Simultaneous Max-Cut Is Harder to Approximate Than Max-Cut

Authors Amey Bhangale, Subhash Khot



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Author Details

Amey Bhangale
  • Department of Computer Science and Engineering, University of California, Riverside, CA, USA
Subhash Khot
  • Department of Computer Science, Courant Institute of Mathematical Sciences, New York University, NY, USA

Acknowledgements

Our numerical calculations involve minor modifications of the prover code [Austrin et al.], written by [Austrin et al., 2016], which uses interval arithmetic to get a computer generated proof. We are indebted to the authors of [Austrin et al., 2016] for making it available online. We are also thankful to the anonymous reviewers whose comments helped greatly in improving the presentation of the paper.

Cite AsGet BibTex

Amey Bhangale and Subhash Khot. Simultaneous Max-Cut Is Harder to Approximate Than Max-Cut. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 9:1-9:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.CCC.2020.9

Abstract

A systematic study of simultaneous optimization of constraint satisfaction problems was initiated by Bhangale et al. [ICALP, 2015]. The simplest such problem is the simultaneous Max-Cut. Bhangale et al. [SODA, 2018] gave a .878-minimum approximation algorithm for simultaneous Max-Cut which is almost optimal assuming the Unique Games Conjecture (UGC). For single instance Max-Cut, Goemans-Williamson [JACM, 1995] gave an α_GW-approximation algorithm where α_GW ≈ .87856720... which is optimal assuming the UGC. It was left open whether one can achieve an α_GW-minimum approximation algorithm for simultaneous Max-Cut. We answer the question by showing that there exists an absolute constant ε₀ ≥ 10^{-5} such that it is NP-hard to get an (α_GW- ε₀)-minimum approximation for simultaneous Max-Cut assuming the Unique Games Conjecture.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Simultaneous CSPs
  • Unique Games hardness
  • Max-Cut

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References

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