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Algorithmica

, Volume 80, Issue 9, pp 2637–2655 | Cite as

Dynamic Parameterized Problems

  • R. KrithikaEmail author
  • Abhishek Sahu
  • Prafullkumar Tale
Article
  • 210 Downloads
Part of the following topical collections:
  1. Special Issue dedicated to the 60th Birthday of Gregory Gutin

Abstract

We study the parameterized complexity of various graph theoretic problems in the dynamic framework where the input graph is being updated by a sequence of edge additions and deletions. Vertex subset problems on graphs typically deal with finding a subset of vertices having certain properties. In real world applications, the graph under consideration often changes over time and due to this dynamics, the solution at hand might lose the desired properties. The goal in the area of dynamic graph algorithms is to efficiently maintain a solution under these changes. Recomputing a new solution on the new graph is an expensive task especially when the number of modifications made to the graph is significantly smaller than the size of the graph. In the context of parameterized algorithms, two natural parameters are the size k of the symmetric difference of the edge sets of the two graphs (on n vertices) and the size r of the symmetric difference of the two solutions. We study the Dynamic \(\Pi \)-Deletion problem which is the dynamic variant of the classical \(\Pi \)-Deletion problem and show NP-hardness, fixed-parameter tractability and kernelization results. For specific cases of Dynamic \(\Pi \)-Deletion such as Dynamic Vertex Cover and Dynamic Feedback Vertex Set, we describe improved algorithms and linear kernels. Specifically, we show that Dynamic Vertex Cover has a deterministic algorithm with \(1.0822^k n^{\mathcal {O}(1)}\) running time and Dynamic Feedback Vertex Set has a randomized algorithm with \(1.6667^k n^{\mathcal {O}(1)}\) running time. We also show that Dynamic Connected Feedback Vertex Set can be solved in \(2^{\mathcal {O}(k)} n^{\mathcal {O}(1)}\) time. For each of Dynamic Connected Vertex Cover, Dynamic Dominating Set and Dynamic Connected Dominating Set, we describe an algorithm with \(2^k n^{\mathcal {O}(1)}\) running time and show that this is the optimal running time (up to polynomial factors) assuming the Set Cover Conjecture.

Keywords

Dynamic problems Fixed-parameter tractability kernelization 

Notes

Acknowledgements

We are grateful to Saket Saurabh for the invaluable discussions and for providing several useful pointers.

