We show optimal (up to constant factor) NP-hardness for Max-k-CSP over any domain,
whenever k is larger than the domain size. This follows from our main result concerning predicates
over abelian groups. We show that a predicate is approximation resistant if it contains a subgroup that
is balanced pairwise independent. This gives an unconditional analogue of Austrin–Mossel hardness
result, taking away their Unique-Games Conjecture assumption in exchange for an abelian subgroup
structure.
Our main ingredient is a new technique to reduce soundness, which is inspired by XOR-lemmas.
Using this technique, we also improve the NP-hardness of approximating Independent-Set on bounded
degree graphs, Almost-Coloring, and Two-Prover-One-Round-Game.