Advertisement

Fast Dynamic Graph Algorithms for Parameterized Problems

  • Yoichi Iwata
  • Keigo Oka
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8503)

Abstract

Fully dynamic graph is a data structure that (1) supports edge insertions and deletions and (2) answers problem specific queries. The time complexity of (1) and (2) are referred to as the update time and the query time respectively. There are many researches on dynamic graphs whose update time and query time are o(|G|), that is, sublinear in the graph size. However, almost all such researches are for problems in P. In this paper, we investigate dynamic graphs for NP-hard problems exploiting the notion of fixed parameter tractability (FPT).

We give dynamic graphs for Vertex Cover and Cluster Vertex Deletion parameterized by the solution size k. These dynamic graphs achieve almost the best possible update time O(poly(k)logn) and the query time O(f(poly(k),k)), where f(n,k) is the time complexity of any static graph algorithm for the problems. We obtain these results by dynamically maintaining an approximate solution which can be used to construct a small problem kernel. Exploiting the dynamic graph for Cluster Vertex Deletion, as a corollary, we obtain a quasilinear-time (polynomial) kernelization algorithm for Cluster Vertex Deletion. Until now, only quadratic time kernelization algorithms are known for this problem.

Keywords

Time Complexity Vertex Cover Query Time Static Algorithm Cluster 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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Boral, A., Cygan, M., Kociumaka, T., Pilipczuk, M.: Fast branching algorithm for cluster vertex deletion. CoRR, abs/1306.3877 (2013)Google Scholar
  2. 2.
    Chen, J., Kanj, I.A., Xia, G.: Improved upper bounds for vertex cover. Theor. Comput. Sci. 411(40-42), 3736–3756 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Demetrescu, C., Italiano, G.F.: Fully dynamic transitive closure: Breaking through the o(n2) barrier. In: FOCS, pp. 381–389 (2000)Google Scholar
  4. 4.
    Demetrescu, C., Italiano, G.F.: A new approach to dynamic all pairs shortest paths. In: STOC, pp. 159–166 (2003)Google Scholar
  5. 5.
    Driscoll, J.R., Sarnak, N., Sleator, D.D., Tarjan, R.E.: Making data structures persistent. J. Comput. Syst. Sci. 38(1), 86–124 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Dvorak, Z., Kupec, M., Tuma, V.: Dynamic data structure for tree-depth decomposition. CoRR, abs/1307.2863 (2013)Google Scholar
  7. 7.
    Dvořák, Z., Tůma, V.: A dynamic data structure for counting subgraphs in sparse graphs. In: Dehne, F., Solis-Oba, R., Sack, J.-R. (eds.) WADS 2013. LNCS, vol. 8037, pp. 304–315. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  8. 8.
    Eppstein, D., Galil, Z., Italiano, G.F., Nissenzweig, A.: Sparsification-a technique for speeding up dynamic graph algorithms (extended abstract). In: FOCS, pp. 60–69 (1992)Google Scholar
  9. 9.
    Gary, M.R., Johnson, D.S.: Computers and intractability: A guide to the theory of np-completeness (1979)Google Scholar
  10. 10.
    Henzinger, M.R., King, V.: Randomized dynamic graph algorithms with polylogarithmic time per operation. In: STOC, pp. 519–527 (1995)Google Scholar
  11. 11.
    Holm, J., de Lichtenberg, K., Thorup, M.: Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity. J. ACM 48(4), 723–760 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Hüffner, F., Komusiewicz, C., Moser, H., Niedermeier, R.: Fixed-parameter algorithms for cluster vertex deletion. Theory Comput. Syst. 47(1), 196–217 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Italiano, G.F., Poutré, J.A.L., Rauch, M.H.: Fully dynamic planarity testing in planar embedded graphs (extended abstract). In: Lengauer, T. (ed.) ESA 1993. LNCS, vol. 726, pp. 212–223. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  14. 14.
    Iwata, Y., Oka, K.: Fast dynamic graph algorithms for parameterized problems (2014) (manuscript)Google Scholar
  15. 15.
    Patrascu, M., Demaine, E.D.: Lower bounds for dynamic connectivity. In: STOC, pp. 546–553 (2004)Google Scholar
  16. 16.
    Poutré, J.A.L.: Alpha-algorithms for incremental planarity testing (preliminary version). In: STOC, pp. 706–715 (1994)Google Scholar
  17. 17.
    Protti, F., da Silva, M.D., Szwarcfiter, J.L.: Applying modular decomposition to parameterized cluster editing problems. Theory Comput. Syst. 44(1):91–104 (2009)Google Scholar
  18. 18.
    Roditty, L.: A faster and simpler fully dynamic transitive closure. In: SODA, pp. 404–412 (2003)Google Scholar
  19. 19.
    Roditty, L., Zwick, U.: Improved dynamic reachability algorithms for directed graphs. In: FOCS, pp. 679– (2002)Google Scholar
  20. 20.
    Roditty, L., Zwick, U.: Dynamic approximate all-pairs shortest paths in undirected graphs. In: FOCS, pp. 499–508 (2004)Google Scholar
  21. 21.
    Roditty, L., Zwick, U.: A fully dynamic reachability algorithm for directed graphs with an almost linear update time. In: STOC, pp. 184–191 (2004)Google Scholar
  22. 22.
    Roditty, L., Zwick, U.: On dynamic shortest paths problems. In: Albers, S., Radzik, T. (eds.) ESA 2004. LNCS, vol. 3221, pp. 580–591. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  23. 23.
    Sankowski, P.: Dynamic transitive closure via dynamic matrix inverse (extended abstract). In: FOCS, pp. 509–517 (2004)Google Scholar
  24. 24.
    Thorup, M.: Near-optimal fully-dynamic graph connectivity. In: STOC, pp. 343–350 (2000)Google Scholar
  25. 25.
    Thorup, M.: Fully-dynamic all-pairs shortest paths: Faster and allowing negative cycles. In: Hagerup, T., Katajainen, J. (eds.) SWAT 2004. LNCS, vol. 3111, pp. 384–396. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  26. 26.
    Thorup, M.: Worst-case update times for fully-dynamic all-pairs shortest paths. In: STOC, pp. 112–119 (2005)Google Scholar
  27. 27.
    Wulff-Nilsen, C.: Faster deterministic fully-dynamic graph connectivity. In: SODA, pp. 1757–1769 (2013)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Yoichi Iwata
    • 1
  • Keigo Oka
    • 1
  1. 1.Department of Computer Science Graduate School of Information Science and TechnologyThe University of TokyoJapan

Personalised recommendations