Document Open Access Logo

Online Dominating Set

Authors Joan Boyar, Stephan J. Eidenbenz, Lene M. Favrholdt, Michal Kotrbcik, Kim S. Larsen



PDF
Thumbnail PDF

File

LIPIcs.SWAT.2016.21.pdf
  • Filesize: 480 kB
  • 15 pages

Document Identifiers

Author Details

Joan Boyar
Stephan J. Eidenbenz
Lene M. Favrholdt
Michal Kotrbcik
Kim S. Larsen

Cite AsGet BibTex

Joan Boyar, Stephan J. Eidenbenz, Lene M. Favrholdt, Michal Kotrbcik, and Kim S. Larsen. Online Dominating Set. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 21:1-21:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.SWAT.2016.21

Abstract

This paper is devoted to the online dominating set problem and its variants on trees, bipartite, bounded-degree, planar, and general graphs, distinguishing between connected and not necessarily connected graphs. We believe this 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. In most cases, we obtain tight bounds on the competitive ratios. Our results show that requiring the graphs to be presented in a connected fashion allows the online algorithms to obtain provably better solutions. Furthermore, 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
  • graph classes
  • connected graphs

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. J. Alber, H. L. Bodlaender, H. Fernau, T. Kloks, and R. Niedermeier. Fixed parameter algorithms for dominating set and related problems on planar graphs. Algorithmica, 33(4):461-493, 2002. Google Scholar
  2. B. S. Baker. Approximation algorithms for NP-complete problems on planar graphs. Journal of the ACM, 41(1):153-180, 1994. Google Scholar
  3. C. Berge. Theory of Graphs and its Applications. Meuthen, London, 1962. Google Scholar
  4. M. Böhm, J. Sgall, and P. Veselý. Online colored bin packing. In E. Bampis and O. Svensson, editors, 12th International Workshop on Approximation and Online Algorithms (WAOA), volume 8952 of Lecture Notes in Computer Science, pages 35-46. Springer, 2015. Google Scholar
  5. A. Borodin and R. El-Yaniv. Online Computation and Competitive Analysis. Cambridge University Press, 1998. Google Scholar
  6. J. Boyar, S. J. Eidenbenz, L. M. Favrholdt, M. Kotrbčík, and K. S. Larsen. Online dominating set. Technical Report arXiv:1604.05172 [cs.DS], arXiv, 2016. Google Scholar
  7. J. Boyar and K. S. Larsen. The seat reservation problem. Algorithmica, 25(4):403-417, 1999. Google Scholar
  8. M. Chrobak, J. Sgall, and G. J. Woeginger. Two-bounded-space bin packing revisited. In C. Demetrescu and M. M. Halldórsson, editors, 19th Annual European Symposium (ESA), volume 6942 of Lecture Notes in Computer Science, pages 263-274. Springer, 2011. Google Scholar
  9. V. Chvátal. A greedy heuristic for the set-covering problem. Mathematics of Operations Research, 4(3):233-235, 1979. Google Scholar
  10. B. Das and V. Bharghavan. Routing in ad-hoc networks using minimum connected dominating sets. In IEEE International Conference on Communications (ICC), volume 1, pages 376-380, 1997. Google Scholar
  11. R. G. Downey and M. R. Fellows. Fixed-parameter tractability and completeness I: Basic results. SIAM Journal on Computing, 24(4):873-921, 1995. Google Scholar
  12. D.-Z. Du and P.-J. Wan. Connected Dominating Set: Theory and Applications. Springer, New York, 2013. Google Scholar
  13. U. Feige. A threshold of ln n for approximating set cover. Journal of the ACM, 45(4):634-652, 1998. Google Scholar
  14. T. W. Haynes, S. Hedetniemi, and P. Slater. Fundamentals of Domination in Graphs. Marcel Dekker, New York, 1998. Google Scholar
  15. M. Henning and A. Yao. Total Domination in Graphs. Springer, New York, 2013. Google Scholar
  16. A. R. Karlin, M. S. Manasse, L. Rudolph, and D. D. Sleator. Competitive snoopy caching. Algorithmica, 3:79-119, 1988. Google Scholar
  17. R. M. Karp. Reducibility among combinatorial problems. In R. E. Miller and J. W. Thatcher, editors, Complexity of Computer Computations, The IBM Research Symposia Series, pages 85-103. Plenum Press, New York, 1972. Google Scholar
  18. G.-H. King and W.-G. Tzeng. On-line algorithms for the dominating set problem. Information Processing Letters, 61(1):11-14, 1997. Google Scholar
  19. D. König. Theorie der Endlichen und Unendlichen Graphen. Chelsea, New York, 1950. Google Scholar
  20. C. L. Liu. Introduction to Combinatorial Mathematics. McGraw-Hill, New York, 1968. Google Scholar
  21. O. Ore. Theory of Graphs, volume 38 of Colloquium Publications. American Mathematical Society, Providence, 1962. Google Scholar
  22. D. D. Sleator and R. E. Tarjan. Amortized efficiency of list update and paging rules. Communications of the ACM, 28(2):202-208, 1985. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail