Constant-Query Testability of Assignments to Constraint Satisfaction Problems

For each finite relational structure $A$, let $CSP(A)$ denote the CSP instances whose constraint relations are taken from $A$. The resulting family of problems $CSP(A)$ has been considered heavily in a variety of computational contexts. In this article, we consider this family from the perspective of property testing: given a CSP instance and query access to an assignment, one wants to decide whether the assignment satisfies the instance or is far from doing so. While previous work on this scenario studied concrete templates or restricted classes of structures, this article presents a comprehensive classification theorem. Our main contribution is a dichotomy theorem completely characterizing the finite structures $A$ such that $CSP(A)$ is constant-query testable: (i) If $A$ has a majority polymorphism and a Maltsev polymorphism, then $CSP(A)$ is constant-query testable with one-sided error. (ii) Otherwise, testing $CSP(A)$ requires a superconstant number of queries.