We study minimum cost constraint satisfaction problems (MinCostCSP) through the algebraic lens. We show that for any constraint language $Γ$ which has the dual discriminator operation as a polymorphism, there exists a $|D|$-approximation algorithm for MinCostCSP$(Γ)$ where $D$ is the domain. Complementing our algorithmic result, we show that any constraint language $Γ$ where MinCostCSP$(Γ)$ admits a constant-factor approximation must have a \emph{near-unanimity} (NU) polymorphism unless P = NP, extending a similar result by Dalmau et al. on MinCSPs. These results imply a dichotomy of constant-factor approximability for constraint languages that contain all permutation relations (a natural generalization for Boolean CSPs that allow variable negation): either MinCostCSP$(Γ)$ has an NU polymorphism and is $|D|$-approximable, or it does not have any NU polymorphism and is NP-hard to approximate within any constant factor. Finally, we present a constraint language which has a majority polymorphism, but is nonetheless NP-hard to approximate within any constant factor assuming the Unique Games Conjecture, showing that the condition of having an NU polymorphism is in general not sufficient unless UGC fails.

22 pages