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

## TL;DR

This paper demonstrates that under the Unique Games Conjecture, certain Max-2-CSPs with cardinality constraints are harder to approximate than previously known, establishing tight hardness bounds.

## Contribution

It proves new UG-hardness bounds for Max-Cut and Max-2-Sat with cardinality constraints, showing these are more difficult to approximate than unconstrained versions.

## Key findings

- Max-Cut with constraints is UG-hard to approximate within 0.858
- Max-2-Sat with constraints is UG-hard to approximate within 0.929
- Results apply to monotone Max-2-Sat and Max-k-Vertex-Cover

## Abstract

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.