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Approximating Max-Cut on Bounded Degree Graphs: Tighter Analysis of the FKL Algorithm

Authors Jun-Ting Hsieh, Pravesh K. Kothari

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ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Max-Cut
  • approximation algorithm
  • semidefinite programming

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Abstract

In this note, we describe a α_GW + Ω̃(1/d²)-factor approximation algorithm for Max-Cut on weighted graphs of degree ⩽ d. Here, α_GW ≈ 0.878 is the worst-case approximation ratio of the Goemans-Williamson rounding for Max-Cut. This improves on previous results for unweighted graphs by Feige, Karpinski, and Langberg [Feige et al., 2002] and Florén [Florén, 2016]. Our guarantee is obtained by a tighter analysis of the solution obtained by applying a natural local improvement procedure to the Goemans-Williamson rounding of the basic SDP strengthened with triangle inequalities.

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Jun-Ting Hsieh and Pravesh K. Kothari. Approximating Max-Cut on Bounded Degree Graphs: Tighter Analysis of the FKL Algorithm. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 77:1-77:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ICALP.2023.77

Author Details

Jun-Ting Hsieh
  • Carnegie Mellon University, Pittsburgh, PA, USA
Pravesh K. Kothari
  • Carnegie Mellon University, Pittsburgh, PA, USA

Funding

  • Hsieh, Jun-Ting: Supported by NSF CAREER Award #2047933.
  • Kothari, Pravesh K.: Supported by NSF CAREER Award #2047933, Alfred P. Sloan Fellowship and a Google Research Scholar Award.

Acknowledgements

We thank Prasad Raghavendra for his Simons Institute talk on approximation algorithms for bounded degree constraint satisfaction problems and various related discussions that directly motivated this work. We thank Simons Institute Berkeley for hosting us in the Fall 2021 research program on Computational Complexity of Statistical Inference.

References

  1. Uriel Feige, Marek Karpinski, and Michael Langberg. Improved approximation of max-cut on graphs of bounded degree. Journal of Algorithms, 43(2):201-219, 2002. Google Scholar
  2. Noah Fleming, Pravesh Kothari, and Toniann Pitassi. Semialgebraic Proofs and Efficient Algorithm Design. Foundations and Trends in Theoretical Computer Science, 14(1-2):1-221, 2019. Google Scholar
  3. Mikael Florén. Approximation of max-cut on graphs of bounded degree, 2016. Google Scholar
  4. Michel X Goemans and David P Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM (JACM), 42(6):1115-1145, 1995. Google Scholar
  5. Subhash Khot. On the power of unique 2-prover 1-round games. In Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, pages 767-775, 2002. Google Scholar
  6. Subhash Khot, Guy Kindler, Elchanan Mossel, and Ryan O’Donnell. Optimal inapproximability results for max-cut and other 2-variable csps? SIAM Journal on Computing, 37(1):319-357, 2007. Google Scholar
  7. Luca Trevisan. Non-approximability results for optimization problems on bounded degree instances. In Proceedings of the thirty-third annual ACM symposium on Theory of computing, pages 453-461, 2001. Google Scholar
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