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Colouring, constraint satisfaction, and complexity. (English) Zbl 1302.68251

Summary: Constraint satisfaction problems have enjoyed much attention since the early seventies, and in the last decade have become also a focus of attention amongst theoreticians. Graph colourings are a special class of constraint satisfaction problems; they offer a microcosm of many of the considerations that occur in constraint satisfaction. From the point of view of theory, they are well known to exhibit a dichotomy of complexity – the \(k\)-colouring problem is polynomial-time solvable when \(k \leq 2\), and NP-complete when \(k \geq 3\). Similar dichotomy has been proved for the class of graph homomorphism problems, which are intermediate problems between graph colouring and constraint satisfaction. However, for general constraint satisfaction problems, dichotomy has only been conjectured. Although the conjecture remains unproven to this day, it has been driving much of the theoretical research on constraint satisfaction problems, which combines methods of logic, universal algebra, analysis, and combinatorics. Currently, this is a very active area of research, and it is our goal here to present some of the recent developments, updating some of the information in existing books and surveys, while focusing on both the mathematical and the computational aspects of the theory. Given the level of activity, we are only able to survey a fraction of the new work, with emphasis on our own areas of interest.

MSC:

68T20 Problem solving in the context of artificial intelligence (heuristics, search strategies, etc.)
05C15 Coloring of graphs and hypergraphs
68Q25 Analysis of algorithms and problem complexity
68-02 Research exposition (monographs, survey articles) pertaining to computer science
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