Complexity of constraint satisfaction

The complexity of constraint satisfaction is the application of computational complexity theory on constraint satisfaction. It has mainly been studied for discriminating between tractable and intractable classes of constraint satisfaction problems on finite domains. Solving a constraint satisfaction problem on a finite domain is an NP-complete problem in general. Research has shown a number of polynomial-time subcases, mostly obtained by restricting either the allowed domains or constraints or the way constraints can be placed over the variables. Research has also established a relationship between the constraint satisfaction problem and problems in other areas such as finite model theory and databases. Establishing whether a constraint satisfaction problem on a finite domain has solutions is an NP complete problem in general. This is an easy consequence of a number of other NP complete problems being expressible as constraint satisfaction problems. Such other problems include propositional satisfiability and three-colorability. Tractability can be obtained by considering specific classes of constraint satisfaction problems. As an example, if the domain is binary and all constraints are binary, establishing satisfiability is a polynomial-time problem because this problem is equivalent to 2-SAT, which is tractable. Research has shown a number of tractable subcases. Some of these classes are based on restricting the allowed domains or relations, some on restricting the way constraints are placed over variables, and some on both kinds of restrictions. One line of research used a correspondence between constraint satisfaction problem and the problem of establishing the existence of a homomorphism between two relational structures. This correspondence has been used to link constraint satisfaction with topics traditionally related to database theory. A considered research problem is about the existence of dichotomies among sets of restrictions. This is the question of whether a set of restrictions contains only polynomial-time restrictions and NP-complete restrictions.