Advertisement

The Complexity of Finding a Large Subgraph under Anonymity Constraints

  • Robert Bredereck
  • Sepp Hartung
  • André Nichterlein
  • Gerhard J. Woeginger
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8283)

Abstract

We define and analyze an anonymization problem in undirected graphs, which is motivated by certain privacy issues in social networks. The goal is to remove a small number of vertices from the graph such that in the resulting subgraph every occurring vertex degree occurs many times.

We prove that the problem is NP-hard for trees, and also for a number of other highly structured graph classes. Furthermore we provide polynomial time algorithms for other graph classes (like threshold graphs), and thereby establish a sharp borderline between hard and easy cases of the problem. Finally we perform a parametrized analysis, and we concisely characterize combinations of natural parameters that allow FPT algorithms.

Keywords

Maximum Degree Vertex Cover Graph Class Perfect Graph Split Graph 
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.
    Barabási, A., Albert, R.: Emergence of scaling in random networks. Science 286(5439), 509 (1999)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: a Survey. SIAM Monographs on Discrete Mathematics and Applications, vol. 3. SIAM (1999)Google Scholar
  3. 3.
    Chester, S., Kapron, B., Srivastava, G., Venkatesh, S.: Complexity of social network anonymization. Social Network Analysis and Mining, 1–16 (2012a)Google Scholar
  4. 4.
    Chester, S., Kapron, B.M., Ramesh, G., Srivastava, G., Thomo, A., Venkatesh, S.: Why Waldo befriended the dummy? k-anonymization of social networks with pseudo-nodes. In: Social Network Analysis and Mining, pp. 1–19 (2012b) ISSN 1869-5450Google Scholar
  5. 5.
    Chvátal, V., Hammer, P.L.: Aggregation of inequalities in integer programming. Annals of Discrete Mathematics 1, 145–162 (1977)CrossRefGoogle Scholar
  6. 6.
    Dom, M., Lokshtanov, D., Saurabh, S.: Incompressibility through colors and IDs. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 378–389. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  7. 7.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer (1999)Google Scholar
  8. 8.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman (1979)Google Scholar
  9. 9.
    Golumbic, M.C.: Algorithmic graph theory and perfect graphs, 2nd edn. Annals of Discrete Mathematics, vol. 57. Elsevier B.V. (2004), 1st edn. Academic Press (1980)Google Scholar
  10. 10.
    Hartung, S., Nichterlein, A., Niedermeier, R., Suchý, O.: A refined complexity analysis of degree anonymization in graphs. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part II. LNCS, vol. 7966, pp. 594–606. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  11. 11.
    Liu, K., Terzi, E.: Towards identity anonymization on graphs. In: Proc. ACM SIGMOD International Conference on Management of Data, SIGMOD 2008, pp. 93–106. ACM (2008)Google Scholar
  12. 12.
    Mathieson, L., Szeider, S.: Editing graphs to satisfy degree constraints: A parameterized approach. J. Comput. Syst. Sci. 78(1), 179–191 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press (2006)Google Scholar
  14. 14.
    Vazirani, V.V.: Approximation Algorithms. Springer (2001)Google Scholar
  15. 15.
    Wu, X., Ying, X., Liu, K., Chen, L.: A survey of privacy-preservation of graphs and social networks. In: Managing and Mining Graph Data, pp. 421–453. Springer (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Robert Bredereck
    • 1
  • Sepp Hartung
    • 1
  • André Nichterlein
    • 1
  • Gerhard J. Woeginger
    • 2
  1. 1.Institut für Softwaretechnik und Theoretische InformatikTU BerlinGermany
  2. 2.Department of Mathematics and Computer ScienceTU EindhovenThe Netherlands

Personalised recommendations