Document Open Access Logo

Parameterized Complexity of Perfectly Matched Sets

Authors Akanksha Agrawal , Sutanay Bhattacharjee, Satyabrata Jana , Abhishek Sahu

PDF
Thumbnail PDF

File

LIPIcs.IPEC.2022.2.pdf
  • Filesize: 0.8 MB
  • 13 pages

Document Identifiers

Author Details

Akanksha Agrawal
  • Indian Institute of Technology Madras, Chennai, India
Sutanay Bhattacharjee
  • Indian Institute of Technology Madras, Chennai, India
Satyabrata Jana
  • The Institute of Mathematical Sciences, HBNI, Chennai, India
Abhishek Sahu
  • Indian Institute of Technology Madras, Chennai, India

Funding

  • Agrawal, Akanksha: Supported by New Faculty Initiation Grant no. NFIG008972.

Cite AsGet BibTex

Akanksha Agrawal, Sutanay Bhattacharjee, Satyabrata Jana, and Abhishek Sahu. Parameterized Complexity of Perfectly Matched Sets. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 2:1-2:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.IPEC.2022.2

Abstract

For an undirected graph G, a pair of vertex disjoint subsets (A, B) is a pair of perfectly matched sets if each vertex in A (resp. B) has exactly one neighbor in B (resp. A). In the above, the size of the pair is |A| (= |B|). Given a graph G and a positive integer k, the Perfectly Matched Sets problem asks whether there exists a pair of perfectly matched sets of size at least k in G. This problem is known to be NP-hard on planar graphs and W[1]-hard on general graphs, when parameterized by k. However, little is known about the parameterized complexity of the problem in restricted graph classes. In this work, we study the problem parameterized by k, and design FPT algorithms for: i) apex-minor-free graphs running in time 2^O(√k)⋅ n^O(1), and ii) K_{b,b}-free graphs. We obtain a linear kernel for planar graphs and k^𝒪(d)-sized kernel for d-degenerate graphs. It is known that the problem is W[1]-hard on chordal graphs, in fact on split graphs, parameterized by k. We complement this hardness result by designing a polynomial-time algorithm for interval graphs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
Keywords
  • Perfectly Matched Sets
  • Parameterized Complexity
  • Apex-minor-free graphs
  • d-degenerate graphs
  • Planar graphs
  • Interval Graphs

