate assumptions their facet-defining property for the convex hull of the integer solutions. Based on the proposed model and the strong inequalities, we develop a new branch-and-cut algorithm. Being a relaxation of the higher-dimensional problem, each of the smaller-dimensional problems is also relevant for different areas, e.g. for scheduling. To tackle any of these smaller-dimensional problems, we use a Gilmore-Gomory model, which is a Dantzig-Wolfe decomposition of the position-indexed formulation. In order to obtain a contiguous structure for the optimal solution, its basis matrix must have a consecutive 1's property. For construction of such matrices, we develop new branch-and-price algorithms which are distinguished by various strategies for the enumeration of partial solutions. We also prove some characteristics of partial solutions, which tighten the slave problem of column generation. For a nonlinear modeling of the higher-dimensional packing problems, we investigate state-of-the-art constraint programming approaches, modify them, and propose new dichotomy and intersection branching strategies. To tighten the constraint propagation, we introduce new pruning rules. For that, we apply 1D relaxation with intervals and forbidden pairs, an advanced bar relaxation, 2D slice relaxation, and 1D slice-bar relaxation with forbidden pairs. The new rules are based on the relaxation by the smaller-dimensional problems which, in turn, are replaced by a linear programming relaxation of the Gilmore-Gomory model. We conclude with a discussion of implementation issues and numerical studies of all proposed approaches. / Es werden NP-schwere höherdimensionale orthogonale Packungsprobleme betrachtet. Wir untersuchen ihre logische Struktur genauer und zeigen, dass sie sich in Probleme kleinerer Dimension mit einer speziellen Nachbarschaftsstruktur zerlegen lassen. Dies beeinflusst die Modellierung des Packungsprozesses, die ihreseits zu drei neuen Lösungsansätzen führt. Unter Beachtung dieser Zerlegung modellieren wir die Probleme kleinerer Dimension in einer einzigen positionsindizierten Formulierung mit Nichtüberlappungsungleichungen, die als Bindungsbedingungen dienen. Damit entwickeln wir ein neues Modell der ganzzahligen linearen Optimierung und unterziehen dies einer Polyederanalyse. Weiterhin geben wir allgemeine Nichtüberlappungs- und Dichtheitsungleichungen an und beweisen unter geeigneten Annahmen ihre facettendefinierende Eigenschaft für die konvexe Hülle der ganzzahligen Lösungen. Basierend auf dem vorgeschlagenen Modell und den starken Ungleichungen entwickeln wir einen neuen Branch-and-Cut-Algorithmus. Jedes Problem kleinerer Dimension ist eine Relaxation des höherdimensionalen Problems. Darüber hinaus besitzt es Anwendungen in verschiedenen Bereichen, wie zum Beispiel im Scheduling. Für die Behandlung der Probleme kleinerer Dimension setzen wir das Gilmore-Gomory-Modell ein, das eine Dantzig-Wolfe-Dekomposition der positionsindizierten Formulierung ist. Um eine Nachbarschaftsstruktur zu erhalten, muss die Basismatrix der optimalen Lösung die consecutive-1’s-Eigenschaft erfüllen. Für die Konstruktion solcher Matrizen entwickeln wir neue Branch-and-Price-Algorithmen, die sich durch Strategien zur Enumeration von partiellen Lösungen unterscheiden. Wir beweisen auch einige Charakteristiken von partiellen Lösungen, die das Hilfsproblem der Spaltengenerierung verschärfen. Für die nichtlineare Modellierung der höherdimensionalen Packungsprobleme untersuchen wir moderne Ansätze des Constraint Programming, modifizieren diese und schlagen neue Dichotomie- und Überschneidungsstrategien für die Verzweigung vor. Für die Verstärkung der Constraint Propagation stellen wir neue Ablehnungskriterien vor. Wir nutzen dabei 1D Relaxationen mit Intervallen und verbotenen Paaren, erweiterte Streifen-Relaxation, 2D Scheiben-Relaxation und 1D Scheiben-Streifen-Relaxation mit verbotenen Paaren. Alle vorgestellten Kriterien basieren auf Relaxationen durch Probleme kleinerer Dimension, die wir weiter durch die LP-Relaxation des Gilmore-Gomory-Modells abschwächen. Wir schließen mit Umsetzungsfragen und numerischen Experimenten aller vorgeschlagenen Ansätze.

