Schaefer's theorem for graphs

Abstract

Schaefer's theorem is a complexity classification result for so-called Boolean constraint satisfaction problems: it states that every Boolean constraint satisfaction problem is either contained in one out of six classes and can be solved in polynomial time, or is NP-complete. We present an analog of this dichotomy result for the propositional logic of graphs instead of Boolean logic. In this generalization of Schaefer's result, the input consists of a set W of variables and a conjunction Phi of statements (constraints) about these variables in the language of graphs, where each statement is taken from a fixed finite set Psi of allowed quantifier-free first-order formulas; the question is whether Phi is satisfiable in a graph.We prove that either Psi is contained in one out of 17 classes of graph formulas and the corresponding problem can be solved in polynomial time, or the problem is NP-complete. This is achieved by a universal-algebraic approach, which in turn allows us to use structural Ramsey theory. To apply the universal-algebraic approach, we formulate the computational problems under consideration as constraint satisfaction problems (CSPs) whose templates are first-order definable in the countably infinite random graph. Our method to classify the computational complexity of those CSPs produces many statements of independent mathematical interest.