Global Cardinality Constraints Make Approximating Some Max-2-CSPs Harder
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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 ≈ 0.858, and that Max-2-Sat with cardinality constraints is UG-hard to approximate within ≈ 0.929. In both cases, the previous best hardness results were the same as the hardness of the corresponding unconstrained Max-2-CSP (≈ 0.878 for Max-Cut, and ≈ 0.940 for Max-2-Sat). The hardness 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.
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