Promise Constraint Satisfaction: Algebraic Structure and a Symmetric Boolean Dichotomy

A classic result due to Schaefer (1978) classifies all constraint satisfaction problems (CSPs) over the Boolean domain as being either in $\mathsf{P}$ or $\mathsf{NP}$-hard. This paper considers a promise-problem variant of CSPs called PCSPs. A PCSP over a finite set of pairs of constraints $\Gamma$ consists of a pair $(\Psi_P, \Psi_Q)$ of CSPs with the same set of variables such that for every $(P, Q) \in \Gamma$, $P(x_{i_1}, ..., x_{i_k})$ is a clause of $\Psi_P$ if and only if $Q(x_{i_1}, ..., x_{i_k})$ is a clause of $\Psi_Q$. The promise problem $\operatorname{PCSP}(\Gamma)$ is to distinguish, given $(\Psi_P, \Psi_Q)$, between the cases $\Psi_P$ is satisfiable and $\Psi_Q$ is unsatisfiable. Many natural problems including approximate graph and hypergraph coloring can be placed in this framework. This paper is motivated by the pursuit of understanding the computational complexity of Boolean promise CSPs. As our main result, we show that $\operatorname{PCSP}(\Gamma)$ exhibits a dichotomy (it is either polynomial time solvable or $\mathsf{NP}$-hard) when the relations in $\Gamma$ are symmetric and allow for negations of variables. We achieve our dichotomy theorem by extending the weak polymorphism framework of Austrin, Guruswami, and H\aa stad [FOCS '14] which itself is a generalization of the algebraic approach to study CSPs. In both the algorithm and hardness portions of our proof, we incorporate new ideas and techniques not utilized in the CSP case. Furthermore, we show that the computational complexity of any promise CSP (over arbitrary finite domains) is captured entirely by its weak polymorphisms, a feature known as Galois correspondence, as well as give necessary and sufficient conditions for the structure of this set of weak polymorphisms. Such insights call us to question the existence of a general dichotomy for Boolean PCSPs.