Promise Constraint Satisfaction: Algebraic Structure and a Symmetric Boolean Dichotomy
Joshua Brakensiek, Venkatesan Guruswami · SIAM Journal on Computing · 2021
A classic result due to Schaefer [ Proceedings of STOC 78, ACM, 1978, pp. 216--226] classifies all constraint satisfaction problems (CSPs) over the Boolean domain as being either in ${P}$ or ${NP}$-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 $(P, Q) \in \Gamma$, $P(x_{i_1}, \hdots, x_{i_k})$ is a clause of $\Psi_P$ if and only if $Q(x_{i_1}, \hdots, x_{i_k})$ is a clause of $\Psi_Q$. The promise problem ${PCSP}(\Gamma)$ is to distinguish, given $(\Psi_P, \Psi_Q)$, between the cases $\Psi_P$ is satisfiable and $\Psi_Q$ is unsatisfiable. Many problems such as approximate graph and hypergraph coloring as well as the $(2+\epsilon)$-SAT problem due to Austrin, Guruswami, and H\aastad [ SIAM J. Comput., 46 (2017), pp. 1554--1573] can be placed in this framework. This paper is motivated by the pursuit of understanding the computational complexity of Boolean PCSPs, determining for which $\Gamma$ the associated PCSP is polynomial-time tractable or ${NP}$-hard. As our main result, we show that ${PCSP}(\Gamma)$ exhibits a dichotomy (it is either polynomial-time tractable or ${NP}$-hard) when the relations in $\Gamma$ are symmetric and allow for negations of variables. In particular, we show that every such polynomial-time tractable $\Gamma$ can be solved via either Gaussian elimination over $\mathbb F_2$ or a linear programming relaxation. We achieve our dichotomy theorem by extending the (weak) polymorphism framework of Austrin, Guruswami, and H\aastad which itself is a generalization of the algebraic approach used by polymorphisms 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.