Theory of Computing Systems

, Volume 61, Issue 3, pp 795–819 | Cite as

Parameterized Complexity of Secluded Connectivity Problems

  • Fedor V. Fomin
  • Petr A. Golovach
  • Nikolay Karpov
  • Alexander S. Kulikov
Article
  • 78 Downloads

Abstract

The Secluded Path problem models a situation where sensitive information has to be transmitted between a pair of nodes along a path in a network. The measure of the quality of a selected path is its exposure cost, which is the total cost of vertices in its closed neighborhood. The task is to select a secluded path, i.e., a path with a small exposure cost. Similarly, the Secluded Steiner Tree problem is to find a tree in a graph connecting a given set of terminals such that the exposure cost of the tree is minimized. In this paper we present a systematic study of the parameterized complexity of Secluded Steiner Tree. In particular, we establish the tractability of Secluded Path being parameterized by “above guarantee” value, which in this case is the length of a shortest path between vertices. We also show how to extend this result for Secluded Steiner Tree, in this case we parameterize above the size of an optimal Steiner tree and the number of terminals. We also consider various parameterization of the problems such as by the treewidth, the size of a vertex cover, feedback vertex set, or the maximum vertex degree and establish kernelization complexity of the problem subject to different choices of parameters.

Keywords

Secluded path Secluded Steiner tree Parameterized complexity Kernelization 

References

  1. 1.
    Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM 42, 844–856 (1995)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bafna, V., Berman, P., Fujito, T.: A 2-approximation algorithm for the undirected feedback vertex set problem. SIAM J. Discret. Math. 12, 289–297 (1999)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    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)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Fourier meets Möbius: fast subset convolution. In: Proceedings of the 39th Annual ACM Symposium on Theory of Computing, pp 67–74. ACM, California, USA (2007)Google Scholar
  5. 5.
    Bodlaender, H. L., Downey, R. G., Fellows, M. R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci. 75, 423–434 (2009)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bodlaender, H. L., Jansen, B. M. P., Kratsch, S.: Kernelization lower bounds by cross-composition. SIAM J. Discret. Math. 28, 277–305 (2014)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cai, L.: Parameterized complexity of cardinality constrained optimization problems. Comput. J. 51, 102–121 (2008)CrossRefGoogle Scholar
  8. 8.
    Cai, L., Chan, S. M., Chan, S. O.: Random separation: a new method for solving fixed-cardinality optimization problems. In: IWPEC, vol. 4169 of Lecture Notes in Computer Science, Springer, pp 239–250 (2006)Google Scholar
  9. 9.
    Chechik, S., Johnson, M. P., Parter, M., Peleg, D.: Secluded connectivity problems. CoRR, abs/1212, 6176 (2012)Google Scholar
  10. 10.
    Chechik, S., Johnson, M. P., Parter, M., Peleg, D.: Secluded connectivity problems. In: Proceedings of the 21st Annual European Symposium Algorithms (ESA), vol. 8125 of Lecture Notes in Computer Science, pp 301–312. Springer (2013)Google Scholar
  11. 11.
    Cygan, M., Fomin, F. V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized algorithms. Springer (2015)Google Scholar
  12. 12.
    Downey, R. G., Fellows, M. R.: Fundamentals of parameterized complexity. Texts in Computer Science, Springer (2013)Google Scholar
  13. 13.
    Dreyfus, S. E., Wagner, R. A.: The Steiner problem in graphs. Networks 1, 195–207 (1971)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Fellows, M. R., Hermelin, D., Rosamond, F. A., Vialette, S.: On the parameterized complexity of multiple-interval graph problems. Theor. Comput. Sci. 410, 53–61 (2009)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Fomin, F. V., Golovach, P. A., Karpov, N., Kulikov, A.S.: Parameterized complexity of secluded connectivity problems. In: FSTTCS 2015, vol. 45 of LIPIcs, pp 408–419Google Scholar
  16. 16.
    Fomin, F. V., Kratsch, D.: Exact exponential algorithms, Texts in Theoretical Computer Science. An EATCS Series, Springer-Verlag (2010)Google Scholar
  17. 17.
    Fomin, F. V., Villanger, Y.: Treewidth computation and extremal combinatorics. Combinatorica 32, 289–308 (2012)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Gao, J., Zhao, Q., Swami, A.: The thinnest path problem for secure communications: A directed hypergraph approach. In: Proceedings of the 50th Annual Allerton Conference on Communication, Control, and Computing, pp 847–852. IEEEGoogle Scholar
  19. 19.
    Garey, M. R., Johnson, D. S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman (1979)Google Scholar
  20. 20.
    Gilbers, A.: Visibility domains and complexity. PhD thesis, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn (2013)MATHGoogle Scholar
  21. 21.
    Gutin, G., Jones, M., Yeo, A.: Kernels for below-upper-bound parameterizations of the hitting set and directed dominating set problems. Theor. Comput. Sci. 412, 5744–5751 (2011)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Johnson, M. P., Liu, O., Rabanca, G.: Secluded path via shortest path. In: SIROCCO 2014, vol. 8576 of Lecture Notes in Computer Science, pp 108–120. Springer (2014)Google Scholar
  23. 23.
    Karp, R. M.: Reducibility among combinatorial problems. In: Proceedings of a symposium on the Complexity of Computer Computations, The IBM Research Symposia Series, pp 85–103. Plenum Press, New York (1972)Google Scholar
  24. 24.
    Naor, M., Schulman, L., Srinivasan, A.: Splitters and nearoptimal derandomization. In: 36th Annual Symposium on Foundations of Computer Science (FOCS 1995), pp 182–191. IEEE (1995)Google Scholar
  25. 25.
    Nederlof, J.: Fast polynomial-space algorithms using inclusion-exclusion. Algorithmica 65, 868–884 (2013)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Pietrzak, K.: On the parameterized complexity of the fixed alphabet shortest common supersequence and longest common subsequence problems. J. Comput. Syst. Sci. 67, 757–771 (2003)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Fedor V. Fomin
    • 1
    • 2
  • Petr A. Golovach
    • 1
    • 2
  • Nikolay Karpov
    • 2
  • Alexander S. Kulikov
    • 2
  1. 1.Department of InformaticsUniversity of BergenBergenNorway
  2. 2.Steklov Institute of Mathematics at St. Petersburg, Russian Academy of SciencesSaint PetersburgRussia

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