Skip to main content
SpringerLink
Log in

On the Complexity of Minimum Maximal Acyclic Matchings

  • Conference paper
  • First Online:
Computing and Combinatorics (COCOON 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13595))

Included in the following conference series:

Abstract

Given a graph G, Min-Max-Acy-Matching is the problem of finding a maximal matching M in G of minimum cardinality such that the set of M-saturated vertices induces an acyclic subgraph in G. Min-Max-Acy-Matching is known to be \({\textsf{NP}}\)-hard. In this paper, we strengthen this result by proving that the decision version of Min-Max-Acy-Matching is \({\textsf{NP}}\)-complete for planar perfect elimination bipartite graphs. We also prove that Min-Max-Acy-Matching for bipartite graphs cannot be approximated within a ratio of \(n^{1-\epsilon }\), for any \(\epsilon >0\) unless \({\textsf{P}}={\textsf{NP}}\). Finally, we show that Min-Max-Acy-Matching is \(\textsf{APX}\)-hard for 4-regular graphs.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Spaccamela, A.M., Protasi, M.: Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties. Springer, Heidelberg (2012)

    MATH  Google Scholar 

  2. Bazgan, C., et al.: The many facets of upper domination. Theoret. Comput. Sci. 717, 2–25 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  3. Boria, N., Croce, F.D., Paschos, V.T.: On the max min vertex cover problem. Discret. Appl. Math. 196, 62–71 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chaudhary, J., Panda, B.S.: On the complexity of minimum maximal uniquely restricted matching. Theoret. Comput. Sci. 882, 15–28 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  5. Damaschke, P.: Parameterized algorithms for double hypergraph dualization with rank limitation and maximum minimal vertex cover. Discret. Optim. 8(1), 18–24 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dublois, L., Hanaka, T.,. Ghadikolaei, M.K, Lampis, M., Melissinos, N.: (In)approximability of maximum minimal FVS. In: Proceedings of the 31st International Symposium on Algorithms and Computation (ISAAC). LIPIcs, vo. 181, pp. 3:1–3:14 (2020)

    Google Scholar 

  7. Edmonds, J.: Paths, trees, and flowers. Can. J. Math. 17, 449–467 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fürst, M.: Restricted matchings. Ph.D. thesis, Universität Ulm (2019)

    Google Scholar 

  9. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, New York (1979)

    MATH  Google Scholar 

  10. Goddard, W., Hedetniemi, S.M., Hedetniemi, S.T., Laskar, R.: Generalized subgraph-restricted matchings in graphs. Discret. Math. 293(1–3), 129–138 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Goddard, W., Henning, M.A.: Independent domination in graphs: a survey and recent results. Discret. Math. 313(7), 839–854 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Golumbic, M.C., Goss, C.F.: Perfect elimination and chordal bipartite graphs. J. Graph Theory 2(2), 155–163 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kann, V.: On the approximability of NP-complete optimization problems. Ph.D. thesis, Royal Institute of Technology Stockholm (1992)

    Google Scholar 

  14. Krishnamoorthy, M.S., Deo, N.S.: Node-deletion NP-complete problems. SIAM J. Comput. 8(4), 619–625 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lepin, V.V.: A linear algorithm for computing of a minimum weight maximal induced matching in an edge-weighted tree. Electron. Notes Discrete Math. 24, 111–116 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lovász, L., Plummer, M.D.: Matching Theory, vol. 367. American Mathematical Society (2009)

    Google Scholar 

  17. Orlovich, Y.L., Finke, G., Gordon, V., Zverovich, I.: Approximability results for the maximum and minimum maximal induced matching problems. Discret. Optim. 5(3), 584–593 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Orlovich, Y.L., Zverovich, I.E.: Maximal induced matchings of minimum/maximum size. Technical report, DIMACS TR 2004-26 (2004)

    Google Scholar 

  19. Panda, B.S., Chaudhary, J.: Acyclic matching in some subclasses of graphs. In: Gąsieniec, L., Klasing, R., Radzik, T. (eds.) IWOCA 2020. LNCS, vol. 12126, pp. 409–421. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-48966-3_31

    Chapter  Google Scholar 

  20. Panda, B.S., Pandey, A.: On the complexity of minimum cardinality maximal uniquely restricted matching in graphs. In: Arumugam, S., Bagga, J., Beineke, L.W., Panda, B.S. (eds.) ICTCSDM 2016. LNCS, vol. 10398, pp. 218–227. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-64419-6_29

    Chapter  Google Scholar 

  21. Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. J. Comput. Syst. Sci. 43(3), 425–440 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  22. Tovey, C.A.: A simplified NP-complete satisfiability problem. Discret. Appl. Math. 8(1), 85–89 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  23. Yannakakis, M., Gavril, F.: Edge dominating sets in graphs. SIAM J. Appl. Math. 38(3), 364–372 (1980)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Ben-Gurion University of the Negev, Be’er Sheva, Israel

    Juhi Chaudhary

  2. Department of Mathematics, Indian Institute of Technology Madras, Chennai, India

    Sounaka Mishra

  3. Department of Mathematics, Indian Institute of Technology Delhi, New Delhi, India

    B. S. Panda

Authors
  1. Juhi Chaudhary
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sounaka Mishra
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. B. S. Panda
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Juhi Chaudhary .

Editor information

Editors and Affiliations

  1. Shenzhen Institute of Advanced Technology, Shenzhen, China

    Yong Zhang

  2. Harbin Institute of Technology, Harbin, China

    Dongjing Miao

  3. Technical University of Berlin, Berlin, Germany

    Rolf Möhring

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Chaudhary, J., Mishra, S., Panda, B.S. (2022). On the Complexity of Minimum Maximal Acyclic Matchings. In: Zhang, Y., Miao, D., Möhring, R. (eds) Computing and Combinatorics. COCOON 2022. Lecture Notes in Computer Science, vol 13595. Springer, Cham. https://doi.org/10.1007/978-3-031-22105-7_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-22105-7_10

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-22104-0

  • Online ISBN: 978-3-031-22105-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation