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Assuming the Unique Games Conjecture, we show that existing approximationalgorithms for some Boolean Max-2-CSPs with cardinality constraints areoptimal. In particular, we prove that Max-Cut with cardinality constraints isUG-hard to approximate within \approx 0.858, and that Max-2-Sat withcardinality constraints is UG-hard to approximate within \approx 0.929. In bothcases, the previous best hardness results were the same as the hardness of thecorresponding unconstrained Max-2-CSP (\approx 0.878 for Max-Cut, and \approx0.940 for Max-2-Sat). The hardness obtained for Max-2-Sat applies to monotoneMax-2-Sat instances, meaning that we also obtain tight inapproximability forthe Max-k-Vertex-Cover problem.

Global Cardinality Constraints Make Approximating Some Max-2-CSPs Harder