NP-hard problems of higher-dimensional orthogonal packing are considered. We look closer at their logical structure and show that they can be decomposed into problems of a smaller dimension with a special contiguous structure. This decomposition influences the modeling of the packing process, which results in three new solution approaches. Keeping this decomposition in mind, we model the smaller-dimensional problems in a single position-indexed formulation with non-overlapping inequalities serving as binding constraints. Thus, we come up with a new integer linear programming model, which we subject to polyhedral analysis. Furthermore, we establish general non-overlapping and density inequalities and prove under appropriate assumptions their facet-defining property for the convex hull of the integer solutions. Based on the proposed model and the strong inequalities, we develop a new branch-and-cut algorithm. Being a relaxation of the higher-dimensional problem, each of the smaller-dimensional problems is also relevant for different areas, e.g. for scheduling. To tackle any of these smaller-dimensional problems, we use a Gilmore-Gomory model, which is a Dantzig-Wolfe decomposition of the position-indexed formulation. In order to obtain a contiguous structure for the optimal solution, its basis matrix must have a consecutive 1's property. For construction of such matrices, we develop new branch-and-price algorithms which are distinguished by various strategies for the enumeration of partial solutions. We also prove some characteristics of partial solutions, which tighten the slave problem of column generation. For a nonlinear modeling of the higher-dimensional packing problems, we investigate state-of-the-art constraint programming approaches, modify them, and propose new dichotomy and intersection branching strategies. To tighten the constraint propagation, we introduce new pruning rules. For that, we apply 1D relaxation with intervals and forbidden pairs, an advanced bar relaxation, 2D slice relaxation, and 1D slice-bar relaxation with forbidden pairs. The new rules are based on the relaxation by the smaller-dimensional problems which, in turn, are replaced by a linear programming relaxation of the Gilmore-Gomory model. We conclude with a discussion of implementation issues and numerical studies of all proposed approaches. / Es werden NP-schwere höherdimensionale orthogonale Packungsprobleme betrachtet. Wir untersuchen ihre logische Struktur genauer und zeigen, dass sie sich in Probleme kleinerer Dimension mit einer speziellen Nachbarschaftsstruktur zerlegen lassen. Dies beeinflusst die Modellierung des Packungsprozesses, die ihreseits zu drei neuen Lösungsansätzen führt. Unter Beachtung dieser Zerlegung modellieren wir die Probleme kleinerer Dimension in einer einzigen positionsindizierten Formulierung mit Nichtüberlappungsungleichungen, die als Bindungsbedingungen dienen. Damit entwickeln wir ein neues Modell der ganzzahligen linearen Optimierung und unterziehen dies einer Polyederanalyse. Weiterhin geben wir allgemeine Nichtüberlappungs- und Dichtheitsungleichungen an und beweisen unter geeigneten Annahmen ihre facettendefinierende Eigenschaft für die konvexe Hülle der ganzzahligen Lösungen. Basierend auf dem vorgeschlagenen Modell und den starken Ungleichungen entwickeln wir einen neuen Branch-and-Cut-Algorithmus. Jedes Problem kleinerer Dimension ist eine Relaxation des höherdimensionalen Problems. Darüber hinaus besitzt es Anwendungen in verschiedenen Bereichen, wie zum Beispiel im Scheduling. Für die Behandlung der Probleme kleinerer Dimension setzen wir das Gilmore-Gomory-Modell ein, das eine Dantzig-Wolfe-Dekomposition der positionsindizierten Formulierung ist. Um eine Nachbarschaftsstruktur zu erhalten, muss die Basismatrix der optimalen Lösung die consecutive-1’s-Eigenschaft erfüllen. Für die Konstruktion solcher Matrizen entwickeln wir neue Branch-and-Price-Algorithmen, die sich durch Strategien zur Enumeration von partiellen Lösungen unterscheiden. Wir beweisen auch einige Charakteristiken von partiellen Lösungen, die das Hilfsproblem der Spaltengenerierung verschärfen. Für die nichtlineare Modellierung der höherdimensionalen Packungsprobleme untersuchen wir moderne Ansätze des Constraint Programming, modifizieren diese und schlagen neue Dichotomie- und Überschneidungsstrategien für die Verzweigung vor. Für die Verstärkung der Constraint Propagation stellen wir neue Ablehnungskriterien vor. Wir nutzen dabei 1D Relaxationen mit Intervallen und verbotenen Paaren, erweiterte Streifen-Relaxation, 2D Scheiben-Relaxation und 1D Scheiben-Streifen-Relaxation mit verbotenen Paaren. Alle vorgestellten Kriterien basieren auf Relaxationen durch Probleme kleinerer Dimension, die wir weiter durch die LP-Relaxation des Gilmore-Gomory-Modells abschwächen. Wir schließen mit Umsetzungsfragen und numerischen Experimenten aller vorgeschlagenen Ansätze.

