The Complexity of König Subgraph Problems and Above-Guarantee Vertex Cover | Algorithmica Skip to main content
Springer Nature Link
Log in

The Complexity of König Subgraph Problems and Above-Guarantee Vertex Cover

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

A graph is König-Egerváry if the size of a minimum vertex cover equals that of a maximum matching in the graph. These graphs have been studied extensively from a graph-theoretic point of view. In this paper, we introduce and study the algorithmic complexity of finding König-Egerváry subgraphs of a given graph. In particular, given a graph G and a nonnegative integer k, we are interested in the following questions:

  1. 1.

    does there exist a set of k vertices (edges) whose deletion makes the graph König-Egerváry?

  2. 2.

    does there exist a set of k vertices (edges) that induce a König-Egerváry subgraph?

We show that these problems are NP-complete and study their complexity from the points of view of approximation and parameterized complexity. Towards this end, we first study the algorithmic complexity of Above Guarantee Vertex Cover, where one is interested in minimizing the additional number of vertices needed beyond the maximum matching size for the vertex cover. Further, while studying the parameterized complexity of the problem of deleting k vertices to obtain a König-Egerváry graph, we show a number of interesting structural results on matchings and vertex covers which could be useful in other contexts.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agarwal, A., Charikar, M., Makarychev, K., Makarychev, Y.: \(O(\sqrt{\log n})\) approximation algorithms for Min UnCut, min-2CNF deletion, and directed cut problems. In: Proceedings of the 37th ACM Symposium on Theory of Computing (STOC 2005), pp. 573–581 (2005)

  2. Bourjolly, J.M., Pulleyblank, W.R.: König-Egerváry graphs, 2-bicritical graphs and fractional matchings. Discrete Appl. Math. 24, 63–82 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chlebik, M., Chlebikova, J.: On approximation hardness of the minimum 2-SAT deletion problem. Discrete Appl. Math. 155(2), 172–179 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, J., Kanj, I.A.: On approximating minimum vertex cover for graphs with perfect matching. Theor. Comput. Sci. 337, 305–318 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Deming, R.W.: Independence numbers of graphs—an extension of the König-Egerváry theorem. Discrete Math. 27, 23–33 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dinur, I., Safra, S.: On the hardness of approximating minimum vertex cover. Ann. Math. 162(1), 439–485 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Downey, R., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999)

    Book  Google Scholar 

  8. Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, New York (2006)

    Google Scholar 

  9. Fomin, F.V., Grandoni, F., Kratsch, D.: Measure and conquer: a simple O(20.288n) independent set algorithm. In: Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2006), pp. 18–25 (2006)

  10. Gavril, F.: Testing for equality between maximum matching and minimum node covering. Inf. Process. Lett. 6(6), 199–202 (1977)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Guo, J., Gramm, J., Hüffner, F., Niedermeier, R., Wernicke, S.: Compression-based fixed-parameter algorithms for feedback vertex set and edge bipartization. J. Comput. Syst. Sci. 72(8), 1386–1396 (2006)

    Article  MATH  Google Scholar 

  13. Håstad, J.: Clique is hard to approximate to within n 1−ε. Acta Math. 182, 105–142 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. Khot, S.: On the power of unique 2-prover 1-round games. In: Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC 2002), pp. 767–775 (2002)

  15. Khot, S., Regev, O.: Vertex cover might be hard to approximate to within 2−ε. J. Comput. Syst. Sci. 74(3), 335–349 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Klein, P.N., Plotkin, S.A., Rao, S., Tardos, E.: Approximation algorithms for Steiner and directed multicuts. J. Algorithms 22(2), 241–269 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. Korach, E., Nguyen, T., Peis, B.: Subgraph characterization of red/blue-split graphs and König-Egerváry graphs. In: Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2006), pp. 842–850 (2006)

  18. Lovász, L.: Ear-decompositions of matching-covered graphs. Combinatorica 3, 105–118 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lovász, L., Plummer, M.D.: Matching Theory. North Holland, Amsterdam (1986)

    MATH  Google Scholar 

  20. Mahajan, M., Raman, V.: Parameterizing above guaranteed values: max sat and max cut. J. Algorithms 31(2), 335–354 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  21. Moser, H., Thilikos, D.M.: Parameterized complexity of finding regular induced subgraphs. In: Proceedings of the 2nd Workshop on Algorithms and Complexity in Durham (ACiD 2006), pp. 107–118 (2006)

  22. Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  23. Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, London (2006)

    Book  MATH  Google Scholar 

  24. Razgon, I., O’Sullivan, B.: Almost 2-SAT is fixed-parameter tractable. J. Comput. Syst. Sci. 75(8), 435–450 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Reed, B., Smith, K., Vetta, A.: Finding odd cycle transversals. Oper. Res. Lett. 32, 299–301 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Schröder, H., May, A.E., Sýkora, O., Vrt’o, I.: Approximation algorithms for the vertex bipartization problem. In: Proceedings of the 24th Seminar on Current Trends in Theory and Practice of Informatics (SOFSEM 1997). LNCS, vol. 1338, pp. 547–554. Springer, Berlin (1997)

    Google Scholar 

  27. Sterboul, F.: A characterization of graphs in which the transversal number equals the matching number. J. Comb. Theory, Ser. B 27, 228–229 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  28. Subramanian, C.R.: Vertex covers: parameterizing above the requirement. IMSc Technical Report, February 2001

  29. Vazirani, V.V.: A theory of alternating paths and blossoms for proving correctness of the \(O(\sqrt{V}E)\) general graph maximum matching algorithm. Combinatorica 14(1), 71–109 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  30. Vishwanathan, S.: On hard instances of approximate vertex cover. ACM Trans. Algorithms 5(1) (2008). doi:10.1145/1435375.1435382

  31. Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. In: Proceedings of the 38th ACM Symposium on Theory of Computing (STOC 2006), pp. 681–690 (2006)

Download references

Author information

Authors and Affiliations

  1. The Institute of Mathematical Sciences, Chennai, India

    Venkatesh Raman, Saket Saurabh & C. R. Subramanian

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

    Sounaka Mishra

  3. Department of Computer Science, RWTH Aachen University, 52056, Aachen, Germany

    Somnath Sikdar

Authors
  1. Sounaka Mishra
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Venkatesh Raman
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Saket Saurabh
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Somnath Sikdar
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. C. R. Subramanian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Saket Saurabh.

Additional information

Preliminary versions of this paper appeared in the proceedings of the 18th and 19th International Symposium on Algorithms and Computation (ISAAC 2007 and ISAAC 2008).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mishra, S., Raman, V., Saurabh, S. et al. The Complexity of König Subgraph Problems and Above-Guarantee Vertex Cover. Algorithmica 61, 857–881 (2011). https://doi.org/10.1007/s00453-010-9412-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-010-9412-2

Keywords

Search

Navigation