[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Log in

New Results on Polynomial Inapproximabilityand Fixed Parameter Approximability of Edge Dominating Set

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

An edge dominating set in a graph G = (V, E) is a subset S of edges such that each edge in ES is adjacent to at least one edge in S. The edge dominating set problem, to find an edge dominating set of minimum size, is a basic and important NP-hard problem that has been extensively studied in approximation algorithms and parameterized complexity. In this paper, we present improved hardness results and parameterized approximation algorithms for edge dominating set. More precisely, we first show that it is NP-hard to approximate edge dominatingset in polynomial time within a factor better than 1.18. Next, we give a parameterized approximation schema (with respect to the standard parameter) for the problem and, finally, we develop an O (1.821τ)-time exact algorithm where τ is the size of a minimum vertex cover of G.

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

Access this article

Subscribe and save

Springer+ Basic
£29.99 /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 (United Kingdom)

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Binkele-Raible, D., Fernau, H.: Enumerate and measure: improving parameter budget management. In: Raman, V., Saurabh, S. (eds.) Proc. International Symposium on Parameterized and Exact Computation, IPEC’10, volume 6478 of Lecture Notes in Computer Science, pp 38–49. Springer-Verlag (2010)

  2. Bourgeois, N., Escoffier, B., Paschos, V. Th.: Approximation of max independent set, min vertex cover and related problems by moderately exponential algorithms. Discret. Appl. Math. 159(17), 1954–1970 (2011)

    MATH  MathSciNet  Google Scholar 

  3. Brankovic, L., Fernau, H.: A novel parameterised approximation algorithm for minimum vertex cover. Theor. Comput. Sci. 511, 85–108 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cai, L., Huang, X.: Fixed-parameter approximation: conceptual framework and approximability results. In: Bodlaender, H.L., Langston, M.A. (eds.) Proc. International Workshop on Parameterized and Exact Computation, IWPEC’06, volume 4169 of Lecture Notes in Computer Science, pp 96–108. Springer-Verlag (2006)

  5. Cardinal, J., Langerman, S., Levy, E.: Improved approximation bounds for edge dominating set in dense graphs. Theoret. Comput. Sci. 410(8-10), 949–957 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Carr, R., Fujito, T., Konjevod, G., Parekh, O.: A \((2+\frac {1}{10})\)-approximation algorithm for a generalization of the weighted edge-dominating set problem. J. Comb. Optim. 5, 317–326 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chen, J., Kanj, I.A., Xia, G.: Improved upper bounds for vertex cover. Theoret. Comput. Sci. 411(40-42), 3736–3756 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chlebik, M., Chlebikova, J.: Approximation hardness of edge dominating set problems. J. Comb. Optim. 11(3), 279–290 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Dinur, I., Safra, M.: The importance of being biased. Proc. STOC’02, 33–42 (2002)

  10. Downey, R.G., Fellows, M.R., McCartin, C., Rosamond, F.A.: Parameterized approximation of dominating set problems. Inform. Process. Lett. 109(1), 68–70 (2008)

    Article  MathSciNet  Google Scholar 

  11. Fellows, M.R., Kulik, A., Rosamond, F.A., Shachnai, H.: Parameterized approximation via fidelity preserving transformations. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) Proc. ICALP’12, volume 7391 of Lecture Notes in Computer Science, pp 351–362. Springer-Verlag (2012)

  12. Fernau, H.: Edge dominating set: efficient enumeration-based exact algorithms. In: Bodlaender, H.L., Langston, M.A. (eds.) Proc. International Workshop on Parameterized and Exact Computation, IWPEC’06, volume 4169 of Lecture Notes in Computer Science, pp 142–153. Springer-Verlag (2006)

  13. Fomin, F.V., Gaspers, S., Saurabh, S., Stepanov, A.A.: On two techniques of combining branching and treewidth. Algorithmica 54(2), 181–207 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  14. Fujito, T., Nagamochi, H.: A 2-approximation algorithm for the minimum weight edge dominating set problem. Discret. Appl. Math. 118, 199–207 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  15. Garey, M.R., Johnson, D.S.: Computers and intractability. A guide to the theory of NP-completeness. W.H. Freeman, San Francisco (1979)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  17. Marx, D.: Parameterized complexity and approximation algorithms. Comput. J. 51(1), 60–78 (2008)

    Article  Google Scholar 

  18. Raman, V., Saurabh, S., Sikdar, S.: Efficient exact algorithms through enumerating maximal independent sets and other techniques. Theory Comput. Syst. 41(3), 563–587 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  19. Schmied, R., Viehmann, C.: Approximating edge dominating set in dense graphs. Theoret. Comput. Sci. 414(1), 92–99 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  20. van Rooij, J.M.M., Bodlaender, H.L.: Exact Algorithms for Edge Domination. Algorithmica 64(4), 535–563 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  21. Xiao, M., Kloks, T., Poon, S.-H.: New parameterized algorithms for the edge dominating set problem. Theor. Comput. Sci. 511, 147–158 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  22. Xiao, M., Nagamochi, H.: Parameterized edge dominating set in graphs with degree bounded by 3. Theor. Comput. Sci. 508, 2–15 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  23. Xiao, M., Nagamochi, H.: A refined exact algorithm for edge dominating set. In: Agrawal, M., Barry Cooper, S., Li, A. (eds.) Proc. Theory and Applications of Models of Computation, TAMC’12, volume 7287 of Lecture Notes in Computer Science, pp 360–372. Springer-Verlag (2012)

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

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Bruno Escoffier.

Additional information

Research partially supported by the French Agency for Research under the DEFIS program TODO, ANR-09-EMER-010, the National Natural Science Foundation of China under the Grant 61370071 and Fundamental Research Funds for the Central Universities under the Grant ZYGX2012J069. An extended abstract of the paper appears in the proceedings of IPEC’12, LNCS, volume 7535

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Escoffier, B., Monnot, J., Paschos, V.T. et al. New Results on Polynomial Inapproximabilityand Fixed Parameter Approximability of Edge Dominating Set . Theory Comput Syst 56, 330–346 (2015). https://doi.org/10.1007/s00224-014-9549-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-014-9549-5

Keywords

Navigation