Schaefer's theorem for graphs

TL;DR

This paper extends Schaefer's complexity classification from Boolean logic to graph logic, establishing a dichotomy for graph constraint satisfaction problems with a universal-algebraic and Ramsey-theoretic approach.

Contribution

It introduces a complexity dichotomy for graph-based CSPs, identifying 17 classes solvable in polynomial time and proving NP-completeness otherwise, using novel algebraic and Ramsey-theoretic methods.

Findings

Classifies graph CSPs into polynomial-time solvable and NP-complete cases.

Develops a universal-algebraic framework for analyzing graph CSP complexity.

Introduces new Ramsey-theoretic tools for studying functions on the random graph.

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 is based on a Ramsey-theoretic analysis of functions acting on the random graph, and we develop general tools suitable for such an analysis which are of independent mathematical interest.