Many real life problems can be formulated as constraint satisfaction problems <i>(CSPs)</i>. Backtracking search algorithms are usually employed to solve <i>CSPs</i> and in backtracking search the choice of branching strategies can be critical since they specify how a search algorithm can instantiate a variable and how a problem can be reduced into subproblems; that is, they define a search tree. In spite of the apparent importance of the branching strategy, there have been only a few empirical studies about different branching strategies and they all have been tested exclusively for numerical constraints. In this thesis, we employ the three most commonly used branching strategies in solving finite domain <i>CSPs</i>. These branching strategies are described as follows: first, a branching strategy with strong commitment assigns its variables in the early stage of the search as in k-Way branching; second, 2-Way branching guides a search by branching one side with assigning a variable and the other with eliminating the assigned value; third, the domain splitting strategy, based on the least commitment principle, branches by dividing a variable's domain rather than by assigning a single value to a variable. In our experiments, we compared the efficiency of different branching strategies in terms of their execution times and the number of choice points in solving finite domain <i>CSPs</i>. Interestingly, our experiments provide evidence that the choice of branching strategy for finite domain problems does not matter much in most cases--provided we are using an effective variable ordering heuristic--as domain splitting and 2-Way branching end up simulating k-Way branching. However, for an optimization problem with large domain size, the branching strategy with the least commitment principle can be more efficient than the other strategies. This empirical study will hopefully interest other practitioners to take different branching schemes into consideration in designing heuristics.

A Geração de Língua Natural (GLN) ocupa-se de atribuir forma lingüística a dados em representação não-lingüística (Reiter & Dale, 2000); a Realização Lingüística (RL), por sua vez, reúne as subtarefas da GLN estritamente dependentes das especificidades da língua-alvo. Este trabalho objetiva a investigação em RL, uma de cujas aplicações mais proeminentes é a construção de módulos geradores de língua-alvo na tradução automática baseada em transferência semântica. Partimos da identificação de três requisitos fundamentais para modelos de RL quais sejam generalidade, instanciabilidade e complexidade e da tensão entre esses requisitos no estado da arte. Argumentamos pela relevância da avaliação formal dos modelos da literatura contra esses critérios e focalizamos em modelos baseados em restrições (Schulte, 2002) como promissores para reconciliar os três requisitos. Nesta classe de modelos, identificamos o recente modelo de Debusmann (2006) Extensible Dependency Grammar (XDG) e sua implementação - o XDG Development Toolkit (XDK) - como uma plataforma especialmente promissora para o desenvolvimento em RL, apesar de jamais utilizada para tal. Nossas contribuições práticas se resumem ao esforço de tornar o XDK mais eficiente e uma formulação da disjunção inerente à lexicalização adequada à XDG, demonstrando suas potenciais vantagens numa sistema de GLN mais completo / Natural Language Generation (NLG) concerns assigning linguistic form to data in nonlinguistic representation (Reiter & Dale, 2000); Linguistic Realization (LR), in turn, comprises all strictly target language-dependent NLG tasks. This work looks into RL systems from the perspective of three fundamental requirements - namely generality, instantiability, and complexity and the tension between them in the state of the art. We argue for the formal evaluation of models against these criteria and focus on constraint-based models (Schulte, 2002) as tools to reconcile them. In this class of models we identify the recent development of Debusmann (2006) - Extensible Dependency Grammar (XDG) - and its implementation - the XDG Development Toolkit (XDK) - as an especially promising platform for RL work, in spite of never having been used as such. Our practical contributions comprehend a successful effort to make the XDK more efficient and a formulation of lexicalization disjunction suitable to XDG, illustrating its potential advantages in a full-fledged NLG system