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. N. R. Aravind and Roopam Saxena. Perfectly Matched Sets in Graphs: Hardness, Kernelization Lower Bound, and FPT and Exact Algorithms. CoRR, abs/2107.08584, 2021. URL: http://arxiv.org/abs/2107.08584.
  2. Paul S. Bonsma. The complexity of the matching-cut problem for planar graphs and other graph classes. J. Graph Theory, 62(2):109-126, 2009. Google Scholar
  3. Leizhen Cai, Siu Man Chan, and Siu On Chan. Random Separation: A New Method for Solving Fixed-Cardinality Optimization Problems. In Hans L. Bodlaender and Michael A. Langston, editors, Parameterized and Exact Computation, Second International Workshop, IWPEC 2006, Zürich, Switzerland, September 13-15, 2006, Proceedings, volume 4169 of Lecture Notes in Computer Science, pages 239-250. Springer, 2006. Google Scholar
  4. Kathie Cameron. Induced matchings in intersection graphs. Discret. Math., 278(1-3):1-9, 2004. Google Scholar
  5. Yixin Cao and Dániel Marx. Interval Deletion Is Fixed-Parameter Tractable. ACM Trans. Algorithms, 11(3):21:1-21:35, 2015. Google Scholar
  6. Chi-Yeh Chen, Sun-Yuan Hsieh, Hoàng-Oanh Le, Van Bang Le, and Sheng-Lung Peng. Matching Cut in Graphs with Large Minimum Degree. Algorithmica, 83(5):1238-1255, 2021. Google Scholar
  7. Vasek Chvátal. Recognizing decomposable graphs. J. Graph Theory, 8(1):51-53, 1984. Google Scholar
  8. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  9. Konrad K. Dabrowski, Marc Demange, and Vadim V. Lozin. New results on maximum induced matchings in bipartite graphs and beyond. Theor. Comput. Sci., 478:33-40, 2013. Google Scholar
  10. Erik D. Demaine, MohammadTaghi Hajiaghayi, and Dimitrios M. Thilikos. The Bidimensional Theory of Bounded-Genus Graphs. SIAM J. Discret. Math., 20(2):357-371, 2006. Google Scholar
  11. Reinhard Diestel. Graph theory. Graduate texts in mathematics, 173, 2017. Google Scholar
  12. Rodney G. Downey and Michael R. Fellows. Parameterized Complexity. Monographs in Computer Science. Springer, 1999. Google Scholar
  13. Shimon Even, Oded Goldreich, Shlomo Moran, and Po Tong. On the NP-completeness of certain network testing problems. Networks, 14(1):1-24, 1984. Google Scholar
  14. Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, 2006. Google Scholar
  15. Fedor V. Fomin, Daniel Lokshtanov, Venkatesh Raman, and Saket Saurabh. Subexponential algorithms for partial cover problems. Inf. Process. Lett., 111(16):814-818, 2011. Google Scholar
  16. Martin Grohe, Stephan Kreutzer, and Sebastian Siebertz. Deciding first-order properties of nowhere dense graphs. CoRR, abs/1311.3899, 2013. URL: http://arxiv.org/abs/1311.3899.
  17. Stasys Jukna. Extremal Combinatorics - With Applications in Computer Science. Texts in Theoretical Computer Science. An EATCS Series. Springer, 2011. Google Scholar
  18. Iyad A. Kanj, Michael J. Pelsmajer, Marcus Schaefer, and Ge Xia. On the induced matching problem. J. Comput. Syst. Sci., 77(6):1058-1070, 2011. Google Scholar
  19. Johannes Köbler, Sebastian Kuhnert, and Osamu Watanabe. Interval graph representation with given interval and intersection lengths. Journal of Discrete Algorithms, 34:108-117, 2015. Google Scholar
  20. Łukasz Kowalik, Matjaž Krnc, Tomasz Waleń, et al. Improved induced matchings in sparse graphs. Discrete Applied Mathematics, 158(18):1994-2003, 2010. Google Scholar
  21. Hoàng-Oanh Le and Van Bang Le. On the Complexity of Matching Cut in Graphs of Fixed Diameter. In 27th International Symposium on Algorithms and Computation, ISAAC 2016, December 12-14, 2016, Sydney, Australia, volume 64 of LIPIcs, pages 50:1-50:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. Google Scholar
  22. Van Bang Le and Jan Arne Telle. The Perfect Matching Cut Problem Revisited. In Graph-Theoretic Concepts in Computer Science - 47th International Workshop, WG 2021, Warsaw, Poland, June 23-25, 2021, Revised Selected Papers, volume 12911 of Lecture Notes in Computer Science, pages 182-194. Springer, 2021. Google Scholar
  23. Silvio Micali and Vijay V. Vazirani. An O(sqrt(|v|) |E|) Algorithm for Finding Maximum Matching in General Graphs. In 21st Annual Symposium on Foundations of Computer Science, Syracuse, New York, USA, 13-15 October 1980, pages 17-27. IEEE Computer Society, 1980. Google Scholar
  24. Hannes Moser and Somnath Sikdar. The parameterized complexity of the induced matching problem. Discret. Appl. Math., 157(4):715-727, 2009. Google Scholar
  25. Augustine M. Moshi. Matching cutsets in graphs. J. Graph Theory, 13(5):527-536, 1989. Google Scholar
  26. Rolf Niedermeier. Invitation to Fixed-Parameter Algorithms. Oxford University Press, 2006. Google Scholar
  27. Larry J. Stockmeyer and Vijay V. Vazirani. NP-Completeness of Some Generalizations of the Maximum Matching Problem. Inf. Process. Lett., 15(1):14-19, 1982. Google Scholar
  28. Michele Zito. Induced matchings in regular graphs and trees. In Graph-Theoretic Concepts in Computer Science, pages 89-101, Berlin, Heidelberg, 1999. Springer Berlin Heidelberg. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact