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, which we refer to as sign-symmetric languages Γ. Via an algebraic connection, this reduces to the study of restricted partial polymorphisms we refer to as pSDI-operations (for partial, self-dual and idempotent), under which the language Γ is invariant. First, we focus on 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 levels, corresponding to a rough notion of difficulty, where every level k has an easiest language class, containing the language for (k − 1)-SAT, and a hardest language class, containing (among other things) constraints encoded as roots of multivariate polynomials of degree (k − 1). Particular classes in each level correspond to the natural partially defined versions of previously studied total algebraic invariants. In particular, the easiest class on level k ≥ 3 corresponds to the partial k-ary near-unanimity (k-NU) operation, and a larger class corresponds to the partial k-edge operation. The largest class at each level corresponds to a partial operation uk we call k-universal. Furthermore, every sign-symmetric language Γ not preserved by uk implements all k-clauses, hence SAT(Γ) is at least as hard as k-SAT; and if Γ is not preserved by uk for any k, then SAT(Γ) is trivially SETH-hard (i.e., takes time O(2) under SETH). Second, we consider implications of this for the complexity of SAT(Γ). We find that particular classes in the hierarchy correspond to previously known algorithmic strategies. In particular, languages preseved by the partial 2-edge operation can be solved via Subset Sum-style meet in the middle, and languages preserved by the partial 3-NU operation can be solved via fast matrix multiplication. These results also hold for the correspondning non-Boolean CSP problems. We also find that symmetric 3-edge languages reduce to finding a monochromatic triangle in an edge-coloured graph, which can be done using algorithms for sparse matrix multiplication; and if the sunflower conjecture holds for sunflowers with k petals, then the partial k-NU language has an improved algorithm via Schöning-style local search. Complementing this, we show a lower bound, showing that for every level k there is a constant ck such that for every partial operation p on level k, the problem SAT(Γ)with Γ = Inv(p) cannot be solved faster thanO(c k ) unless SETH fails. In particular, when Γ = Inv(2-edge), this gives us the first NP-hard SAT problem which simultaneously has non-trivial upper and lower bounds on the running time, assuming SETH. Finally, we note a possible conjecture: It is consistent with our present knowledge that SAT(Γ) admits an improved algorithm if and only if Γ is preserved by uk for some constant k. However, to show this in the positive poses some significant difficulty. [email protected] [email protected]