The complexity of satisfiability problems: Refining Schaefer's theorem

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Abstract

Schaefer proved in 1978 that the Boolean constraint satisfaction problem for a given constraint language is either in P or is NP-complete, and identified all tractable cases. Schaefer's dichotomy theorem actually shows that there are at most two constraint satisfaction problems, up to polynomial-time isomorphism (and these isomorphism types are distinct if and only if PNP). We show that if one considers AC0 isomorphisms, then there are exactly six isomorphism types (assuming that the complexity classes NP, P, ⊕L, NL, and L are all distinct). A similar classification holds for quantified constraint satisfaction problems.

Keywords

Computational complexity
Constraint satisfaction problem
Propositional satisfiability
Clone theory

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Supported in part by DFG Grants Vo 630/5-1/2, Vo 630/6-1, and NSF Grants CCF-0514155, DMS-0652582, CCF-0830133, CCF-0832787, and CCF-0514621.