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Clustering with Local Restrictions

  • Daniel Lokshtanov
  • Dániel Marx
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)

Abstract

We study a family of graph clustering problems where each cluster has to satisfy a certain local requirement. Formally, let μ be a function on the subsets of vertices of a graph G. In the (μ,p,q)-Partition problem, the task is to find a partition of the vertices into clusters where each cluster C satisfies the requirements that (1) at most q edges leave C and (2) μ(C) ≤ p. Our first result shows that if μ is an arbitrary polynomial-time computable monotone function, then (μ,p,q)-Partition can be solved in time n O(q), i.e., it is polynomial-time solvable for every fixed q. We study in detail three concrete functions μ (number of nonedges in the cluster, maximum number of non-neighbours a vertex has in the cluster, the number of vertices in the cluster), which correspond to natural clustering problems. For these functions, we show that (μ,p,q)-Partition can be solved in time 2 O(p)·n O(1) and in randomized time 2 O(q)·n O(1), i.e., the problem is fixed-parameter tractable parameterized by p or by q.

Keywords

Cluster Problem Local Restriction Parallel Edge Correlation Cluster Satellite Problem 
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.

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References

  1. 1.
    Ailon, N., Charikar, M., Newman, A.: Aggregating inconsistent information: ranking and clustering. In: STOC 2005, pp. 684–693 (2005)Google Scholar
  2. 2.
    Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM 42(4), 844–856 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bansal, N., Blum, A., Chawla, S.: Correlation clustering. Machine Learning 56(1-3), 89–113 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bellman, R.: Dynamic programming treatment of the travelling salesman problem. J. ACM 9(1), 61–63 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen, J., Liu, Y., Lu, S.: An improved parameterized algorithm for the minimum node multiway cut problem. In: Dehne, F., Sack, J.-R., Zeh, N. (eds.) WADS 2007. LNCS, vol. 4619, pp. 495–506. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  6. 6.
    Chen, J., Liu, Y., Lu, S., O’Sullivan, B., Razgon, I.: A fixed-parameter algorithm for the directed feedback vertex set problem. J. ACM 55(5) (2008)Google Scholar
  7. 7.
    Cormen, T., Leiserson, C., Rivest, R., Stein, C.: Introduction to algorithms (2001)Google Scholar
  8. 8.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)CrossRefzbMATHGoogle Scholar
  9. 9.
    Heggernes, P., Lokshtanov, D., Nederlof, J., Paul, C., Telle, J.A.: Generalized graph clustering: Recognizing (o,q)-cluster graphs. In: Thilikos, D.M. (ed.) WG 2010. LNCS, vol. 6410, pp. 171–183. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  10. 10.
    Langston, M.A., Plaut, B.C.: On algorithmic applications of the immersion order: An overview of ongoing work presented at the third slovenian international conference on graph theory. Discrete Mathematics 182(1-3), 191–196 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lokshtanov, D., Marx, D.: Clustering with local restrictions. In: preparation, http://www.ii.uib.no/~daniello/papers/clusteringLocal.pdf
  12. 12.
    Marx, D.: Parameterized graph separation problems. Theoret. Comput. Sci. 351(3), 394–406 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Marx, D., Razgon, I.: Fixed-parameter tractability of multicut parameterized by the size of the cutset. To appear in STOC (2011)Google Scholar
  14. 14.
    Mathieu, C., Sankur, O., Schudy, W.: Online correlation clustering. In: STACS, pp. 573–584 (2010)Google Scholar
  15. 15.
    Mathieu, C., Schudy, W.: Correlation clustering with noisy input. In: SODA, pp. 712–728 (2010)Google Scholar
  16. 16.
    Naor, M., Schulman, L.J., Srinivasan, A.: Splitters and near-optimal derandomization. In: FOCS, pp. 182–191 (1995)Google Scholar
  17. 17.
    Razgon, I., O’Sullivan, B.: Almost 2-sat is fixed-parameter tractable (extended abstract). In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 551–562. Springer, Heidelberg (2008)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Daniel Lokshtanov
    • 1
  • Dániel Marx
    • 2
  1. 1.University of CaliforniaSan DiegoUSA
  2. 2.Humboldt-Universität zu BerlinBerlinGermany

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