Theory of Computing Systems

, Volume 55, Issue 1, pp 61–83 | Cite as

Constant Thresholds Can Make Target Set Selection Tractable

  • Morgan Chopin
  • André Nichterlein
  • Rolf Niedermeier
  • Mathias Weller
Article

Abstract

Target Set Selection, which is a prominent NP-hard problem occurring in social network analysis and distributed computing, is notoriously hard both in terms of achieving useful polynomial-time approximation as well as fixed-parameter algorithms. Given an undirected graph, the task is to select a minimum number of vertices into a “target set” such that all other vertices will become active in the course of a dynamic process (which may go through several activation rounds). A vertex, equipped with a threshold value t, becomes active once at least t of its neighbors are active; initially, only the target set vertices are active. We contribute further insights into the existence of islands of tractability for Target Set Selection by spotting new parameterizations characterizing some sparse graphs as well as some “cliquish” graphs and developing corresponding fixed-parameter tractability and (parameterized) hardness results. In particular, we demonstrate that upper-bounding the thresholds by a constant may significantly alleviate the search for efficiently solvable, but still meaningful special cases of Target Set Selection.

Keywords

Spread of influence Dynamics in social networks Parameterized complexity Kernelization 

References

  1. Ackerman et al.(2010).
    Ackerman, E., Ben-Zwi, O., Wolfovitz, G.: Combinatorial model and bounds for target set selection. Theor. Comput. Sci. 411(44–46), 4017–4022 (2010) CrossRefMATHMathSciNetGoogle Scholar
  2. Adams et al.(2011).
    Adams, S.S., Troxell, D.S., Zinnen, S.L.: Dynamic monopolies and feedback vertex sets in hexagonal grids. Comput. Math. Appl. 62(11), 4049–4057 (2011) CrossRefMATHMathSciNetGoogle Scholar
  3. Adams et al.(2012).
    Adams, S.S., Booth, P., Troxell, D.S., Zinnen, S.L.: Modeling the spread of fault in majority-based network systems: dynamic monopolies in triangular grids. Discrete Appl. Math. 160(10–11), 1624–1633 (2012) CrossRefMATHMathSciNetGoogle Scholar
  4. Balogh et al.(2010).
    Balogh, J., Bollobás, B., Morris, R.: Bootstrap percolation in high dimensions. Comb. Probab. Comput. 19(5–6), 643–692 (2010) CrossRefMATHGoogle Scholar
  5. Bazgan et al.(2013).
    Bazgan, C., Chopin, M., Nichterlein, A., Sikora, F.: Parameterized approximability of influence in social networks. In: Proc. 19th COCOON. LNCS, vol. 7936, pp. 543–554. Springer, Berlin (2013) Google Scholar
  6. Ben-Zwi et al.(2011).
    Ben-Zwi, O., Hermelin, D., Lokshtanov, D., Newman, I.: Treewidth governs the complexity of target set selection. Discrete Optim. 8(1), 87–96 (2011) CrossRefMATHMathSciNetGoogle Scholar
  7. Bodlaender(2009).
    Bodlaender, H.L.: Kernelization: New upper and lower bound techniques. In: Proc. 4th IWPEC. LNCS, vol. 5917, pp. 17–37. Springer, Berlin (2009) Google Scholar
  8. Centeno et al.(2011).
    Centeno, C.C., Dourado, M.C., Penso, L.D., Rautenbach, D., Szwarcfiter, J.L.: Irreversible conversion of graphs. Theor. Comput. Sci. 412(29), 3693–3700 (2011) CrossRefMATHMathSciNetGoogle Scholar
  9. Chang and Lyuu(2009).
    Chang, C.-L., Lyuu, Y.-D.: Spreading messages. Theor. Comput. Sci. 410(27–29), 2714–2724 (2009) CrossRefMATHMathSciNetGoogle Scholar
  10. Chen(2009).
    Chen, N.: On the approximability of influence in social networks. SIAM J. Discrete Math. 23(3), 1400–1415 (2009) CrossRefMATHMathSciNetGoogle Scholar
  11. Chiang et al.(2013).
    Chiang, C.-Y., Huang, L.-H., Li, B.-J., Wu, J., Yeh, H.-G.: Some results on the target set selection problem. J. Comb. Optim. 25(4), 702–715 (2013) CrossRefMATHMathSciNetGoogle Scholar
  12. Diestel(2010).
    Diestel, R.: Graph Theory, 4th edn. Graduate Texts in Mathematics, vol. 173. Springer, Berlin (2010) CrossRefGoogle Scholar
  13. Doucha and Kratochvíl(2012).
    Doucha, M., Kratochvíl, J.: Cluster vertex deletion: a parameterization between vertex cover and clique-width. In: Proc. 37th MFCS. LNCS, vol. 7464, pp. 348–359. Springer, Berlin (2012) Google Scholar
