The Complexity of Degree Anonymization by Vertex Addition

  • Robert Bredereck
  • Vincent Froese
  • Sepp Hartung
  • André Nichterlein
  • Rolf Niedermeier
  • Nimrod Talmon
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8546)


Motivated by applications in privacy-preserving data publishing, we study the problem of making an undirected graph k-anonymous by adding few vertices (together with incident edges). That is, after adding these “dummy vertices”, for every vertex degree d in the resulting graph, there shall be at least k vertices with degree d. We explore three variants of vertex addition (justified by real-world considerations) and study their (parameterized) computational complexity. We derive mostly (worst-case) intractability results, even for very restricted cases (including trees or bounded-degree graphs) but also obtain a few encouraging fixed-parameter tractability results.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bredereck, R., Hartung, S., Nichterlein, A., Woeginger, G.J.: The complexity of finding a large subgraph under anonymity constraints. In: Cai, L., Cheng, S.-W., Lam, T.-W. (eds.) Algorithms and Computation. LNCS, vol. 8283, pp. 152–162. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  2. 2.
    Casas-Roma, J., Herrera-Joancomartí, J., Torra, V.: An algorithm for k-degree anonymity on large networks. In: Proc. ASONAM 2013, pp. 671–675. ACM Press (2013)Google Scholar
  3. 3.
    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. Social Netw. Analys. Mining 3(3), 381–399 (2013)CrossRefGoogle Scholar
  4. 4.
    Chester, S., Kapron, B.M., Srivastava, G., Venkatesh, S.: Complexity of social network anonymization. Social Netw. Analys. Mining 3(2), 151–166 (2013)CrossRefGoogle Scholar
  5. 5.
    Clarkson, K.L., Liu, K., Terzi, E.: Towards identity anonymization in social networks. In: Link Mining: Models, Algorithms, and Applications, pp. 359–385. Springer (2010)Google Scholar
  6. 6.
    Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Springer (2013)Google Scholar
  7. 7.
    Erdős, P., Kelly, P.: The minimal regular graph containing a given graph. Amer. Math. Monthly 70, 1074–1075 (1963)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Fellows, M.R., Jansen, B.M.P., Rosamond, F.A.: Towards fully multivariate algorithmics: Parameter ecology and the deconstruction of computational complexity. Eur. J. Combin. 34(3), 541–566 (2013)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer (2006)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); To appear in Information and ComputationGoogle Scholar
  11. 11.
    Lenstra, H.W.: Integer programming with a fixed number of variables. Math. Oper. Res. 8, 538–548 (1983)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Liu, K., Terzi, E.: Towards identity anonymization on graphs. In: ACM SIGMOD Conference, SIGMOD 2008, pp. 93–106. ACM (2008)Google Scholar
  13. 13.
    Lovász, L., Plummer, M.D.: Matching Theory. Annals of Discrete Mathematics, vol. 29. North-Holland (1986)Google Scholar
  14. 14.
    Lu, X., Song, Y., Bressan, S.: Fast identity anonymization on graphs. In: Liddle, S.W., Schewe, K.-D., Tjoa, A.M., Zhou, X. (eds.) DEXA 2012, Part I. LNCS, vol. 7446, pp. 281–295. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  15. 15.
    Lueker, G.S.: Two NP-complete problems in nonnegative integer programming. Technical report. Computer Science Laboratory, Princeton University (1975)Google Scholar
  16. 16.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press (2006)Google Scholar
  17. 17.
    Niedermeier, R.: Reflections on multivariate algorithmics and problem parameterization. In: Proc. 27th STACS. LIPIcs, vol. 5, pp. 17–32. Schloss Dagstuhl–Leibniz-Zentrum für Informatik (2010)Google Scholar
  18. 18.
    Zhou, B., Pei, J.: The k-anonymity and l-diversity approaches for privacy preservation in social networks against neighborhood attacks. Knowl. Inf. Syst. 28(1), 47–77 (2011)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Robert Bredereck
    • 1
  • Vincent Froese
    • 1
  • Sepp Hartung
    • 1
  • André Nichterlein
    • 1
  • Rolf Niedermeier
    • 1
  • Nimrod Talmon
    • 1
  1. 1.Institut für Softwaretechnik und Theoretische InformatikTU BerlinGermany

Personalised recommendations