## AbstractLet $A$ be a finite set and let $\varphi:A^k\rightarrow A$ with $k\geq 3$ be a $k$-ary operation on $A$. We say that $\varphi$ is a generalized majority-minority (GMM) operation if for all $a,b\in A$ we have that \noindent \begin{center} $\varphi(x,y,..,y)=\varphi(y,x,..,y)=\cdots=\varphi(y,y,..,x)=y$ \hspace{5cm} for all $x,y\in\{a,b\}$ or $\varphi(x,y,..,y)=\varphi(y,y,..,x)=x \;\; \text{ for all } x,y\in\{a,b\}$ \end{center} 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.
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