Testing Assignments to Constraint Satisfaction Problems
Hubie Chen, Matt Valeriote, Yuichi Yoshida

TL;DR
This paper classifies the query complexity of property testing for constraint satisfaction problems (CSPs) based on the algebraic properties of their templates, providing a complete dichotomy and trichotomy theorems.
Contribution
It offers the first comprehensive classification of the query complexity for testing CSPs and their existential extensions, based on polymorphism conditions.
Findings
CSP(A) is constant-query testable if A has a majority and Maltsev polymorphism.
Testing CSP(A) requires super-constant queries if A lacks these polymorphisms.
Existential CSPs exhibit a trichotomy: constant, sublinear, or linear query complexity depending on algebraic properties.
Abstract
For a 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 an instance of a CSP and query access to an assignment, one wants to decide whether the assignment satisfies the instance, or is far from so doing. While previous works on this scenario studied concrete templates or restricted classes of structures, this article presents comprehensive classification theorems. Our first contribution is a dichotomy theorem completely characterizing the 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) Else,…
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Videos
Testing Assignments to Constraint Satisfaction Problems· youtube
Taxonomy
TopicsAdvanced Graph Theory Research · Complexity and Algorithms in Graphs · Optimization and Search Problems
