Computer Science – Computational Complexity
Scientific paper
2012-01-04
Computer Science
Computational Complexity
M\'emoire pour l'obtention d'une HDR \`a l'universit\'e Paris 7. 255 pages, + 25 pages appendix containing CV, publications li
Scientific paper
A constraint satisfaction problem (CSP) is a computational problem where the input consists of a finite set of variables and a finite set of constraints, and where the task is to decide whether there exists a satisfying assignment of values to the variables. Depending on the type of constraints that we allow in the input, a CSP might be tractable, or computationally hard. In recent years, general criteria have been discovered that imply that a CSP is polynomial-time tractable, or that it is NP-hard. Finite-domain CSPs have become a major common research focus of graph theory, artificial intelligence, and finite model theory. It turned out that the key questions for complexity classification of CSPs are closely linked to central questions in universal algebra. This thesis studies CSPs where the variables can take values from an infinite domain. This generalization enhances dramatically the range of computational problems that can be modeled as a CSP. Many problems from areas that have so far seen no interaction with constraint satisfaction theory can be formulated using infinite domains, e.g. problems from temporal and spatial reasoning, phylogenetic reconstruction, and operations research. It turns out that the universal-algebraic approach can also be applied to study large classes of infinite-domain CSPs, yielding elegant complexity classification results. A new tool in this thesis that becomes relevant particularly for infinite domains is Ramsey theory. We demonstrate the feasibility of our approach with two complete complexity classification results: one on CSPs in temporal reasoning, the other on a generalization of Schaefer's theorem for propositional logic to logic over graphs. We also study the limits of complexity classification, and present classes of computational problems provably do not exhibit a complexity dichotomy into hard and easy problems.
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