References

  1. 1.
    Abu-Khzam, F.N., Cai, S., Egan, J., Shaw, P., Wang. K.: Turbo-charging cominating set with an FPT subroutine: Further Improvements and Experimental Analysis. In: Proceedings of the 14th International Conference on Theory and Applications of Models of Computation pp. 59–70. Springer (2017)Google Scholar
  2. 2.
    Abu-Khzam, F.N., Egan, J., Fellows, M.R., Rosamond, F.A.: Shaw. P.: on the parameterized complexity of dynamic problems. Theor. Comput. Sci. 607(3), 426–434 (2015)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bodlaender, H.L., Cygan, M., Kratsch, S., Nederlof, J.: Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth. Inf. Comput. 243, 86–111 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Beigel, R.: Finding maximum independent sets in sparse and general graphs. In: Proceedings of the \(10th\) Annual ACM-SIAM Symposium on Discrete Algorithms SODA ’99, pp. 856–857. SIAM (1999)Google Scholar
  5. 5.
    Bonsma, P., Lokshtanov, D.: Feedback vertex set in mixed graphs. In: Proceedings of the 12th International Conference on Algorithms and Data Structures WADS ’11, pp. 122–133. Springer (2011)Google Scholar
  6. 6.
    Bar-Yehuda, R., Geiger, D., Naor, J., Roth, R.M.: Approximation algorithms for the vertex feedback set problem with applications to constraint satisfaction and bayesian inference. In: Proceedings of the 5th Annual ACM–SIAM Symposium on Discrete Algorithms SODA ’94, pp. 344–354. SIAM (1994)Google Scholar
  7. 7.
    Cai, L.: Fixed-parameter tractability of graph modification problems for hereditary properties. Inf. Process. Lett. 58(4), 171–176 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cygan, M., Dell, H., Lokshtanov, D., Marx, D., Nederlof, J., Okamoto, Y., Paturi, R., Saurabh, S., Wahlstrom, M.: On problems as hard as CNF-SAT. ACM Trans. Algorithms 12(3), 41 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Cygan, M., Fomin, F.V., Łukasz, K., Lokshtanov, D., Marx, D., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Berlin (2015)CrossRefzbMATHGoogle Scholar
  10. 10.
    Chen, J., Kanj, I.A., Jia, W.: Vertex cover: further observations and further improvements. J. Algorithms 41(2), 280–301 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chen, J., Kanj, I.A., Xia, G.: Improved upper bounds for vertex cover. Theor. Comput. Sci. 411(40–42), 3736–3756 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cygan, M., Nederlof, J., Pilipczuk, M., van Rooij, J.M.M., Wojtaszczyk, J.O.: Solving connectivity problems parameterized by treewidth in single exponential time. In: Proceedings of the 52nd Annual Symposium on Foundations of Computer Science FOCS ’11, pp. 150–159. IEEE (2011)Google Scholar
  13. 13.
    Downey, R.G., Egan, J., Fellows, M.R., Rosamond, F.A.: Shaw. P.: dynamic dominating set and turbo-charging greedy heuristics. Tsinghua. Sci. Technol. 19(4), 329–337 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Downey, R.G., Fellows, R.G.: Fundamentals of Parameterized Complexity. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
  15. 15.
    Diestel, R.: Raph Theory. Springer, Berlin (2005)Google Scholar
  16. 16.
    Dom, M., Lokshtanov, D., Saurabh, S.: Incompressibility through colors and IDs. In: Proceedings of the 36th International Colloquium on Automata, Languages and Programming (ICALP 2009) pp. 378–389. Springer (2009)Google Scholar
  17. 17.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Berlin (2006)zbMATHGoogle Scholar
  18. 18.
    Fomin, F.V., Gaspers, S., Lokshtanov, D., Saurabh, S.: Exact algorithms via monotone local search. In: Proceedings of the 48th Annual ACM Symposium on Theory of Computing STOC ’16, pp. 764–775. ACM (2016)Google Scholar
  19. 19.
    Fomin, F.V., Gaspers, S., Saurabh, S., Stepanov, A.A.: On two techniques of combining branching and treewidth. Algorithmica 54(2), 181–207 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Fomin, F.V., Kratsch, D., Woeginger, G.J.: Exact (exponential) algorithms for the dominating set problem. In: Proceedings of the 30th Workshop on Graph Theoretic Concepts in Computer Science (WG) pp. 245–256. Springer (2005)Google Scholar
  21. 21.
    Gaspers, S., Gudmundsson, J., Jones, M., Mestre, J., Rümmele, S.: Turbocharging treewidth heuristics. In: Proceedings of the 11th International Symposium on Parameterized and Exact Computation (IPEC 2016) volume 63 of Leibniz International Proceedings in Informatics (LIPIcs) pp. 13:1–13:13. Schloss Dagstuhl–Leibniz–Zentrum fuer Informatik (2017)Google Scholar
  22. 22.
    Hartung, S., Niedermeier, R.: Incremental list coloring of graphs, parameterized by conservation. Theor. Comput. Sci. 494, 86–98 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kociumaka, T., Pilipczuk, M.: Faster deterministic feedback vertex set. Inf. Process. Lett. 114(10), 556–560 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Khot, S., Raman, V.: Parameterized complexity of finding subgraphs with hereditary properties. Theor. Comput. Sci. 289(2), 997–1008 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lewis, J.M., Yannakakis, M.: The node-deletion problem for hereditary properties is NP-complete. J. Comput. Syst. Sci. 20(2), 219–230 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Misra, N., Philip, G., Raman, V., Saurabh, S., Sikdar, S.: FPT algorithms for connected feedback vertex set. J. Comb. Optim. 24(2), 131–146 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Thomassé, S.: A \(4k^2\) kernel for feedback vertex set. ACM Trans. Algorithms 6(2), 32 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.The Institute of Mathematical SciencesHBNIChennaiIndia

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