  Assuming the Unique Games Conjecture, we show that existing approximation
algorithms for some Boolean Max-2-CSPs with cardinality constraints are
optimal. In particular, we prove that Max-Cut with cardinality constraints is
UG-hard to approximate within \approx 0.858, and that Max-2-Sat with
cardinality constraints is UG-hard to approximate within \approx 0.929. In both
cases, the previous best hardness results were the same as the hardness of the
corresponding unconstrained Max-2-CSP (\approx 0.878 for Max-Cut, and \approx
0.940 for Max-2-Sat). The hardness obtained for Max-2-Sat applies to monotone
Max-2-Sat instances, meaning that we also obtain tight inapproximability for
the Max-k-Vertex-Cover problem.
