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

Authors Amey Bhangale, Subhash Khot

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LIPIcs.CCC.2020.9.pdf
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ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Simultaneous CSPs
  • Unique Games hardness
  • Max-Cut

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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.

Cite As Get 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

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

Funding

  • Bhangale, Amey: Research supported by Irit Dinur’s ERC-CoG grant 772839.
  • Khot, Subhash: Supported by the NSF Award CCF-1813438, the Simons Collaboration on Algorithms and Geometry, and the Simons Investigator Award.

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.

Supplementary Materials

References

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