Schaefer's theorem for graphs

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 first-order 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 of statements ("constraints") about these variables in the language of graphs, where each statement is taken from a fixed finite set of allowed formulas; the question is whether is satisfiable in a graph. We prove that either 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...

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Manuel Bodirsky, Michael Pinsker
• Keywords: Constraint Satisfaction Problems, CORR 2010, Education, Polynomial Time, Satisfaction Problems