Generalized majority-minority operations are tractable

Let A be a finite set and let φ : A^k→A with k≥3 be a k-ary operation on A. We say that φ is a generalized majority-minority (GMM) operation if for all a, b ∈ A we have that φ(x, y,...,y) = φ(y, x,..,y) =...=φ(y, y,..,x) = y for all x, y ∈ {a, b} or φ{x, y,..,y) = φ(y, y,..,x) = x for all x, y ∈ {a, b}. Near-unanimity and Mal´tsev operations are particular instances of GMM operations. We prove that every CSP instance where all constraint relations are invariant under a (fixed) GMM operation is solvable in polynomial time. This constitutes one of the largest tractable cases of the CSP.