Abstract
We study the computational complexity of \(k\)-anonymizing a given graph by as few graph contractions as possible. A graph is said to be \(k\)-anonymous if for every vertex in it, there are at least \(k-1\) other vertices with exactly the same degree. The general degree anonymization problem is motivated by applications in privacy-preserving data publishing, and was studied to some extent for various graph operations (most notable operations being edge addition, edge deletion, vertex addition, and vertex deletion). We complement this line of research by studying several variants of graph contractions, which are operations of interest, for example, in the contexts of social networks and clustering algorithms. We show that the problem of degree anonymization by graph contractions is \({\mathsf {NP}}\)-hard even for some very restricted inputs, and identify some fixed-parameter tractable cases.
A full version is available at http://fpt.akt.tu-berlin.de/talmon/abcfv.pdf.
N. Talmon—Supported by DFG Research Training Group MDS (GRK 1408).
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References
Asano, T., Hirata, T.: Edge-contraction problems. J. Comput. Syst. Sci. 26(2), 197–208 (1983)
Bazgan, C., Nichterlein, A.: Parameterized inapproximability of degree anonymization. In: Cygan, M., Heggernes, P. (eds.) IPEC 2014. LNCS, vol. 8894, pp. 75–84. Springer, Heidelberg (2014)
Belmonte, R., Golovach, P.A., Hof, P., Paulusma, D.: Parameterized complexity of three edge contraction problems with degree constraints. Acta Informatica 51(7), 473–497 (2014)
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)
Bredereck, R., Froese, V., Hartung, S., Nichterlein, A., Niedermeier, R., Talmon, N.: The complexity of degree anonymization by vertex addition. In: Gu, Q., Hell, P., Yang, B. (eds.) AAIM 2014. LNCS, vol. 8546, pp. 44–55. Springer, Heidelberg (2014)
Cai, L.: Parameterized complexity of cardinality constrained optimization problems. Comput. J. 51(1), 102–121 (2008)
Cai, L., Chan, S.M., Chan, S.O.: Random separation: a new method for solving fixed-cardinality optimization problems. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 239–250. Springer, Heidelberg (2006)
Casas-Roma, J., Herrera-Joancomartí, J., Torra, V.: An algorithm for \(k\)-degree anonymity on large networks. In: Proceedings of ASONAM 2013, pp. 671–675. ACM Press (2013)
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. Soc. Netw. Analys. Min. 3(3), 381–399 (2013)
Clarkson, K.L., Liu, K., Terzi, E.: Towards identity anonymization in social networks. In: Yu, P.S., Han, J., Faloutsos, C. (eds.) Link Mining: Models, Algorithms, and Applications, pp. 359–385. Springer, New York (2010)
Delling, D., Görke, R., Schulz, C., Wagner, D.: Orca reduction and contraction graph clustering. In: Goldberg, A.V., Zhou, Y. (eds.) AAIM 2009. LNCS, vol. 5564, pp. 152–165. Springer, Heidelberg (2009)
Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173, 4th edn. Springer, Heidelberg (2010)
Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Springer, Heidelberg (2013)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)
Golovach, P.A., van ’t Hof, P., Paulusma, D.: Obtaining planarity by contracting few edges. In: Rovan, B., Sassone, V., Widmayer, P. (eds.) MFCS 2012. LNCS, vol. 7464, pp. 455–466. Springer, Heidelberg (2012)
Guillemot, S., Marx, D.: A faster FPT algorithm for bipartite contraction. Inf. Process. Lett. 113(22), 906–912 (2013)
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)
Heggernes, P., Hof, P., Lévêque, B., Lokshtanov, D., Paul, C.: Contracting graphs to paths and trees. Algorithmica 68(1), 109–132 (2014)
Hulett, H., Will, T.G., Woeginger, G.J.: Multigraph realizations of degree sequences: maximization is easy, minimization is hard. Oper. Res. Lett. 36(5), 594–596 (2008)
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)
Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)
Oxley, J.G.: Matroid theory, vol. 3. Oxford University Press, Oxford (2006)
Wolle, T., Bodlaender, H.L.: A note on edge contraction. Technical report, Technical Report UU-CS-2004 (2004)
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Hartung, S., Talmon, N. (2015). The Complexity of Degree Anonymization by Graph Contractions. In: Jain, R., Jain, S., Stephan, F. (eds) Theory and Applications of Models of Computation. TAMC 2015. Lecture Notes in Computer Science(), vol 9076. Springer, Cham. https://doi.org/10.1007/978-3-319-17142-5_23
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DOI: https://doi.org/10.1007/978-3-319-17142-5_23
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