Algorithmica

, Volume 68, Issue 3, pp 715–738 | Cite as

On Making a Distinguished Vertex of Minimum Degree by Vertex Deletion

  • Nadja Betzler
  • Hans L. Bodlaender
  • Robert Bredereck
  • Rolf Niedermeier
  • Johannes Uhlmann
Article
  • 228 Downloads

Abstract

For directed and undirected graphs, we study how to make a distinguished vertex the unique minimum-(in)degree vertex through deletion of a minimum number of vertices. The corresponding NP-hard optimization problems are motivated by applications concerning control in elections and social network analysis. Continuing previous work for the directed case, we show that the problem is W[2]-hard when parameterized by the graph’s feedback arc set number, whereas it becomes fixed-parameter tractable when combining the parameters “feedback vertex set number” and “number of vertices to delete”. For the so far unstudied undirected case, we show that the problem is NP-hard and W[1]-hard when parameterized by the “number of vertices to delete”. On the positive side, we show fixed-parameter tractability for several parameterizations measuring tree-likeness. In particular, we provide a dynamic programming algorithm for graphs of bounded treewidth and a vertex-linear problem kernel with respect to the parameter “feedback edge set number”. On the contrary, we show a non-existence result concerning polynomial-size problem kernels for the combined parameter “vertex cover number and number of vertices to delete”, implying corresponding non-existence results when replacing vertex cover number by treewidth or feedback vertex set number.

