Advertisement

Journal of Combinatorial Optimization

, Volume 11, Issue 3, pp 279–290 | Cite as

Approximation hardness of edge dominating set problems

  • Miroslav ChlebíkEmail author
  • Janka Chlebíková
Article

Abstract

We provide the first interesting explicit lower bounds on efficient approximability for two closely related optimization problems in graphs, MINIMUM EDGE DOMINATING SET and MINIMUM MAXIMAL MATCHING. We show that it is NP-hard to approximate the solution of both problems to within any constant factor smaller than \({\frac{7}{6}}\). The result extends with negligible loss to bounded degree graphs and to everywhere dense graphs.

Keywords

Minimum edge dominating set Minimum maximal matching Approximation lower bound Bounded degree graphs Everywhere dense graphs 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Baker BS (1994) Approximation algorithms for NP-complete problems on planar graphs. J ACM 41:153–180CrossRefzbMATHGoogle Scholar
  2. Carr R, Fujito T, Konjevod G, Parekh O (2001) A 2110-approximation algorithm for a generalization of the weighted edge-dominating set problem. J Comb Optim 5:317–326CrossRefMathSciNetGoogle Scholar
  3. Chung FRK (1997) Spectral Graph Theory. CBMS Regional Conference Series in Mathematics, American Mathematical Society, ISSN 0160-7642, ISBN 0-8218-0315-8Google Scholar
  4. Chlebík M, ChlebíkovéJ (2003a) Approximation hardness for small occurrence instances of NP-hard problems, In: Proc. of the 5th CIAC, LNCS 2653. Springer, pp. 152–164 (also ECCC Report TR02-73, 2002)Google Scholar
  5. Chlebík M, ChlebíkovéJ (2003b) Complexity of approximating bounded variants of optimization problems. Theor Comp Sci 354:320–338CrossRefGoogle Scholar
  6. Chlebík M, ChlebíkovéJ (2003b) Inapproximability results for bounded variants of optimization problems. In: Proc. of the 14th Inter. Symp. on Fundamentals of Computation Theory, FCT 2003, Malmö, Sweden, pp. 12–15 LNCS 2751, 2003, Springer, pp. 27–38, also ECCC Report TR03-26 (to appear in Theoretical Computer Science)Google Scholar
  7. Clementi A, Trevisan L (1999) Improved non-approximability results for minimum vertex cover with density constraints. Theor Comp Sci 225:113–128CrossRefMathSciNetGoogle Scholar
  8. Dinur I, Safra S (2002) The importance of being biased. In: Proc. of the 34th ACM Symposium on Theory of Computing, STOC, pp. 33–42Google Scholar
  9. Duckworth W, Wormald NC (preprint) Linear Programming and the Worst-Case Analysis of Greedy Algorithms on Cubic GraphsGoogle Scholar
  10. Feige U, Goldwasser S, Lovész L, Safra S, Szegedy M (1991) Approximating clique is almost NP-complete. In: Proc. of the 32nd IEEE Symposium on Foundations of Computer Science, pp. 2–12Google Scholar
  11. Fujito T, Nagamochi H (2002) A 2-approximation algorithm for the minimum weight edge dominating set problem. Discrete Appl Math 118:199–207CrossRefMathSciNetGoogle Scholar
  12. Gavril F (1978) A recognition algorithms for the total graph. Networks 8:121–133zbMATHMathSciNetGoogle Scholar
  13. Hå stad J (2001) Some optimal inapproximability results. J ACM 48:798–859CrossRefMathSciNetGoogle Scholar
  14. Horton JD, Kilakos K (1993) Minimum edge dominating sets. SIAM J Discrete Math 6:375–387CrossRefMathSciNetGoogle Scholar
  15. Hunt III HB, Marathe MV, Radhakrishnan V, Ravi SS, Rosenkrantz DJ, Stearns RE (1994) A unified approach to approximation schemes for NP- and PSPACE-hard problems for geometric graphs. In: Proc. of the 2nd Annual European Symposium on Algorithms, ESA 1994, LNCS 855:424–435MathSciNetGoogle Scholar
  16. Mitchell S, Hedetniemi S (1977) Edge domination in trees. In: Proc. of the 8th Southearn Conference on Combinatorics, Graph Theory, and Computing, pp. 489–509Google Scholar
  17. Parekh O (2002) Edge dominating and hypomatchable sets. In: Proc. of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pp. 287–291Google Scholar
  18. Srinivasan A, Madhukar K, Nagavamsi P, Pandu Rangan C, Chang M-S (1995) Edge domination on bipartite permutation graphs and cotriangulated graphs. Inf Proc Letters 56:165–171CrossRefMathSciNetGoogle Scholar
  19. Yannakakis M, Gavril F (1980) Edge dominating sets in graphs. SIAM J Appl Math 38:364–372CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  1. 1.Max Planck Institute for Mathematics in the SciencesLeipzigGermany
  2. 2.Department of Informatics Education, Faculty of Mathematics, Physics, and InformaticsComenius UniversityBratislavaSlovakia

Personalised recommendations