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A Faster Algorithm for Vertex Cover Parameterized by Solution Size

Authors David G. Harris , N. S. Narayanaswamy

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Author Details

David G. Harris
  • Department of Computer Science, University of Maryland, College Park, MD, USA
N. S. Narayanaswamy
  • Department of Computer Science and Engineering, Indian Institute of Technology Madras, India

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David G. Harris and N. S. Narayanaswamy. A Faster Algorithm for Vertex Cover Parameterized by Solution Size. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 40:1-40:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.STACS.2024.40

Abstract

We describe a new algorithm for vertex cover with runtime O^*(1.25284^k), where k is the size of the desired solution and O^* hides polynomial factors in the input size. This improves over the previous runtime of O^*(1.2738^k) due to Chen, Kanj, & Xia (2010) standing for more than a decade. The key to our algorithm is to use a measure which simultaneously tracks k as well as the optimal value λ of the vertex cover LP relaxation. This allows us to make use of prior algorithms for Maximum Independent Set in bounded-degree graphs and Above-Guarantee Vertex Cover. The main step in the algorithm is to branch on high-degree vertices, while ensuring that both k and μ = k - λ are decreased at each step. There can be local obstructions in the graph that prevent μ from decreasing in this process; we develop a number of novel branching steps to handle these situations.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
Keywords
  • Vertex cover
  • FPT
  • Graph algorithm

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References

  1. J. Chen, I.A. Kanj, and G. Xia. Improved upper bounds for vertex cover. Theoretical Computer Science, 411(40-42):3736-3756, 2010. Google Scholar
  2. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  3. R.G. Downey and M.R. Fellows. Parameterized complexity. Springer, 1999. Google Scholar
  4. J. Flum and M. Grohe. Parameterized complexity theory. Springer, 2006. Google Scholar
  5. F. V. Fomin and D. Kratsch. Exact Exponential Algorithms. Springer-Verlag Berlin Heidelberg, 2010. Google Scholar
  6. M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H.Freeman and Company, 1979. Google Scholar
  7. Shivam Garg and Geevarghese Philip. Raising the bar for vertex cover: Fixed-parameter tractability above a higher guarantee. In Proc. 26th annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1152-1166, 2016. Google Scholar
  8. Leon Kellerhals, Tomohiro Koana, and Pascal Kunz. Vertex cover and feedback vertex set above and below structural guarantees. In Proc. 17th International Symposium on Parameterized and Exact Computation (IPEC), 2022. Google Scholar
  9. D. Lokshtanov, N. S. Narayanaswamy, V. Raman, M. S. Ramanujan, and S. Saurabh. Faster parameterized algorithms using linear programming. ACM Trans. Algorithms, 11(2):15:1-15:31, 2014. Google Scholar
  10. M. Mahajan and V. Raman. Parameterizing above guaranteed values: MaxSat and MaxCut. Journal of Algorithms, 31:335-354, 1999. Google Scholar
  11. M. Mahajan, V. Raman, and S. Sikdar. Parameterizing above or below guaranteed values. Journal of Computer and System Sciences, 75(2):137-153, 2009. Google Scholar
  12. G. L. Nemhauser and L. E. Trotter. Properties of vertex packing and independence system polyhedra. Mathematical Programming, 6(1):48-61, 1974. Google Scholar
  13. G. L. Nemhauser and L. E. Trotter. Vertex packings: Structural properties and algorithms. Mathematical Programming, 8(1):232-248, 1975. Google Scholar
  14. R. Niedermeier. Invitation to fixed-parameter algorithms. Oxford University Press, 2006. Google Scholar
  15. Dekel Tsur. Above guarantee parameterization for vertex cover on graphs with maximum degree 4. Journal of Combinatorial Optimization, 45(1):34, 2023. Google Scholar
  16. Mingyu Xiao and Hiorshi Nagamochi. A refined algorithm for maximum independent set in degree-4 graphs. Journal of Combinatorial Optimization, 34(3):830-873, 2017. Google Scholar
  17. Mingyu Xiao and Hiroshi Nagamochi. Confining sets and avoiding bottleneck cases: A simple maximum independent set algorithm in degree-3 graphs. Theoretical Computer Science, 469:92-104, 2013. Google Scholar
  18. Mingyu Xiao and Hiroshi Nagamochi. An exact algorithm for maximum independent set in degree-5 graphs. Discrete Applied Mathematics, 199:137-155, 2016. Google Scholar
  19. Mingyu Xiao and Hiroshi Nagamochi. Exact algorithms for maximum independent set. Inf. Comput., 255:126-146, 2017. URL: https://doi.org/10.1016/j.ic.2017.06.001.
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