Refined complexity of constraint satisfaction problems

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In a constraint satisfaction problem (CSP), one must assign values to variables that must satisfy various constraints; typical real world examples include scheduling problems, database queries, image-processing, and frequency assignment problems. In general, determining whether a CSP admits a solution is an algorithmic challenge, but it often happens in practice that the constraints are of a very restricted form, allowing the use of efficient methods to solve the CSP. Our long-term goal is to classify precisely what kinds of restrictions lead to these tractable CSP's. Our approach is based on an unexpected and fruitful connection between CSP's and universal algebra that was uncovered in the late 90's, and which has led to major breakthroughs in our understanding of the complexity of CSPs over the past 20 years. In short, every family of constraints is transformed into a mathematical object whose algebraic properties reflect the difficulty of solving the CSP. Several precise conjectures have been formulated, predicting which equations should lead***to solvability with given time and space restrictions. The goal of this program is to investigate and solve these conjectures in various important special cases. The investigation of special cases of the refined dichotomy conjectures is bound to provide insights into an eventual solution of the full L- and NL- conjectures. This will give us a complete classification of the complexity of CSPs of bounded width, and hence a much deeper understanding of the complexity of non-uniform CSPs, which are ubiquitous in the theory of computing, with wide-ranging applications from artificial intelligence to database theory.