The complexity of the constraint satisfaction problem and its variants

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The constraint satisfaction problem (CSP) provides a general framework in which it is possible to express, in a natural way, a wide variety of problems. The aim in a CSP is to find an assignment of values to a given set of variables, subject to constraints on the values which can be simultaneously assigned to certain specified subsets of variables; in the counting constraint satisfaction problem the objective is to find the number of such assignments. The CSP is therefore very important in a number of applications, both theoretical and practical. In theoretical applications we look for new algorithmic ideas that can help to solve problems that could not be solved before, to improve the existing algorithms, to design more general and uniform algorithms, and to find evidence that certain problems are not efficiently solvable in the general case. In practical applications, constraint and satisfiability solvers are now standard tools in a wide range of areas from scheduling to hardware verification. Advances in the study of the CSP will speed up such solvers and expand their area of applicability.*** This project will contribute in both aspects of constraint problems, although its main focus is on the theoretical side. One of its main goals is to precisely understand which CSPs can be solved efficiently and design an efficient solution algorithm. This is one of the long standing open problems in computer science, but there is evidence that we may be approaching its resolution. Another component of the project is the study of counting constraint satisfaction problems. This area has unlikely connections to statistical physics, where such numbers affect the properties of large ensembles of particle, and, in particular,  help to locate phase transition thresholds, at which, for instance, liquid turns to gas. Finally, this project proposes to study the efficiency and to design new algorithms in the framework that is closer to practical circumstances than the standard worst case approach. It is known that practical problems tend to have a certain property called the power law distribution, and we plan to mathematically study the behaviour of various algorithms on instance of the CSP satisfying this property.**