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On the Threshold of Intractability

  • Pål Grønås Drange
  • Markus Sortland Dregi
  • Daniel Lokshtanov
  • Blair D. Sullivan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9294)

Abstract

We study the computational complexity of the graph modification problems Open image in new window and Open image in new window , adding and deleting as few edges as possible to transform the input into a threshold (or chain) graph. In this article, we show that both problems are Open image in new window -hard, resolving a conjecture by Natanzon, Shamir, and Sharan (2001). On the positive side, we show that these problems admit quadratic vertex kernels. Furthermore, we give a subexponential time parameterized algorithm solving Open image in new window in Open image in new window time, making it one of relatively few natural problems in this complexity class on general graphs. These results are of broader interest to the field of social network analysis, where recent work of Brandes (2014) posits that the minimum edit distance to a threshold graph gives a good measure of consistency for node centralities. Finally, we show that all our positive results extend to Open image in new window , as well as the completion and deletion variants of both problems.

Keywords

Chordal Graph Split Graph Discrete Apply Mathematic Chain Graph Threshold 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.

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References

  1. 1.
    Brandes, U.: Social network algorithmics. ISAAC, Invited talk (2014)Google Scholar
  2. 2.
    Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes. A Survey. SIAM, Philadelphia (1999)Google Scholar
  3. 3.
    Burzyn, P., Bonomo, F., Durán, G.: NP-completeness results for edge modification problems. Discrete Applied Mathematics 154(13), 1824–1844 (2006)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Cai, L.: Fixed-parameter tractability of graph modification problems for hereditary properties. Information Processing Letters 58(4), 171–176 (1996)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Cao, Y., Marx, D.: Chordal editing is fixed-parameter tractable. In: STACS. LIPIcs, vol. 25, pp. 214–225 (2014)Google Scholar
  6. 6.
    Dehne, F., Langston, M., Luo, X., Pitre, S., Shaw, P., Zhang, Y.: The cluster editing problem: Implementations and experiments. In: IPEC (2006)Google Scholar
  7. 7.
    Drange, P.G., Fomin, F.V., Pilipczuk, M., Villanger, Y.: Exploring subexponential parameterized complexity of completion problems. In: STACS (2014)Google Scholar
  8. 8.
    Drange, P.G., Pilipczuk, M.: A polynomial kernel for trivially perfect editing. In: ESA (to appear, 2015)Google Scholar
  9. 9.
    Drange, P.G., Dregi, M.S., Lokshtanov, D., Sullivan, B.D.: On the threshold of intractability. CoRR, abs/1505.00612 (2015)Google Scholar
  10. 10.
    Feder, T., Mannila, H., Terzi, E.: Approximating the minimum chain completion problem. Information Processing Letters 109(17), 980–985 (2009)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Fomin, F.V., Villanger, Y.: Subexponential parameterized algorithm for minimum fill-in. SIAM J. Comput. 42(6), 2197–2216 (2013)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980)MATHGoogle Scholar
  13. 13.
    Guo, J.: Problem kernels for NP-complete edge deletion problems: Split and related graphs. In: Tokuyama, T. (ed.) ISAAC 2007. LNCS, vol. 4835, pp. 915–926. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  14. 14.
    Liu, Y., Wang, J., Guo, J.: An overview of kernelization algorithms for graph modification problems. Tsinghua Science and Technology 19(4), 346–357 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Liu, Y., Wang, J., Guo, J., Chen, J.: Complexity and parameterized algorithms for cograph editing. TCS 461, 45–54 (2012)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Liu, Y., Wang, J., You, J., Chen, J., Cao, Y.: Edge deletion problems: Branching facilitated by modular decomposition. Theoretical Computer Science 573, 63–70 (2015)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Mahadev, N., Peled, U.: Threshold graphs and related topics, vol. 56. Elsevier (1995)Google Scholar
  18. 18.
    Mancini, F.: Graph modification problems related to graph classes. PhD thesis, University of Bergen (2008)Google Scholar
  19. 19.
    Nastos, J., Gao, Y.: Familial groups in social networks. Social Networks 35(3), 439–450 (2013)CrossRefGoogle Scholar
  20. 20.
    Natanzon, A.: Complexity and approximation of some graph modification problems. PhD thesis, Tel Aviv University (1999)Google Scholar
  21. 21.
    Natanzon, A., Shamir, R., Sharan, R.: Complexity classification of some edge modification problems. Discrete Applied Mathematics 113(1), 109–128 (2001)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Schoch, D., Brandes, U.: Stars, neighborhood inclusion, and network centrality. In: SIAM Workshop on Network Science (2015)Google Scholar
  23. 23.
    Shamir, R., Sharan, R., Tsur, D.: Cluster graph modification problems. Discrete Applied Mathematics 144(1), 173–182 (2004)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Sharan, R.: Graph modification problems and their applications to genomic research. PhD thesis, Tel-Aviv University (2002)Google Scholar
  25. 25.
    Yannakakis, M.: Computing the minimum fill-in is NP-complete. SIAM Journal on Algebraic and Discrete Methods 2(1), 77–79 (1981)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Pål Grønås Drange
    • 1
  • Markus Sortland Dregi
    • 1
  • Daniel Lokshtanov
    • 1
  • Blair D. Sullivan
    • 2
  1. 1.Dept. InformaticsUniv. BergenBergenNorway
  2. 2.Dept. Computer ScienceNorth Carolina State UniversityRaleighUSA

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