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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Boral, A., Cygan, M., Kociumaka, T., Pilipczuk, M.: Fast branching algorithm for cluster vertex deletion. CoRR, abs/1306.3877 (2013)
Chen, J., Kanj, I.A., Xia, G.: Improved upper bounds for vertex cover. Theor. Comput. Sci. 411(40-42), 3736–3756 (2010)
Demetrescu, C., Italiano, G.F.: Fully dynamic transitive closure: Breaking through the o(n2) barrier. In: FOCS, pp. 381–389 (2000)
Demetrescu, C., Italiano, G.F.: A new approach to dynamic all pairs shortest paths. In: STOC, pp. 159–166 (2003)
Driscoll, J.R., Sarnak, N., Sleator, D.D., Tarjan, R.E.: Making data structures persistent. J. Comput. Syst. Sci. 38(1), 86–124 (1989)
Dvorak, Z., Kupec, M., Tuma, V.: Dynamic data structure for tree-depth decomposition. CoRR, abs/1307.2863 (2013)
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)
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)
Gary, M.R., Johnson, D.S.: Computers and intractability: A guide to the theory of np-completeness (1979)
Henzinger, M.R., King, V.: Randomized dynamic graph algorithms with polylogarithmic time per operation. In: STOC, pp. 519–527 (1995)
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)
Hüffner, F., Komusiewicz, C., Moser, H., Niedermeier, R.: Fixed-parameter algorithms for cluster vertex deletion. Theory Comput. Syst. 47(1), 196–217 (2010)
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)
Iwata, Y., Oka, K.: Fast dynamic graph algorithms for parameterized problems (2014) (manuscript)
Patrascu, M., Demaine, E.D.: Lower bounds for dynamic connectivity. In: STOC, pp. 546–553 (2004)
Poutré, J.A.L.: Alpha-algorithms for incremental planarity testing (preliminary version). In: STOC, pp. 706–715 (1994)
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)
Roditty, L.: A faster and simpler fully dynamic transitive closure. In: SODA, pp. 404–412 (2003)
Roditty, L., Zwick, U.: Improved dynamic reachability algorithms for directed graphs. In: FOCS, pp. 679– (2002)
Roditty, L., Zwick, U.: Dynamic approximate all-pairs shortest paths in undirected graphs. In: FOCS, pp. 499–508 (2004)
Roditty, L., Zwick, U.: A fully dynamic reachability algorithm for directed graphs with an almost linear update time. In: STOC, pp. 184–191 (2004)
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)
Sankowski, P.: Dynamic transitive closure via dynamic matrix inverse (extended abstract). In: FOCS, pp. 509–517 (2004)
Thorup, M.: Near-optimal fully-dynamic graph connectivity. In: STOC, pp. 343–350 (2000)
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)
Thorup, M.: Worst-case update times for fully-dynamic all-pairs shortest paths. In: STOC, pp. 112–119 (2005)
Wulff-Nilsen, C.: Faster deterministic fully-dynamic graph connectivity. In: SODA, pp. 1757–1769 (2013)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Iwata, Y., Oka, K. (2014). Fast Dynamic Graph Algorithms for Parameterized Problems. In: Ravi, R., Gørtz, I.L. (eds) Algorithm Theory – SWAT 2014. SWAT 2014. Lecture Notes in Computer Science, vol 8503. Springer, Cham. https://doi.org/10.1007/978-3-319-08404-6_21
Download citation
DOI: https://doi.org/10.1007/978-3-319-08404-6_21
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-08403-9
Online ISBN: 978-3-319-08404-6
eBook Packages: Computer ScienceComputer Science (R0)