Advertisement

Algorithmica

, Volume 81, Issue 5, pp 1938–1964 | Cite as

Online Dominating Set

  • Joan Boyar
  • Stephan J. Eidenbenz
  • Lene M. Favrholdt
  • Michal Kotrbčík
  • Kim S. LarsenEmail author
Article
  • 51 Downloads

Abstract

This paper is devoted to the online dominating set problem and its variants. We believe the paper represents the first systematic study of the effect of two limitations of online algorithms: making irrevocable decisions while not knowing the future, and being incremental, i.e., having to maintain solutions to all prefixes of the input. This is quantified through competitive analyses of online algorithms against two optimal algorithms, both knowing the entire input, but only one having to be incremental. We also consider the competitive ratio of the weaker of the two optimal algorithms against the other. We consider important graph classes, distinguishing between connected and not necessarily connected graphs. For the classic graph classes of trees, bipartite, planar, and general graphs, we obtain tight results in almost all cases. We also derive upper and lower bounds for the class of bounded-degree graphs. From these analyses, we get detailed information regarding the significance of the necessary requirement that online algorithms be incremental. In some cases, having to be incremental fully accounts for the online algorithm’s disadvantage.

Keywords

Online algorithms Dominating set Competitive analysis Irrevocability 

Notes

Acknowledgements

The authors would like to thank an anonymous referee for constructive suggestions.

References

  1. 1.
    Alber, J., Bodlaender, H.L., Fernau, H., Kloks, T., Niedermeier, R.: Fixed parameter algorithms for dominating set and related problems on planar graphs. Algorithmica 33(4), 461–493 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41(1), 153–180 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berge, C.: Theory of Graphs and its Applications. Meuthen, London (1962)zbMATHGoogle Scholar
  4. 4.
    Böhm, M., Sgall, J., Veselý, P.: Online colored bin packing. In: Bampis, E., Svensson, O. (eds.) 12th International Workshop on Approximation and Online Algorithms (WAOA), Lecture Notes in Computer Science, vol. 8952, pp. 35–46. Springer, Berlin (2015)Google Scholar
  5. 5.
    Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  6. 6.
    Boyar, J., Larsen, K.S.: The seat reservation problem. Algorithmica 25(4), 403–417 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chrobak, M., Sgall, J., Woeginger, G.J.: Two-bounded-space bin packing revisited. In: Demetrescu, C., Halldórsson, M.M. (eds.) 19th Annual European Symposium on Algorithms (ESA), Lecture Notes in Computer Science, vol. 6942, pp. 263–274. Springer, Berlin (2011)Google Scholar
  8. 8.
    Chvátal, V.: A greedy heuristic for the set-covering problem. Math. Oper. Res. 4(3), 233–235 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Das, B., Bharghavan, V.: Routing in ad-hoc networks using minimum connected dominating sets. IEEE Int. Conf. Commun. 1, 376–380 (1997)Google Scholar
  10. 10.
    Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness I: basic results. SIAM J. Comput. 24(4), 873–921 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Du, D.Z., Wan, P.J.: Connected Dominating Set: Theory and Applications. Springer, New York (2013)CrossRefzbMATHGoogle Scholar
  12. 12.
    Feige, U.: A threshold of \(\ln n\) for approximating set cover. J. ACM 45(4), 634–652 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Haynes, T.W., Hedetniemi, S., Slater, P.: Fundamentals of Domination in Graphs. Marcel Dekker, New York (1998)zbMATHGoogle Scholar
  14. 14.
    Henning, M., Yao, A.: Total Domination in Graphs. Springer, New York (2013)CrossRefGoogle Scholar
  15. 15.
    Karlin, A.R., Manasse, M.S., Rudolph, L., Sleator, D.D.: Competitive snoopy caching. Algorithmica 3, 79–119 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations. The IBM Research Symposia Series, pp. 85–103. Plenum Press, New York (1972)CrossRefGoogle Scholar
  17. 17.
    King, G.H., Tzeng, W.G.: On-line algorithms for the dominating set problem. Inf. Process. Lett. 61(1), 11–14 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    König, D.: Theorie der Endlichen und Unendlichen Graphen. Chelsea, New York (1950)zbMATHGoogle Scholar
  19. 19.
    Liu, C.L.: Introduction to Combinatorial Mathematics. McGraw-Hill, New York (1968)zbMATHGoogle Scholar
  20. 20.
    Ore, O.: Theory of graphs. In: Colloquium Publications, vol. 38, American Mathematical Society, Providence (1962)Google Scholar
  21. 21.
    Sleator, D.D., Tarjan, R.E.: Amortized efficiency of list update and paging rules. Commun. ACM 28(2), 202–208 (1985)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Joan Boyar
    • 1
  • Stephan J. Eidenbenz
    • 2
  • Lene M. Favrholdt
    • 1
  • Michal Kotrbčík
    • 3
  • Kim S. Larsen
    • 1
    Email author
  1. 1.Department of Mathematics and Computer ScienceUniversity of Southern DenmarkOdenseDenmark
  2. 2.Los Alamos National LaboratoryLos AlamosUSA
  3. 3.School of Mathematics and PhysicsUniversity of QueenslandBrisbaneAustralia

Personalised recommendations