The Robust Set Problem: Parameterized Complexity and Approximation

  • Cristina Bazgan
  • Morgan Chopin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7464)

Abstract

In this paper, we introduce the Robust Set problem: given a graph G = (V,E), a threshold function t:V → N and an integer k, find a subset of vertices V′ ⊆ V of size at least k such that every vertex v in G has less than t(v) neighbors in V′. This problem occurs in the context of the spread of undesirable agents through a network (virus, ideas, fire, …). Informally speaking, the problem asks to find the largest subset of vertices with the property that if anything bad happens in it then this will have no consequences on the remaining graph. The threshold t(v) of a vertex v represents its reliability regarding its neighborhood; that is, how many neighbors can be infected before v gets himself infected.

We study in this paper the parameterized complexity of Robust Set and the approximation of the associated maximization problem. When the parameter is k, we show that this problem is W[2]-complete in general and W[1]-complete if all thresholds are constant bounded. Moreover, we prove that, if P ≠ NP, the maximization version is not n1 − ε- approximable for any ε > 0 even when all thresholds are at most two. When each threshold is equal to the degree of the vertex, we show that k-Robust Set is fixed-parameter tractable for parameter k and the maximization version is APX-complete. We give a polynomial-time algorithm for graphs of bounded treewidth and a PTAS for planar graphs. Finally, we show that the parametric dual problem (n − k)-Robust Set is fixed-parameter tractable for a large family of threshold functions.

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References

  1. 1.
    Aazami, A., Stilp, M.D.: Approximation Algorithms and Hardness for Domination with Propagation. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds.) RANDOM 2007 and APPROX 2007. LNCS, vol. 4627, pp. 1–15. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  2. 2.
    Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs. Journal of the ACM 41(1), 153–180 (1994)MATHCrossRefGoogle Scholar
  3. 3.
    Ben-Zwi, O., Hermelin, D., Lokshtanov, D., Newman, I.: Treewidth governs the complexity of Target Set Selection. Discrete Optimization 8(1), 87–96 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Berman, P., Karpinski, M.: On Some Tighter Inapproximability Results. In: Wiedermann, J., Van Emde Boas, P., Nielsen, M. (eds.) ICALP 1999. LNCS, vol. 1644, pp. 200–209. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  5. 5.
    Bodlaender, H.L.: A partial k-arboretum of graphs with bounded treewidth. Theoretical Computer Science 209(1-2), 1–45 (1998)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Cesati, M.: The Turing way to parameterized complexity. Journal of Computer and System Sciences 67(4), 654–685 (2003)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Chalermsook, P., Chuzhoy, J.: Resource minimization for fire containment. In: Proceedings of the 21th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2010), pp. 1334–1349 (2010)Google Scholar
  8. 8.
    Chen, N.: On the approximability of influence in social networks. In: Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2008), pp. 1029–1037 (2008)Google Scholar
  9. 9.
    Cockayne, E.J., Dawes, R.M., Hedetniemi, S.T.: Total domination in graphs. Networks 10(3), 211–219 (1980)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness II: On completeness for W[1]. Theoretical Computer Science 141(1-2), 109–131 (1995)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Downey, R.G., Fellows, M.R.: Parameterized complexity. Springer, New York (1999)CrossRefGoogle Scholar
  12. 12.
    Dreyer Jr., P.A., Roberts, F.S.: Irreversible k-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Finbow, S., MacGillivray, G.: The firefighter problem: a survey of results, directions and questions. The Australasian Journal of Combinatorics 43, 57–77 (2009)MathSciNetMATHGoogle Scholar
  14. 14.
    Golovach, P.A., Kratochvil, J., Suchý, O.: Parameterized complexity of generalized domination problems. Discrete Applied Mathematics 160(6), 780–792 (2012)MATHCrossRefGoogle Scholar
  15. 15.
    Kempe, D., Kleinberg, J., Tardos, É.: Maximizing the spread of influence through a social network. In: Proceedings of the 9th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD 2003), pp. 137–146 (2003)Google Scholar
  16. 16.
    Kimura, M., Saito, K., Motoda, H.: Blocking links to minimize contamination spread in a social network. ACM Transactions on Knowledge Discovery from Data 3(2), 9:1–9:23 (2009)CrossRefGoogle Scholar
  17. 17.
    Nichterlein, A., Niedermeier, R., Uhlmann, J., Weller, M.: On Tractable Cases of Target Set Selection. In: Cheong, O., Chwa, K.-Y., Park, K. (eds.) ISAAC 2010. LNCS, vol. 6506, pp. 378–389. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  18. 18.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press (2006)Google Scholar
  19. 19.
    Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. J. Comput. Syst. Sci. 43(3), 425–440 (1991)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Cristina Bazgan
    • 1
    • 2
  • Morgan Chopin
    • 1
  1. 1.LAMSADEUniversité Paris-DauphineFrance
  2. 2.Institut Universitaire de FranceFrance

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