A l’opposé de l’approche consistant à concevoir aucas par cas des contraintes et des algorithmes leur étant dédiés, l’objet de cette thèse concerne d’une part la description de familles de contraintes en termes de composition de fonctions, et d’autre part la synthèse d’objets combinatoires pour de telles contraintes. Les objets concernés sont des bornes précises, des coupes linéaires, des invariants non-linéaires et des automates finis ; leur but principal est de prendre en compte l’aspect combinatoire d’une seule contrainte ou d’une conjonction de contraintes. Ces objets sont obtenus d’une façon systématique et sont paramétrés par une ou plusieurs contraintes, par le nombre de variables dans une séquence, et par les domaines initiaux de ces variables. Cela nous permet d’obtenir des objets indépendants d’une instance considérée. Afin de synthétiser des objets combinatoires nous tirons partie de la vue déclarative de telles contraintes, basée sur les expressions régulières, ainsi que la vue opérationnelle, basée sur les automates à registres et les transducteurs finis. Il y a plusieurs avantages à synthétiser des objets combinatoires par rapport à la conception d’algorithmes dédiés : 1) on peut utiliser ces formules paramétrées dans plusieurs contextes, y compris la programmation par contraintes et la programmation linéaire, ce qui est beaucoup plus difficile avec des algorithmes ; 2) la synergie entre des objets combinatoires nous donne une meilleure performance en pratique ; 3) les quantités calculées par certaines des formules peuvent être utilisées non seulement dans le contexte de l’optimisation mais aussi pour la fouille de données. / Contrary to the standard approach consisting in introducing ad hoc constraints and designing dedicated algorithms for handling their combinatorial aspect, this thesis takes another point of view. On the one hand, it focusses on describing a family of sequence constraints in a compositional way by multiple layers of functions. On the other hand, it addresses the combinatorial aspect of both a single constraint and a conjunction of such constraints by synthesising compositional combinatorial objects, namely bounds, linear inequalities, non-linear constraints and finite automata. These objects are obtained in a systematic way and are not instance-specific: they are parameterised by one or several constraints, by the number of variables in a considered sequence of variables, and by the initial domains of the variables. When synthesising such objects we draw full benefit both from the declarative view of such constraints, based on regular expressions, and from the operational view, based on finite transducers and register automata.There are many advantages of synthesising combinatorial objects rather than designing dedicated algorithms: 1) parameterised formulae can be applied in the context of several resolution techniques such as constraint programming or linear programming, whereas algorithms are typically tailored to a specific technique; 2) combinatorial objects can be combined together to provide better performance in practice; 3) finally, the quantities computed by some formulae cannot just be used in an optimisation setting, but also in the context of data mining.

Many real life problems can be formulated as constraint satisfaction problems <i>(CSPs)</i>. Backtracking search algorithms are usually employed to solve <i>CSPs</i> and in backtracking search the choice of branching strategies can be critical since they specify how a search algorithm can instantiate a variable and how a problem can be reduced into subproblems; that is, they define a search tree. In spite of the apparent importance of the branching strategy, there have been only a few empirical studies about different branching strategies and they all have been tested exclusively for numerical constraints. In this thesis, we employ the three most commonly used branching strategies in solving finite domain <i>CSPs</i>. These branching strategies are described as follows: first, a branching strategy with strong commitment assigns its variables in the early stage of the search as in k-Way branching; second, 2-Way branching guides a search by branching one side with assigning a variable and the other with eliminating the assigned value; third, the domain splitting strategy, based on the least commitment principle, branches by dividing a variable's domain rather than by assigning a single value to a variable. In our experiments, we compared the efficiency of different branching strategies in terms of their execution times and the number of choice points in solving finite domain <i>CSPs</i>. Interestingly, our experiments provide evidence that the choice of branching strategy for finite domain problems does not matter much in most cases--provided we are using an effective variable ordering heuristic--as domain splitting and 2-Way branching end up simulating k-Way branching. However, for an optimization problem with large domain size, the branching strategy with the least commitment principle can be more efficient than the other strategies. This empirical study will hopefully interest other practitioners to take different branching schemes into consideration in designing heuristics.