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On Making a Distinguished Vertex of Minimum Degree by Vertex Deletion

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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.

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Fig. 1
Algorithm 1
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Notes

  1. Observation 1 does not hold for a feedback vertex set containing the distinguished vertex. Hence, the following approach does not transfer to this more general case.

References

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  6. Betzler, N., Uhlmann, J.: Parameterized complexity of candidate control in elections and related digraph problems. Theor. Comput. Sci. 410(52), 5425–5442 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bodlaender, H., Downey, R., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci. 75(8), 423–434 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bodlaender, H.L.: A linear time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25, 1305–1317 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bodlaender, H.L.: A partial k-arboretum of graphs with bounded treewidth. Theor. Comput. Sci. 209(1–2), 1–45 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  11. Bodlaender, H.L., Koster, A.M.C.A.: Treewidth computations II. Lower bounds. Inf. Comput. 209(7), 1103–1119 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  12. Bodlaender, H.L., Thomassé, S., Yeo, A.: Kernel bounds for disjoint cycles and disjoint paths. Theor. Comput. Sci. 412(35), 4570–4578 (2011)

    Article  MATH  Google Scholar 

  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. 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)

    Chapter  Google Scholar 

  15. Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Berlin (1999)

    Book  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  19. Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Berlin (2006)

    Google Scholar 

  20. Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct PCPs for NP. J. Comput. Syst. Sci. 77(1), 91–106 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  21. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)

    MATH  Google Scholar 

  22. Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. SIGACT News 38(1), 31–45 (2007)

    Article  Google Scholar 

  23. Huberman, B.A., Romero, D.M., Wu, F.: Social networks that matter: twitter under the microscope. First Monday 14(1) (2009)

  24. Kloks, T.: Treewidth. Computations and Approximations. Lecture Notes in Computer Science, vol. 842. Springer, Berlin (1994)

    MATH  Google Scholar 

  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. 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. Moser, H., Niedermeier, R., Sorge, M.: Exact combinatorial algorithms and experiments for finding maximum k-plexes. J. Comb. Optim. 24(3), 343–373 (2012)

    MathSciNet  Google Scholar 

  28. Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)

    Book  MATH  Google Scholar 

  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. 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)

    Article  Google Scholar 

  31. Ramachandramurthi, S.: The structure and number of obstructions to treewidth. SIAM J. Discrete Math. 10, 146–157 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  32. Romm-Livermore, C., Setzekorn, K.: Social networking communities and e-dating services: concepts and implications. Information Science Reference (2008)

  33. Seidman, S., Foster, B.: A graph-theoretic generalization of the clique concept. J. Math. Sociol. 6(1), 139–154 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  34. Thomassé, S.: A 4k 2 kernel for feedback vertex set. ACM Trans. Algorithms 6(2), 32:1–32:8 (2010)

    Article  Google Scholar 

  35. Wasserman, S., Faust, K.: Social Network Analysis: Methods and Applications. Cambridge University Press, Cambridge (1994)

    Book  Google Scholar 

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Acknowledgement

We are grateful to two anonymous referees of Algorithmica whose constructive feedback helped to improve the quality of our presentation.

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Authors and Affiliations

  1. Institut für Softwaretechnik und Theoretische Informatik, TU Berlin, Berlin, Germany

    Nadja Betzler, Robert Bredereck, Rolf Niedermeier & Johannes Uhlmann

  2. Department of Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands

    Hans L. Bodlaender

Authors
  1. Nadja Betzler
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  2. Hans L. Bodlaender
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  3. Robert Bredereck
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  4. Rolf Niedermeier
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  5. Johannes Uhlmann
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Corresponding author

Correspondence to Robert Bredereck.

Additional information

An extended abstract of this paper appeared in Proceedings of the 37th International Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM-2011), January 22–28, 2011, Nový Smokovec, Slovakia, volume 6543 in Lecture Notes in Computer Science, pages 123–134, Springer, 2011. Compared to the conference version, the most important change (besides providing missing details) here is that we provide a concrete tree decomposition-based dynamic programming algorithm for Min-Degree Deletion parameterized by treewidth while the conference version just claimed a classification result based on monadic second-order logic.

N. Betzler and R. Bredereck were supported by the DFG, research project PAWS, NI 369/10.

J. Uhlmann was supported by the DFG, research project PABI, NI 369/7.

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Betzler, N., Bodlaender, H.L., Bredereck, R. et al. On Making a Distinguished Vertex of Minimum Degree by Vertex Deletion. Algorithmica 68, 715–738 (2014). https://doi.org/10.1007/s00453-012-9695-6

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