Theory of Computing Systems

, Volume 62, Issue 3, pp 533–556 | Cite as

On Approximating (Connected) 2-Edge Dominating Set by a Tree

  • Toshihiro Fujito
  • Tomoaki Shimoda


The edge dominating set problem (EDS) is to compute a minimum size edge set such that every edge is dominated by some edge in it. This paper considers a variant of EDS with extensions of multiple and connected dominations combined. In the b-EDS problem, each edge needs to be dominated b times. Connected EDS requires an edge dominating set to be connected while it has to form a tree in Tree Cover. Although each of EDS, b-EDS, and Connected EDS (or Tree Cover) has been well studied, each known to be approximable within 2 (or 8/3 for b-EDS in general), nothing is known when these extensions are imposed simultaneously on EDS unlike in the case of the (vertex) dominating set problem. We consider Connected 2-EDS and 2-Tree Cover (i.e., a combination of 2-EDS and Tree Cover), and present a polynomial algorithm approximating each within 2. Moreover, it will be shown that the single tree computed is no larger than twice the optimum for (not necessarily connected) 2-EDS, thus also approximating 2-EDS equally well. It also implies that 2-EDS with clustering properties can be approximated within 2 as well.


Edge dominating sets Approximation algorithms Tree cover Connected dominating sets b-edge domination 



The authors are grateful to the anonymous referees for a number of valuable comments and suggestions.


