Gap theorems for robust satisfiability: Boolean CSPs and beyond

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Abstract

A computational problem exhibits a “gap property” when there is no tractable boundary between two disjoint sets of instances. We establish a Gap Trichotomy Theorem for a family of constraint problem variants, completely classifying the complexity of possible NP-hard gaps in the case of Boolean domains. As a consequence, we obtain a number of dichotomies for the complexity of specific variants of the constraint satisfaction problem: all are either polynomial-time tractable or NP-complete. Schaefer's original dichotomy for SAT variants is a notable particular case.

Universal algebraic methods have been central to recent efforts in classifying the complexity of constraint satisfaction problems. A second contribution of the article is to develop aspects of the algebraic approach in the context of a number of variants of the constraint satisfaction problem. In particular, this allows us to lift our results on Boolean domains to many templates on non-Boolean domains.

Keywords

Constraint satisfaction problem
Robust satisfiability
Clone theory
Dichotomy
Trichotomy
Boolean

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This article is a heavily expanded version of an extended abstract submitted to the 2016 ISAAC proceedings, and differs by the inclusion of full proofs and most results of Section 8.