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MaxCut Above Guarantee

Authors Ivan Bliznets , Vladislav Epifanov

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Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Fixed parameter tractability
Keywords
  • Tripartite
  • 3-colorable
  • chordal
  • maximum cut
  • FPT-algorithm
  • linear kernel

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Abstract

In this paper, we study the computational complexity of the Maximum Cut problem parameterized above guarantee. Our main result provides a linear kernel for the Maximum Cut problem in connected graphs parameterized above the spanning tree. This kernel significantly improves the previous O(k⁵) kernel given by Madathil, Saurabh, and Zehavi [ToCS 2020]. We also provide subexponential running time algorithms for this problem in special classes of graphs: chordal, split, and co-bipartite. We complete the picture by lower bounds under the assumption of the ETH. Moreover, we initiate a study of the Maximum Cut problem above 2/3|E| lower bound in tripartite graphs.

Cite As Get BibTex

Ivan Bliznets and Vladislav Epifanov. MaxCut Above Guarantee. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 22:1-22:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.MFCS.2023.22

Author Details

Ivan Bliznets
  • Utrecht University, The Netherlands
Vladislav Epifanov
  • HSE University, St. Petersburg, Russia

Funding

Supported by the project CRACKNP that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 853234).

Acknowledgements

We want to thank anonymous reviewers for their suggestions that helped to improve the presentation of the paper.

References

  1. Hans L. Bodlaender and Klaus Jansen. On the complexity of the maximum cut problem. In Patrice Enjalbert, Ernst W. Mayr, and Klaus W. Wagner, editors, STACS 94, pages 769-780, Berlin, Heidelberg, 1994. Springer Berlin Heidelberg. Google Scholar
  2. Liming Cai and David Juedes. On the existence of subexponential parameterized algorithms. Journal of Computer and System Sciences, 67(4):789-807, 2003. Google Scholar
  3. Robert Crowston, Gregory Gutin, Mark Jones, and Gabriele Muciaccia. Maximum balanced subgraph problem parameterized above lower bound. Theoretical Computer Science, 513:53-64, 2013. Google Scholar
  4. Robert Crowston, Mark Jones, and Matthias Mnich. Max-cut parameterized above the edwards-erdős bound. In Automata, Languages, and Programming: 39th International Colloquium, ICALP 2012, Warwick, UK, July 9-13, 2012, Proceedings, Part I 39, pages 242-253. Springer, 2012. Google Scholar
  5. Robert Crowston, Mark Jones, and Matthias Mnich. Max-cut parameterized above the edwards-erdős bound. Algorithmica, 72(3):734-757, 2015. Google Scholar
  6. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Daniel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://www.springer.com/gp/book/9783319212746.
  7. R. Diestel. Graph Theory. Springer, 2006. Google Scholar
  8. Michael Etscheid and Matthias Mnich. Linear kernels and linear-time algorithms for finding large cuts. Algorithmica, 80:2574-2615, 2018. Google Scholar
  9. Michael Etscheid and Matthias Mnich. Linear kernels and linear-time algorithms for finding large cuts. Algorithmica, 80:2574-2615, 2018. Google Scholar
  10. MR Garey, DS Johnson, and L Stockmeyer. Some simplified np-complete graph problems. Theoretical Computer Science, 1(3):237-267, 1976. Google Scholar
  11. Fǎnicǎ Gavril. The intersection graphs of subtrees in trees are exactly the chordal graphs. Journal of Combinatorial Theory, Series B, 16(1):47-56, 1974. URL: https://doi.org/10.1016/0095-8956(74)90094-X.
  12. 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
  13. Gregory Gutin and Matthias Mnich. A survey on graph problems parameterized above and below guaranteed values. arXiv preprint arXiv:2207.12278, 2022. Google Scholar
  14. Richard M Karp et al. Complexity of computer computations. Reducibility among combinatorial problems, 23(1):85-103, 1972. Google Scholar
  15. Jayakrishnan Madathil, Saket Saurabh, and Meirav Zehavi. Fixed-parameter tractable algorithm and polynomial kernel for max-cut above spanning tree. Theory of Computing Systems, 64(1):62-100, 2020. Google Scholar
  16. Meena Mahajan and Venkatesh Raman. Parameterizing above guaranteed values: Maxsat and maxcut. Journal of Algorithms, 31(2):335-354, 1999. Google Scholar
  17. Prabhakar Raghavan. Probabilistic construction of deterministic algorithms: approximating packing integer programs. Journal of Computer and System Sciences, 37(2):130-143, 1988. Google Scholar
  18. David P Williamson and David B Shmoys. The design of approximation algorithms. Cambridge university press, 2011. Google Scholar
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