Partial vs. Complete Domination: t-Dominating Set

  • Joachim Kneis
  • Daniel Mölle
  • Peter Rossmanith
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4362)


We examine the parameterized complexity of t -Dominating Set, the problem of finding a set of at most k nodes that dominate at least t nodes of a graph G = (V,E). The classic NP-complete problem Dominating Set, which can be seen to be t -Dominating Set with the restriction that t = n, has long been known to be W[2]-complete when parameterized in k. Whereas this implies W[2]-hardness for t -Dominating Set and the parameter k, we are able to prove fixed-parameter tractability for t -Dominating Set and the parameter t. More precisely, we obtain a quintic problem kernel and a randomized \(O((4+\varepsilon)^t\textit{poly}(n))\) algorithm. The algorithm is based on the divide-and-color method introduced to the community earlier this year, rather intuitive and can be derandomized using a standard framework.


Success Probability Parameterized Complexity Vertex Cover Recursive Call Polynomial Factor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Alon, N., Goldreich, O.: Simple Constructions of Almost k-Wise Independent Random Variables. Journal of Random structures and Algorithms 3(3), 289–304 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Alon, N., Yuster, R., Zwick, U.: Color-Coding. Journal of the ACM 42(4), 844–856 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Cai, L., Chan, S.M., Chan, S.O.: Random Separation: A New Method for Solving Fixed-Cardinality Optimization Problems. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 239–250. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. 4.
    Chen, J., Kanj, I.A., Xia, G.: Simplicity is Beauty: Improved Upper Bounds for Vertex Cover. Technical Report TR05-008, School of CTI, DePaul University (2005)Google Scholar
  5. 5.
    Chen, J., Lu, S., Sze, S., Zhang, F.: Improved Algorithms for Path, Matching, and Packing Problems. In SODA07, to appear (2007)Google Scholar
  6. 6.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)Google Scholar
  7. 7.
    Gandhi, R., Khuller, S., Srinivasan, A.: Approximation Algorithms for Partial Covering Problems. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 225–236. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  8. 8.
    Guo, J., Niedermeier, R., Wernicke, S.: Parameterized Complexity of Generalized Vertex Cover Problems. In: Dehne, F., López-Ortiz, A., Sack, J.-R. (eds.) WADS 2005. LNCS, vol. 3608, pp. 36–48. Springer, Heidelberg (2005)Google Scholar
  9. 9.
    Halperin, E., Srinivasan, A.: Improved Approximation Algorithms for the Partial Vertex Cover Problem. In: Jansen, K., Leonardi, S., Vazirani, V.V. (eds.) APPROX 2002. LNCS, vol. 2462, pp. 161–199. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  10. 10.
    Kneis, J., Mölle, D., Richter, S., Rossmanith, P.: Algorithms Based on the Treewidth of Sparse Graphs. In: Kratsch, D. (ed.) WG 2005. LNCS, vol. 3787, pp. 385–396. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    Kneis, J., Mölle, D., Richter, S., Rossmanith, P.: Divide-and-Color. In: Fomin, F.V. (ed.) WG 2006. LNCS, vol. 4271, pp. 58–67. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  12. 12.
    Kneis, J., Mölle, D., Richter, S., Rossmanith, P.: Intuitive Algorithms and t-Vertex Cover. In: Asano, T. (ed.) ISAAC 2006. LNCS, vol. 4288, pp. 598–607. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  13. 13.
    Schöning, U.: A Probabilistic Algorithm for k-SAT and Constraint Satisfaction Problems. In: FOCS40th, pp. 410–414 (1999)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Joachim Kneis
    • 1
  • Daniel Mölle
    • 1
  • Peter Rossmanith
    • 1
  1. 1.Department of Computer Science, RWTH Aachen UniversityGermany

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