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Constraint satisfaction is a simple but powerful tool. Constraints identify the impossible and reduce the realm of possibilities to effectively focus on the possible, allowing for a natural declarative formulation of what must be satisfied, without expressing how. The field of constraint reasoning has matured over the last three decades with contributions from a diverse community of researchers in artificial intelligence, databases and programming languages, operations research, management science, and applied mathematics. Today, constraint problems are used to model cognitive tasks in vision,
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Constraint Programming aims at solving hard combinatorial problems, with a computation time increasing in practice exponentially. The methods are today efficient enough to solve large industrial problems, in a generic framework. However, solvers are dedicated to a single variable type: integer or real. Solving mixed problems relies on ad hoc transformations. In another field, Abstract Interpretation offers tools to prove program properties, by studying an abstraction of their concrete semantics, that is, the set of possible values of the variables during an execution. Various representations
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This dissertation by Biman Roy explores the worst-case complexity of constraint satisfaction problems (CSP) and its variants, focusing on the application of partial polymorphisms. It delves into fine-grained complexity to classify NP-complete problems based on their worst-case time complexities. The study establishes connections between constraint languages and strong partial clones, demonstrating the use of algebraic methods to analyze and describe these clones. The research also investigates the relationships between weak bases and the inclusion structure of Boolean weak bases, aiming to provide a deeper understanding of the complexity of CSPs. This work is intended for an academic audience, particularly those interested in computer science, complexity theory, and universal algebra.
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