Abstract
We consider the problem of controlling the spread of harmful items in networks, such as the contagion proliferation of diseases or the diffusion of fake news. We assume the linear threshold model of diffusion where each node has a threshold that measures the node’s resistance to the contagion. We study the parameterized complexity of the problem: Given a network, a set of initially contaminated nodes, and two integers k and \(\ell \), is it possible to limit the diffusion to at most k other nodes of the network by immunizing at most \(\ell \) nodes? We consider several parameters associated with the input, including the bounds k and \(\ell \), the maximum node degree \(\varDelta \), the treewidth, and the neighborhood diversity of the network. We first give W[1] or W[2]-hardness results for each of the considered parameters. Then we give fixed-parameter algorithms for some parameter combinations.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
For a positive integer a, we use [a] to denote the set of integers \([a] = \{1, 2, \ldots , a\}\).
References
Albert, R., Jeong, H., Barabási, A.-L.: Error and attack tolerance of complex networks. Nature 404, 378–382 (2000)
Ben-Zwi, O., Hermelin, D., Lokshtanov, D., Newman, I.: Treewidth governs the complexity of target set selection. Discrete Optim. 8(1), 87–96 (2011). ISSN 1572–5286. https://doi.org/10.1016/j.disopt.2010.09.007
Chen, P., David, M., Kempe, D.: Better vaccination strategies for better people. In: Proceedings 11th ACM Conference on Electronic Commerce (EC-2010), Cambridge, Massachusetts, USA, 7–11 June (2010)
Cordasco, G., Gargano, L., Mecchia, M., Rescigno, A.A., Vaccaro, U.: A fast and effective heuristic for discovering small target sets in social networks. In: Lu, Z., Kim, D., Wu, W., Li, W., Du, D.-Z. (eds.) COCOA 2015. LNCS, vol. 9486, pp. 193–208. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-26626-8_15
Cordasco, G., Gargano, L., Rescigno, A.A.: Influence propagation over large scale social networks. In: Proceedings of ASONAM 2015, pp. 1531–1538 (2015)
Cordasco, G., Gargano, L., Rescigno, A.A.: On finding small sets that influence large networks. Soc. Netw. Anal. Min. 6(1), 1–20 (2016). https://doi.org/10.1007/s13278-016-0408-z
Cordasco, G., Gargano, L., Rescigno, A.A.: Active influence spreading in social networks. Theoret. Comput. Sci. 764, 15–29 (2019)
Cordasco, G., Gargano, L., Rescigno, A.A., Vaccaro, U.: Evangelism in social networks: algorithms and complexity. Networks 71(4), 346–357 (2018)
Cygan, M., et al.: Parameterized Algorithms. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-21275-3
Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. TCS, Springer, London (2013). https://doi.org/10.1007/978-1-4471-5559-1
Ehard, S., Rautenbach, D.: Vaccinate your trees! Theoret. Comput. Sci. 772, 46–57 (2019). ISSN 0304–3975. https://doi.org/10.1016/j.tcs.2018.11.018
Feige, U., Krauthgamer, R., Nissim, K.: On cutting a few vertices from a graph. Discret. Appl. Math. 127, 643–649 (2003)
Finbow, S., MacGillivray, G.: The firefighter problem: a survey of results, directions and questions. Australas. J. Comb. 43, 57–77 (2009)
Feige, U., Kogan, S.: Target Set Selection for Conservative Population CoRR abs/1909.03422 (2019)
Fomin, F.V., Golovach, P.A., Korhonen, J.H.: On the parameterized complexity of cutting a few vertices from a graph. In: Chatterjee, K., Sgall, J. (eds.) MFCS 2013. LNCS, vol. 8087, pp. 421–432. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40313-2_38
Gargano, L., Rescigno, A.A.: Complexity of conflict-free colorings of graphs. Theoret. Comput. Sci. 566, 39–49 (2015)
Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)
Gavenciak, T., Knop, D., Koutecký, M.: Integer Programming in Parameterized Complexity: Three Miniatures. In Proceedings of IPEC 2018 (2018). https://doi.org/10.4230/LIPIcs.IPEC.2018.21
Granovetter, M.: Threshold models of collective behaviors. Am. J. Sociol. 83(6), 1420–1443 (1978)
Hayrapetyan, A., Kempe, D., Pál, M., Svitkina, Z.: Unbalanced graph cuts. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 191–202. Springer, Heidelberg (2005). https://doi.org/10.1007/11561071_19
Hanaka, T., Bodlaender, H.L., van der Zanden, T.C., Ono, H.: On the maximum weight minimal separator. Theor. Comput. Sci. 796, 294–308 (2019)
Kempe, D., Kleinberg, J., Tardos, E.: Maximizing the spread of influence through a social network. In: Proceedings of the 9th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, Washington, USA, pp. 137–146 (2003)
Khalil, E.B., Dilkina, B., Song, L.: CuttingEdge: influence minimization in networks. In: Methods, Models, and Applications at NIPS, Workshop on Frontiers of Network Analysis (2013)
Kimura, M., Saito, K., Motoda, H.: Blocking links to minimize contamination spread in a social network. ACM Trans. Knowl. Discovery Data 3(2) (2009)
Kloks, T. (ed.): Treewidth. LNCS, vol. 842. Springer, Heidelberg (1994). https://doi.org/10.1007/BFb0045375ISSN: 0302-9743
Knop, D., Koutecký, M., Masarík, T., Toufar, T.: Simplified algorithmic metatheorems beyond MSO: treewidth and neighborhood diversity. Logical Methods Comput. Sci. 15(4) (2019)
Komusiewicz, C., Sorge, M.: Finding dense subgraphs of sparse graphs. In: Thilikos, D.M., Woeginger, G.J. (eds.) IPEC 2012. LNCS, vol. 7535, pp. 242–251. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-33293-7_23
Lampis, M.: Algorithmic meta-theorems for restrictions of treewidth. Algorithmica 64, 19–37 (2012)
Menczer, F., Fortunato, S., Davis, C.A.: A First Course in Network Science, 1st edn. Cambridge University Press (2020)
Newman, M.E.J., Forrest, S., Balthrop, J.: Email networks and the spread of computer viruses. Phys. Rev. E 66 (2002)
Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press (2006)
Robertson, N., Seymour, P.D.: Graph minors. II. Algorithmic aspects of tree-width. J. Algorithms 7(3), 309–322 (1986)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 Springer Nature Switzerland AG
About this paper
Cite this paper
Cordasco, G., Gargano, L., Rescigno, A.A. (2022). Parameterized Complexity of Immunization in the Threshold Model. In: Mutzel, P., Rahman, M.S., Slamin (eds) WALCOM: Algorithms and Computation. WALCOM 2022. Lecture Notes in Computer Science(), vol 13174. Springer, Cham. https://doi.org/10.1007/978-3-030-96731-4_23
Download citation
DOI: https://doi.org/10.1007/978-3-030-96731-4_23
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-96730-7
Online ISBN: 978-3-030-96731-4
eBook Packages: Computer ScienceComputer Science (R0)