  1. 1.
    Alber, J., Betzler, N., Niedermeier, R.: Experiments on data reduction for optimal domination in networks. Ann. Oper. Res. 146, 105–117 (2006)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Arkin, E. M., Halldórsson, M. M., Hassin, R.: Approximating the tree and tour covers of a graph. Inform. Process. Lett. 47, 275–282 (1993)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Armon, A.: On Min-max r-gatherings. Theor. Comput. Sci. 412(7), 573–582 (2011)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Baker, B. S.: Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41, 153–180 (1994)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Berger, A., Fukunaga, T., Nagamochi, H., Parekh, O.: Approximability of the capacitated b-edge dominating set problem. Theory Comput Syst. 385(1–3), 202–213 (2007)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Binkele-Raible, D, Fernau, H.: Enumerate and Measure: Improving Parameter Budget Management. In: Proceedings International Conference on Parameterized and Exact Computation, pp. 38–49. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  7. 7.
    Blum, J., Ding, M., Thaeler, A., Cheng, X.: Connected Dominating Set in Sensor Networks and MANETs. Springer, New York (2005)CrossRefMATHGoogle Scholar
  8. 8.
    Chellali, M., Favaron, O., Hansberg, A.: Volkmann, l.: k-domination and k-independence in graphs: a survey. Graphs Combin. 28, 1–55 (2012)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Chlebík, M., Chlebíková, J.: Approximation hardness of edge dominating set problems, vol. 11 (2006)Google Scholar
  10. 10.
    Cooper, C., Klasing, R., Zito, M.: Dominating Sets in Web Graphs. In: Algorithms and Models for the Web-Graph, LNCS 3243, pp. 31–43. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  11. 11.
    Dai, F., Wu, J.: On Constructing k-connected k-dominating Set in Wireless Ad Hoc and Sensor Networks. J. Parallel Distrib. Comput. 66(7), 947–958 (2006)CrossRefMATHGoogle Scholar
  12. 12.
    Du, D.-Z., Wan, P.-J.: Connected Dominating Set: Theory and Applications. Springer, New York (2013)CrossRefMATHGoogle Scholar
  13. 13.
    Du, H., Ding, L., Wu, W., Kim, D., Pardalos, P. M., Willson, J.: Connected Dominating Set in Wireless Networks. In: Pardalos, P.M., Du, D.-Z., Graham, R. (eds.) Handbook of Combinatorial Optimization, pp. 783–833. Springer, New York (2013)CrossRefGoogle Scholar
  14. 14.
    Escoffier, B., Monnot, J., Paschos, V.T.h., Xiao, M.: New results on polynomial inapproximabilityand fixed parameter approximability of edge dominating set. Theory comput Syst. 56(2), 330–346 (2015)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Fernau, H.: EDGE DOMINATING SET: Efficient Enumeration-based Exact Algorithms Proceedings 2Nd International Conference on Parameterized and Exact Computation, pp. 142–153. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  16. 16.
    Fernau, H., Fomin, F. V., Philip, G., Saurabh, S.: The Curse of Connectivity: t-Total Vertex (Edge) Cover. In: Proceedings 16Th Annual International Conference on Computing and Combinatorics, pp. 34–43. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  17. 17.
    Fernau, H., Manlove, D. F.: Vertex and edge covers with clustering properties: Complexity and algorithms. J. Discrete Algoritms 7(2), 149–167 (2009)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Fink, J. F., Jacobson, M.S.: n-Domination in Graphs. In: Alavi, Y., Chartrand, G., Lick, D. R., Wall, C. E., Lesniak, L. (eds.) Graph Theory with Applications to Algorithms and Computer Science, pp. 283–300. John Wiley & Sons, Inc., New York (1985)Google Scholar
  19. 19.
    Fink, J. F., Jacobson, M. S.: On N-Domination, N-Dependence and Forbidden Subgraphs. In: Alavi, Y., Chartrand, G., Lick, D. R., Wall, C. E., authorLesniak, L. (eds.) Graph Theory with Applications to Algorithms and Computer Science, pp. 301–311. John Wiley & Sons, Inc., New York (1985)Google Scholar
  20. 20.
    Fomin, F. V., Gaspers, S., Saurabh, S., Stepanov, A. A.: On two techniques of combining branching and treewidth. Algorithmica 54(2), 181–207 (2009)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Fujito, T.: On Matchings and b-Edge Dominating Sets: a 2-Approximation Algorithm for the 3-Edge Dominating Set Problem. In: Algorithm Theory–SWAT 2014, LNCS 8503, pp. 206–216. Springer, Cham (2014)Google Scholar
  22. 22.
    Gao, X., Zou, F., Kim, D., Du, D.-Z.: The Latest Researches on Dominating Problems in Wireless Sensor Network. In: Handbook on Sensor Networks, Pp. 197–226. World Scientific (2010)Google Scholar
  23. 23.
    Harary, F.: Graph theory. Addison-wesley, reading MA (1969)Google Scholar
  24. 24.
    Haynes, T. W., Hedetniemi, S. T., Slater, P. J.: Domination in Graphs, Advanced Topics. Marcel Dekker, New York (1998)MATHGoogle Scholar
  25. 25.
    Haynes, T. W., Hedetniemi, S. T., Slater, P. J.: Fundamantals of Domination in Graphs. Marcel Dekker, New York (1998)MATHGoogle Scholar
  26. 26.
    Hunt III, H.B., Marathe, M.V., Radhakrishnan, V., Ravi, S.S., Rosenkrantz, D.J., Stearns, R.E.: A Unified Approach to Approximation Schemes for NP- and PSPACE-Hard Problems for Geometric Graphs. In: Proceedings 2Nd Ann. European Symp. on Algorithms, LNCS 855, pp. 424–435. Springer (1994)Google Scholar
  27. 27.
    Kim, D., Gao, X., Zou, F., Du, D.-Z.: Construction of Fault-Tolerant Virtual Backbones in Wireless Networks. In: Handbook on Security and Networks, pp. 488–509. World Scientific (2011)Google Scholar
  28. 28.
    Lovász, L., Plummer, M. D.: Matching Theory. North-Holland, Amsterdam (1986)MATHGoogle Scholar
  29. 29.
    Małafiejski, M., Żyliński, P.: Weakly Cooperative Guards in Grids. In: Proceedings 2005 International Conference on Computational Science and Its Applications - Volume Part I, pp. 647–656. Springer, Heidelberg (2005)Google Scholar
  30. 30.
    Savage, C.: Depth-first search and the vertex cover problem. Inf. Process. Lett. 14(5), 233–235 (1982)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Schmied, R., Viehmann, C.: Approximating edge dominating set in dense graphs. Theor. Comput. Sci. 414(1), 92–99 (2012)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Shang, W., Wan, P., Yao, F., Hu, X.: Algorithms for minimum m-connected k-tuple dominating set problem. Theor. Comput. Sci. 381(1–3), 241–247 (2007)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Shi, Y., Zhang, Y., Zhang, Z., Wu, W.: A greedy algorithm for the minimum 2-connected m-fold dominating set problem. J. Comb. Optim. 31(1), 136–151 (2016)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Thai, M. T., Zhang, N., Tiwari, R., Xu, X.: On approximation algorithms of k-connected m-dominating sets in disk graphs. Theor. Comput. Sci. 385(1–3), 49–59 (2007)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Wu, Y., Li, Y.: Construction Algorithms for K-Connected M-Dominating Sets in Wireless Sensor Networks. In: Proceedings 9Th ACM International Symposium on Mobile Ad Hoc Networking and Computing, pp. 83–90. ACM, New York (2008)Google Scholar
  36. 36.
    Xiao, M., Kloks, T., Poon, S.-H.: New parameterized algorithms for the edge dominating set problem. Theor. Comput. Sci. 511, 147–158 (2013)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Yannakakis, M., Gavril, F.: Edge dominating sets in graphs. SIAM J. Appl. Math. 38(3), 364–372 (1980)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Zhou, J., Zhang, Z., Wu, W., Xing, K.: A greedy algorithm for the fault-tolerant connected dominating set in a general graph. J. Comb. Optim. 28(1), 310–319 (2014)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringToyohashi University of TechnologyToyohashiJapan

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