Theorems in computational complexity theory | Constraint programming

Schaefer's dichotomy theorem

In computational complexity theory, a branch of computer science, Schaefer's dichotomy theorem states necessary and sufficient conditions under which a finite set S of relations over the Boolean domain yields polynomial-time or NP-complete problems when the relations of S are used to constrain some of the propositional variables.It is called a dichotomy theorem because the complexity of the problem defined by S is either in P or NP-complete as opposed to one of the classes of intermediate complexity that is known to exist (assuming P ≠ NP) by Ladner's theorem. Special cases of Schaefer's dichotomy theorem include the NP-completeness of SAT (the Boolean satisfiability problem) and its two popular variants 1-in-3 SAT and not-all-equal 3SAT (often denoted by NAE-3SAT). In fact, for these two variants of SAT, Schaefer's dichotomy theorem shows that their monotone versions (where negations of variables are not allowed) are also NP-complete. (Wikipedia).

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Not-all-equal 3-satisfiability | Galois connection | NP-intermediate | P versus NP problem | Constraint satisfaction problem | Max/min CSP/Ones classification theorems | L (complexity) | P-complete | Computational complexity theory | Boolean satisfiability problem | Horn clause | P (complexity) | Decision problem | Reduction (complexity) | Universal algebra | NL-complete | Boolean domain | Propositional variable