Universal algebra and hardness results for constraint satisfaction problems

We present algebraic conditions on constraint languages Gamma that ensure the hardness of the constraint satisfaction problem CSP(Gamma) for complexity classes L, NL, P, NP and Mod(p)L. These criteria also give non-expressibility results for various restrictions of Datalog. Furthermore, we show that if CSP(F) is not first-order definable then it is L-hard. Our proofs rely on tame congruence theory and on a fine-grain analysis of the complexity of reductions used in the algebraic study of CSPs. The results pave the way for a refinement of the dichotomy conjecture stating that each CSP(F) lies in P or is NP-complete and they match the recent classification of [1] for Boolean CSP. We also infer a partial classification theorem for the complexity of CSP(Gamma) when the associated algebra of Gamma is the idempotent reduct of a preprimal algebra.