Abstract
In this paper we show that Feedback Vertex Set on planar graphs has a kernel of size at most . We give a polynomial time algorithm, that given a planar graph G finds a equivalent planar graph G′ with at most vertices, where k * is the size of the minimum Feedback Vertex Set of G. The kernelization algorithm is based on a number of reduction rules. The correctness of most of these rules is shown using a new notion: bases of induced subgraphs. We also show how to use this new notion to automatically prove safeness of reduction rules and obtain tighter bounds for the size of the kernel.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Alber, J., Fellows, M.R., Niedermeier, R.: Polynomial-time data reduction for dominating sets. J. ACM 51, 363–384 (2004)
Bafna, V., Berman, P., Fujito, T.: A 2-approximation algorithm for the undirected feedback vertex set problem. SIAM J. Disc. Math. 12, 289–297 (1999)
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 1994, pp. 344–354 (1994)
Becker, A., Bar-Yehuda, R., Geiger, D.: Randomized algorithms for the loop cutset problem. J. Artificial Intelligence Research 12, 219–234 (2000)
Becker, A., Geiger, D.: Optimization of Pearl’s method of conditioning and greedy-like approximation algorithms for the vertex feedback set problem. Artificial Intelligence 83, 167–188 (1996)
Bodlaender, H.L.: On disjoint cycles. Int. J. Found. Computer Science 5(1), 59–68 (1994)
Bodlaender, H.L.: A cubic kernel for feedback vertex set. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 320–331. Springer, Heidelberg (2007)
Burrage, K., Estivill-Castro, V., Fellows, M.R., Langston, M.A., Mac, S., Rosamond, F.A.: The undirected feedback vertex set problem has a poly(k) kernel. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 192–202. Springer, Heidelberg (2006)
Chen, J., Fomin, F.V., Liu, Y., Lu, S., Villanger, Y.: Improved algorithms for the feedback vertex set problems. In: Dehne, F., Sack, J.-R., Zeh, N. (eds.) WADS 2007. LNCS, vol. 4619, pp. 422–433. Springer, Heidelberg (2007)
Chen, J., Liu, Y., Lu, S., O’Sullivan, B., Razgon, I.: A fixed-parameter algorithm for the directed feedback vertex set problem. In: Proceedings STOC 2008 (to appear, 2008)
Chudak, F., Goemans, M., Hochbaum, D., Williamson, D.: A primal–dual interpretation of two 2-approximation algorithms for the feedback vertex set problem in undirected graphs. Operations Research Letters 22, 111–118 (1998)
Dehne, F.K.H.A., Fellows, M.R., Langston, M.A., Rosamond, F.A., Stevens, K.: An O(2O(k) n 3) FPT algorithm for the undirected feedback vertex set problem. In: Wang, L. (ed.) COCOON 2005. LNCS, vol. 3595, pp. 859–869. Springer, Heidelberg (2005)
Demaine, E.D., Hajiaghayi, M.: Bidimensionality: New connections between FPT algorithms and PTASs. In: Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2005, pp. 590–601 (2005)
Dorn, F.: Dynamic programming and fast matrix multiplication. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. LNCS, vol. 4168, pp. 280–291. Springer, Heidelberg (2006)
Dorn, F., Penninkx, E., Bodlaender, H.L., Fomin, F.V.: Efficient exact algorithms on planar graphs: Exploiting sphere cut branch decompositions. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 95–106. Springer, Heidelberg (2005)
Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness. Congressus Numerantium 87, 161–178 (1992)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1998)
Festa, P., Pardalos, P.M., Resende, M.G.C.: Feedback set problems. In: Handbook of Combinatorial Optimization, Amsterdam, The Netherlands, vol. A, pp. 209–258. Kluwer, Dordrecht (1999)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)
Fomin, F.V., Gaspers, S., Knauer, C.: Finding a minimum feedback vertex set in time O(1.7548n). In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 183–191. Springer, Heidelberg (2006)
Fomin, F.V., Thilikos, D.M.: New upper bounds on the decomposability of planar graphs. J. Graph Theory 51, 53–81 (2006)
Garey, M.R., Johnson, D.S.: Computers and Intractability, A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, New York (1979)
Goemans, M.X., Williamson, D.P.: Primal-dual approximation algorithms for feedback problems in planar graphs. Combinatorica 17, 1–23 (1997)
Guo, J., Gramm, J., Hffner, F., Niedermeier, R., Wernicke, S.: Compression-based fixed-parameter algorithms for feedback vertex set and edge bipartization. Journal of Computer and System Sciences 72(8), 1386–1396 (2006)
Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. ACM SIGACT News 38, 31–45 (2007)
Hackbusch, W.: On the feedback vertex set problem for a planar graph. Computing 58, 129–155 (1997)
Kanj, I.A., Pelsmajer, M.J., Schaefer, M.: Parameterized algorithms for feedback vertex set. In: Downey, R.G., Fellows, M.R., Dehne, F. (eds.) IWPEC 2004. LNCS, vol. 3162, pp. 235–248. Springer, Heidelberg (2004)
Kloks, T., Lee, C.M., Liu, J.: New algorithms for k-face cover, k-feedback vertex set, and k-disjoint cycles on plane and planar graphs. In: Kučera, L. (ed.) WG 2002. LNCS, vol. 2573, pp. 282–295. Springer, Heidelberg (2002)
Mehlhorn, K., Näher, S.: LEDA: A Platform for Combinatorial and Geometric Computing. Cambridge University Press, Cambridge (1995)
Niedermeier, R.: Invitation to fixed-parameter algorithms. Universität Tübingen, Habilitation Thesis (2002)
Raman, V., Saurabh, S., Subramanian, C.R.: Faster fixed parameter tractable algorithms for undirected feedback vertex set. In: Bose, P., Morin, P. (eds.) ISAAC 2002. LNCS, vol. 2518, pp. 241–248. Springer, Heidelberg (2002)
Raman, V., Saurabh, S., Subramanian, C.R.: Faster algorithms for feedback vertex set. In: Proceedings 2nd Brazilian Symposium on Graphs, Algorithms, and Combinatorics, GRACO 2005. Electronic Notes in Discrete Mathematics, vol. 19, pp. 273–279 (2005)
Razgon, I.: Exact computation of maximum induced forest. In: Arge, L., Freivalds, R. (eds.) SWAT 2006. LNCS, vol. 4059, pp. 160–171. Springer, Heidelberg (2006)
Stamm, H.: On feedback problems in planar digraphs. In: Möhring, R.H. (ed.) WG 1990. LNCS, vol. 484, pp. 79–89. Springer, Heidelberg (1991)
van Dijk, T.: Fixed parameter complexity of feedback problems. Master’s thesis, Utrecht University (2007)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bodlaender, H.L., Penninkx, E. (2008). A Linear Kernel for Planar Feedback Vertex Set. In: Grohe, M., Niedermeier, R. (eds) Parameterized and Exact Computation. IWPEC 2008. Lecture Notes in Computer Science, vol 5018. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79723-4_16
Download citation
DOI: https://doi.org/10.1007/978-3-540-79723-4_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-79722-7
Online ISBN: 978-3-540-79723-4
eBook Packages: Computer ScienceComputer Science (R0)