Advertisement

Parameterized Dynamic Variants of Red-Blue Dominating Set

  • Faisal N. Abu-Khzam
  • Cristina Bazgan
  • Henning FernauEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12011)

Abstract

We introduce a parameterized dynamic version of the Red-Blue Dominating Set problem and its partial version. We prove the fixed-parameter tractability of the dynamic versions with respect to the (so called) edit-parameter while they remain \(\mathcal{W}[2]\)-hard with respect to the increment-parameter. We provide a complete study of the complexity of the problem with respect to combinations of the various parameters.

Keywords

Dynamic problems Reoptimization Parameterized complexity Set cover Hitting set 

References

  1. 1.
    Abu-Khzam, F.N., Egan, J., Fellows, M.R., Rosamond, F.A., Shaw, P.: On the parameterized complexity of dynamic problems. Theoret. Comput. Sci. 607, 426–434 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alman, J., Mnich, M., Williams, V.V.: Dynamic parameterized problems and algorithms. In: Chatzigiannakis, I., Indyk, P., Kuhn, F., Muscholl, A. (eds.) 44th International Colloquium on Automata, Languages, and Programming, ICALP. LIPIcs, vol. 80, pp. 41:1–41:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)Google Scholar
  3. 3.
    Bilò, D., Widmayer, P., Zych, A.: Reoptimization of weighted graph and covering problems. In: Bampis, E., Skutella, M. (eds.) WAOA 2008. LNCS, vol. 5426, pp. 201–213. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-540-93980-1_16CrossRefzbMATHGoogle Scholar
  4. 4.
    Bläser, M.: Computing small partial coverings. Inf. Process. Lett. 85(6), 327–331 (2003)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Böckenhauer, H.-J., Hromkovič, J., Mömke, T., Widmayer, P.: On the hardness of reoptimization. In: Geffert, V., Karhumäki, J., Bertoni, A., Preneel, B., Návrat, P., Bieliková, M. (eds.) SOFSEM 2008. LNCS, vol. 4910, pp. 50–65. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-77566-9_5CrossRefzbMATHGoogle Scholar
  6. 6.
    Bodlaender, H.L., Lokshtanov, D., Penninkx, E.: Planar capacitated dominating set is W[1]-hard. In: Chen, J., Fomin, F.V. (eds.) IWPEC 2009. LNCS, vol. 5917, pp. 50–60. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-11269-0_4CrossRefGoogle Scholar
  7. 7.
    Cai, Z., Miao, D., Li, Y.: Deletion propagation for multiple key preserving conjunctive queries: approximations and complexity. In: International Conference on Data Engineering, ICDE, pp. 506–517. IEEE (2019)Google Scholar
  8. 8.
    Cesati, M.: The turing way to parameterized complexity. J. Comput. Syst. Sci. 67(4), 654–685 (2003)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Cygan, M., et al.: On problems as hard as CNF-SAT. ACM Trans. Algorithms 12(3), 41:1–41:24 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Damaschke, P.: Parameterizations of hitting set of bundles and inverse scope. J. Comb. Optim. 29(4), 847–858 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dom, M., Lokshtanov, D., Saurabh, S., Villanger, Y.: Capacitated domination and covering: a parameterized perspective. In: Grohe, M., Niedermeier, R. (eds.) IWPEC 2008. LNCS, vol. 5018, pp. 78–90. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-79723-4_9CrossRefzbMATHGoogle Scholar
  12. 12.
    Downey, R.G., Egan, J., Fellows, M.R., Rosamond, F.A., Shaw, P.: Dynamic dominating set and turbo-charging greedy heuristics. Tsinghua Sci. Technol. 19(4), 329–337 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Fernau, H., Rodríguez-Velázquez, J.A.: A survey on alliances and related parameters in graphs. Electron. J. Graph Theory Appl. 2(1), 70–86 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kao, M., Chen, H., Lee, D.: Capacitated domination: problem complexity and approximation algorithms. Algorithmica 72(1), 1–43 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kearns, M.J.: Computational Complexity of Machine Learning. ACM Distinguished Dissertations. MIT Press, Cambridge (1990)Google Scholar
  16. 16.
    Khuller, S., Purohit, M., Sarpatwar, K.K.: Analyzing the optimal neighborhood: algorithms for budgeted and partial connected dominating set problems. In: Symposium on Discrete Algorithms (SODA), pp. 1702–1713. SIAM (2014)Google Scholar
  17. 17.
    Koutis, I., Williams, R.: Limits and applications of group algebras for parameterized problems. ACM Trans. Algorithms 12(3), 31:1–31:18 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lenstra Jr., H.W.: Integer programming with a fixed number of variables. Math. Oper. Res. 8(4), 538–548 (1983)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lin, B.: The parameterized complexity of k-biclique. In: Symposium on Discrete Algorithms (SODA), pp. 605–615. SIAM (2015)Google Scholar
  20. 20.
    Lin, B.: The parameterized complexity of the k-biclique problem. J. ACM 65(5), 34:1–34:23 (2018)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Mikhailyuk, V.A.: Reoptimization of set covering problems. Cybern. Syst. Anal. 46(6), 879–883 (2010)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Srinivasan, A.: Improved approximation guarantees for packing and covering integer programs. SIAM J. Comput. 29(2), 648–670 (1999)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Computer Science and MathematicsLebanese American UniversityBeirutLebanon
  2. 2.Université Paris-Dauphine, PSL University, CNRS, LAMSADEParisFrance
  3. 3.Universität TrierTrierGermany

Personalised recommendations