  14. Downey and Fellows(1999).
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Berlin (1999) CrossRefGoogle Scholar
  15. Dreyer and Roberts(2009).
    Dreyer, P.A. Jr., 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) CrossRefMATHMathSciNetGoogle Scholar
  16. Easley and Kleinberg(2010).
    Easley, D., Kleinberg, J.: Networks, Crowds, and Markets: Reasoning About a Highly Connected World. Cambridge University Press, Cambridge (2010) CrossRefGoogle Scholar
  17. Fellows et al.(2009).
    Fellows, M.R., Hermelin, D., Rosamond, F.A., Vialette, S.: On the parameterized complexity of multiple-interval graph problems. Theor. Comput. Sci. 410(1), 53–61 (2009) CrossRefMATHMathSciNetGoogle Scholar
  18. Flum and Grohe(2006).
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Berlin (2006) Google Scholar
  19. Guo and Niedermeier(2007).
    Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. ACM SIGACT News 38(1), 31–45 (2007) CrossRefGoogle Scholar
  20. Guo et al.(2004).
    Guo, J., Hüffner, F., Niedermeier, R.: A structural view on parameterizing problems: distance from triviality. In: Proc. 1st IWPEC. LNCS, vol. 3162, pp. 162–173. Springer, Berlin (2004) Google Scholar
  21. Harant et al.(1999).
    Harant, J., Pruchnewski, A., Voigt, M.: On dominating sets and independent sets of graphs. Comb. Probab. Comput. 8(6), 547–553 (1999) CrossRefMATHMathSciNetGoogle Scholar
  22. Hüffner et al.(2010).
    Hüffner, F., Komusiewicz, C., Moser, H., Niedermeier, R.: Fixed-parameter algorithms for cluster vertex deletion. Theory Comput. Syst. 47(1), 196–217 (2010) CrossRefMATHMathSciNetGoogle Scholar
  23. Karp(1972).
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum, New York (1972) CrossRefGoogle Scholar
  24. Kempe et al.(2003).
    Kempe, D., Kleinberg, J., Tardos, É.: Maximizing the spread of influence through a social network. In: Proc. 9th ACM KDD, pp. 137–146. ACM, New York (2003) Google Scholar
  25. Klasing and Laforest(2004).
    Klasing, R., Laforest, C.: Hardness results and approximation algorithms of k-tuple domination in graphs. Inf. Process. Lett. 89(2), 75–83 (2004) CrossRefMATHMathSciNetGoogle Scholar
  26. Komusiewicz and Niedermeier(2012).
    Komusiewicz, C., Niedermeier, R.: New races in parameterized algorithmics. In: Proc. 37th MFCS. LNCS, vol. 7464, pp. 19–30. Springer, Berlin (2012) Google Scholar
  27. Lenstra(1983).
    Lenstra, H.W.: Integer programming with a fixed number of variables. Math. Oper. Res. 8(4), 538–548 (1983) CrossRefMATHMathSciNetGoogle Scholar
  28. McConnell and Spinrad(1994).
    McConnell, R.M., Spinrad, J.: Linear-time modular decomposition and efficient transitive orientation of comparability graphs. In: Proc. 5th SODA, pp. 536–545. ACM/SIAM, New York/Philadelphia (1994) Google Scholar
  29. Nichterlein et al.(2013).
    Nichterlein, A., Niedermeier, R., Uhlmann, J., Weller, M.: On tractable cases of target set selection. Soc. Netw. Anal. Mining 3(2), 233–256 (2013) CrossRefGoogle Scholar
  30. Niedermeier(2006).
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, London (2006) CrossRefMATHGoogle Scholar
  31. Peleg(2002).
    Peleg, D.: Local majorities, coalitions and monopolies in graphs: a review. Theor. Comput. Sci. 282(2), 231–257 (2002) CrossRefMATHMathSciNetGoogle Scholar
  32. Raman et al.(2008).
    Raman, V., Saurabh, S., Srihari, S.: Parameterized algorithms for generalized domination. In: Proc. 2nd COCOA. LNCS, vol. 5165, pp. 116–126. Springer, Berlin (2008) Google Scholar
  33. Reddy et al.(2010).
    Reddy, T., Krishna, D., Rangan, C.: Variants of spreading messages. In: Proc. 4th WALCOM. LNCS, vol. 5942, pp. 240–251. Springer, Berlin (2010) Google Scholar
  34. Reichman(2012).
    Reichman, D.: New bounds for contagious sets. Discrete Math. 312(10), 1812–1814 (2012) CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Morgan Chopin
    • 1
  • André Nichterlein
    • 2
  • Rolf Niedermeier
    • 2
  • Mathias Weller
    • 2
  1. 1.LAMSADEUniversité Paris-DauphineParisFrance
  2. 2.Institut für Softwaretechnik und Theoretische InformatikTU BerlinBerlinGermany

Personalised recommendations