Which NP-Hard SAT and CSP Problems Admit Exponentially Improved Algorithms?

We study the complexity of SAT($Γ$) problems for potentially infinite languages $Γ$ closed under variable negation (sign-symmetric languages). Via an algebraic connection, this reduces to the study of restricted partial polymorphisms of $Γ$ we refer to as \emph{pSDI-operations} (for partial, self-dual and idempotent). First, we study the language classes themselves. We classify the structure of the least restrictive pSDI-operations, corresponding to the most powerful languages $Γ$, and find that these operations can be divided into \emph{levels}, corresponding to a rough notion of difficulty; and that within each level there is a strongest operation (the partial $k$-NU operation, preserving $(k-1)$-SAT) and a weakest operation (the $k$-universal operation $u_k$, preserving problems definable via bounded-degree polynomials). We show that every sign-symmetric $Γ$ not preserved by $u_k$ implements all $k$-clauses; thus if $Γ$ is not preserved by $u_k$ for any $k$, then SAT($Γ$) is trivially SETH-hard and cannot be solved faster than $O^*(2^n)$ unless SETH fails. Second, we study upper and lower bounds for SAT($Γ$) for such languages. We show that several classes in the hierarchy correspond to problems which can be solved faster than $2^n$ using previously known algorithmic strategies such as Subset Sum-style meet-in-the-middle and fast matrix multiplication. Furthermore, if the sunflower conjecture holds for sunflowers with k sets, then the partial k-NU language has an improved algorithm via local search. Complementing this, we show that for every class there is a concrete lower bound $c$ such that SAT($Γ$) cannot be solved faster than $O^*(c^n)$ for all problems in the class unless SETH fails. This gives the first known case of a SAT-problem which simultaneously has non-trivial upper and lower bounds under SETH.