Keywords

Graph modification problem Parameterized complexity Treewidth 

References

  1. 1.
    Bafna, V., Berman, P., Fujito, T.: A 2-approximation algorithm for the undirected feedback vertex set problem. SIAM J. Discrete Math. 12(3), 289–297 (1999) CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Balasundaram, B., Butenko, S., Hicks, I.V.: Clique relaxations in social network analysis: the maximum k-plex problem. Oper. Res. 59(1), 133–142 (2011) CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Bar-Yehuda, R., Geiger, D.: Approximation algorithms for the feedback vertex set problem with applications to constraint satisfaction and Bayesian inference. SIAM J. Comput. 27, 942–959 (1998) CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Becker, A., Geiger, D.: Optimization of pearl’s method of conditioning and greedy-like approximation algorithms for the vertex feedback set problem. Artif. Intell. 83, 167–188 (1996) CrossRefMathSciNetGoogle Scholar
  5. 5.
    Betzler, N., Bredereck, R., Niedermeier, R., Uhlmann, J.: On bounded-degree vertex deletion parameterized by treewidth. Discrete Appl. Math. 160(1–2), 53–60 (2012) CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Betzler, N., Uhlmann, J.: Parameterized complexity of candidate control in elections and related digraph problems. Theor. Comput. Sci. 410(52), 5425–5442 (2009) CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Bodlaender, H., Downey, R., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci. 75(8), 423–434 (2009) CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Bodlaender, H.L.: A linear time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25, 1305–1317 (1996) CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Bodlaender, H.L.: A partial k-arboretum of graphs with bounded treewidth. Theor. Comput. Sci. 209(1–2), 1–45 (1998) CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Bodlaender, H.L.: Kernelization: new upper and lower bound techniques. In: Proceedings of the 4th International Workshop on Parameterized and Exact Computation (IWPEC’09). Lecture Notes in Computer Science, vol. 5917, pp. 17–37. Springer, Berlin (2009) CrossRefGoogle Scholar
  11. 11.
    Bodlaender, H.L., Koster, A.M.C.A.: Treewidth computations II. Lower bounds. Inf. Comput. 209(7), 1103–1119 (2011) CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Bodlaender, H.L., Thomassé, S., Yeo, A.: Kernel bounds for disjoint cycles and disjoint paths. Theor. Comput. Sci. 412(35), 4570–4578 (2011) CrossRefMATHGoogle Scholar
  13. 13.
    Cao, Y., Chen, J., Liu, Y.: On feedback vertex set new measure and new structures. In: Proceedings of the 12th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT’10). Lecture Notes in Computer Science, vol. 6139, pp. 93–104. Springer, Berlin (2010) Google Scholar
  14. 14.
    Dom, M., Lokshtanov, D., Saurabh, S.: Incompressibility through colors and IDs. In: Proceedings of the 36th International Colloquium on Automata, Languages, and Programming (ICALP’09). Lecture Notes in Computer Science, vol. 5555, pp. 378–389. Springer, Berlin (2009) CrossRefGoogle Scholar
  15. 15.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Berlin (1999) CrossRefGoogle Scholar
  16. 16.
    Faliszewski, P., Hemaspaandra, E., Hemaspaandra, L.A., Rothe, J.: Llull and Copeland voting computationally resist bribery and constructive control. Artif. Intell. 35(1), 275–341 (2009) MATHMathSciNetGoogle Scholar
  17. 17.
    Fellows, M.: Towards fully multivariate algorithmics: some new results and directions in parameter ecology. In: Proceedings of the 20th International Workshop (IWOCA’09). Lecture Notes in Computer Science, vol. 5874, pp. 2–10. Springer, Berlin (2009) Google Scholar
  18. 18.
    Fellows, M.R., Guo, J., Moser, H., Niedermeier, R.: A generalization of Nemhauser and Trotter’s local optimization theorem. J. Comput. Syst. Sci. 77(6), 1141–1158 (2011) CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Berlin (2006) Google Scholar
  20. 20.
    Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct PCPs for NP. J. Comput. Syst. Sci. 77(1), 91–106 (2011) CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979) MATHGoogle Scholar
  22. 22.
    Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. SIGACT News 38(1), 31–45 (2007) CrossRefGoogle Scholar
  23. 23.
    Huberman, B.A., Romero, D.M., Wu, F.: Social networks that matter: twitter under the microscope. First Monday 14(1) (2009) Google Scholar
  24. 24.
    Kloks, T.: Treewidth. Computations and Approximations. Lecture Notes in Computer Science, vol. 842. Springer, Berlin (1994) MATHGoogle Scholar
  25. 25.
    Komusiewicz, C., Niedermeier, R.: New races in parameterized algorithmics. In: Proceedings of the 37th International Symposium on Mathematical Foundations of Computer Science (MFCS’12). Lecture Notes in Computer Science, vol. 7464, pp. 19–30. Springer, Berlin (2012) Google Scholar
  26. 26.
    Lokshtanov, D., Misra, N., Saurabh, S.: Kernelization—preprocessing with a guarantee. In: The Multivariate Algorithmic Revolution and Beyond—Essays Dedicated to Michael R. Fellows on the Occasion of His 60th Birthday. Lecture Notes in Computer Science, vol. 7370, pp. 129–161. Springer, Berlin (2012) Google Scholar
  27. 27.
    Moser, H., Niedermeier, R., Sorge, M.: Exact combinatorial algorithms and experiments for finding maximum k-plexes. J. Comb. Optim. 24(3), 343–373 (2012) MathSciNetGoogle Scholar
  28. 28.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006) CrossRefMATHGoogle Scholar
  29. 29.
    Niedermeier, R.: Reflections on multivariate algorithmics and problem parameterization. In: Proceedings of the 27th International Symposium on Theoretical Aspects of Computer Science (STACS’10), Leibniz International Proceedings in Informatics, vol. 5, pp. 17–32. Schloss Dagstuhl—Leibniz-Zentrum fuer Informatik, Wadern (2010) Google Scholar
  30. 30.
    Potterat, J., Phillips-Plummer, L., Muth, S., Rothenberg, R., Woodhouse, D., Maldonado-Long, T., Zimmerman, H., Muth, J.: Risk network structure in the early epidemic phase of HIV transmission in Colorado Springs. Sex. Transm. Infect. 78(Supplement 1), 159–163 (2002) CrossRefGoogle Scholar
  31. 31.
    Ramachandramurthi, S.: The structure and number of obstructions to treewidth. SIAM J. Discrete Math. 10, 146–157 (1997) CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Romm-Livermore, C., Setzekorn, K.: Social networking communities and e-dating services: concepts and implications. Information Science Reference (2008) Google Scholar
  33. 33.
    Seidman, S., Foster, B.: A graph-theoretic generalization of the clique concept. J. Math. Sociol. 6(1), 139–154 (1978) CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Thomassé, S.: A 4k 2 kernel for feedback vertex set. ACM Trans. Algorithms 6(2), 32:1–32:8 (2010) CrossRefGoogle Scholar
  35. 35.
    Wasserman, S., Faust, K.: Social Network Analysis: Methods and Applications. Cambridge University Press, Cambridge (1994) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Nadja Betzler
    • 1
  • Hans L. Bodlaender
    • 2
  • Robert Bredereck
    • 1
  • Rolf Niedermeier
    • 1
  • Johannes Uhlmann
    • 1
  1. 1.Institut für Softwaretechnik und Theoretische InformatikTU BerlinBerlinGermany
  2. 2.Department of Information and Computing SciencesUtrecht UniversityUtrechtThe Netherlands

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