Advertisement

Parameterized Inapproximability of Target Set Selection and Generalizations

  • Cristina Bazgan
  • Morgan Chopin
  • André Nichterlein
  • Florian Sikora
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8493)

Abstract

In this paper, we consider the Target Set Selection problem: given a graph and a threshold value Open image in new window for each vertex v of the graph, find a minimum size vertex-subset to “activate” s.t. all the vertices of the graph are activated at the end of the propagation process. A vertex v is activated during the propagation process if at least Open image in new window of its neighbors are activated. This problem models several practical issues like faults in distributed networks or word-to-mouth recommendations in social networks. We show that for any functions f and ρ this problem cannot be approximated within a factor of ρ(k) in f(k) ·n O(1) time, unless FPT = W[P], even for restricted thresholds (namely constant and majority thresholds). We also study the cardinality constraint maximization and minimization versions of the problem for which we prove similar hardness results.

Keywords

Bipartite Graph Vertex Cover Input Node Computable Function Satisfying Assignment 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aazami, A., Stilp, K.: Approximation algorithms and hardness for domination with propagation. SIAM J. Discrete Math. 23(3), 1382–1399 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Arora, S., Lund, C.: Hardness of approximations. In: Approximation Algorithms for NP-Hard Problems, pp. 399–446. PWS Publishing Company (1996)Google Scholar
  3. 3.
    Bazgan, C., Chopin, M., Nichterlein, A., Sikora, F.: Parameterized approximability of maximizing the spread of influence in networks. In: Du, D.-Z., Zhang, G. (eds.) COCOON 2013. LNCS, vol. 7936, pp. 543–554. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  4. 4.
    Ben-Zwi, O., Hermelin, D., Lokshtanov, D., Newman, I.: Treewidth governs the complexity of target set selection. Discrete Optim. 8(1), 87–96 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Cai, L., Huang, X.: Fixed-parameter approximation: Conceptual framework and approximability results. Algorithmica 57(2), 398–412 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Chang, C.-L., Lyuu, Y.-D.: Spreading messages. Theor. Comput. Sci. 410(27-29), 2714–2724 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Chen, N.: On the approximability of influence in social networks. SIAM J. Discrete Math. 23(3), 1400–1415 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Chen, Y., Grohe, M., Grüber, M.: On parameterized approximability. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 109–120. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  9. 9.
    Chopin, M., Nichterlein, A., Niedermeier, R., Weller, M.: Constant thresholds can make target set selection tractable. In: Even, G., Rawitz, D. (eds.) MedAlg 2012. LNCS, vol. 7659, pp. 120–133. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  10. 10.
    Cicalese, F., Cordasco, G., Gargano, L., Milanič, M., Vaccaro, U.: Latency-bounded target set selection in social networks. In: Bonizzoni, P., Brattka, V., Löwe, B. (eds.) CiE 2013. LNCS, vol. 7921, pp. 65–77. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  11. 11.
    Dinur, I., Safra, S.: The importance of being biased. In: Proc. of STOC 2002, pp. 33–42. ACM (2002)Google Scholar
  12. 12.
    Downey, R.G., Fellows, M.R., McCartin, C.: Parameterized approximation algorithms. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 121–129. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  13. 13.
    Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Springer (2013)Google Scholar
  14. 14.
    Dreyer, P.A., Roberts, F.S.: Irreversible k-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Appl. Math. 157(7), 1615–1627 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer (2006)Google Scholar
  16. 16.
    Kempe, D., Kleinberg, J., Tardos, É.: Maximizing the spread of influence through a social network. In: Proc. of KDD 2003, pp. 137–146. ACM (2003)Google Scholar
  17. 17.
    Marx, D.: Parameterized complexity and approximation algorithms. Comput. J. 51(1), 60–78 (2008)CrossRefGoogle Scholar
  18. 18.
    Marx, D.: Completely inapproximable monotone and antimonotone parameterized problems. J. Comput. Syst. Sci. 79(1), 144–151 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Nichterlein, A., Niedermeier, R., Uhlmann, J., Weller, M.: On tractable cases of target set selection. Soc. Network Anal. Mining 3(2), 233–256 (1869) ISSN 1869-5450Google Scholar
  20. 20.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press (2006)Google Scholar
  21. 21.
    Peleg, D.: Local majorities, coalitions and monopolies in graphs: a review. Theor. Comput. Sci. 282(2), 231–257 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Reddy, T.V.T., Rangan, C.P.: Variants of spreading messages. J. Graph Algorithms Appl. 15(5), 683–699 (2011)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Cristina Bazgan
    • 1
    • 4
  • Morgan Chopin
    • 2
  • André Nichterlein
    • 3
  • Florian Sikora
    • 1
  1. 1.LAMSADE UMR CNRS 7243PSL, Université Paris-DauphineFrance
  2. 2.Institut für Optimierung und Operations ResearchUniversität UlmGermany
  3. 3.Institut für Softwaretechnik und Theoretische InformatikTU BerlinGermany
  4. 4.Institut Universitaire de FranceFrance

Personalised recommendations