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

Authors Per Austrin , Aleksa Stanković

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ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Mathematics of computing → Approximation algorithms
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Constraint satisfaction problems
  • global cardinality constraints
  • semidefinite programming
  • inapproximability
  • Unique Games Conjecture
  • Max-Cut
  • Max-2-Sat

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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 ~~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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Per Austrin and Aleksa Stanković. Global Cardinality Constraints Make Approximating Some Max-2-CSPs Harder. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 24:1-24:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.24

Author Details

Per Austrin
  • KTH Royal Institute of Technology, Stockholm, Sweden
Aleksa Stanković
  • KTH Royal Institute of Technology, Stockholm, Sweden

Funding

Research supported by the Approximability and Proof Complexity project funded by the Knut and Alice Wallenberg Foundation.

Acknowledgements

The authors thank Johan Håstad for helpful suggestions and comments on the manuscript. We also thank anonymous reviewers for their helpful remarks.

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