Abstract
The Feedback Vertex Set problem on unweighted, undirected graphs is considered. Improving upon a result by Burrage et al. (Proceedings 2nd International Workshop on Parameterized and Exact Computation, pp. 192–202, 2006), we show that this problem has a kernel with O(k 3) vertices, i.e., there is a polynomial time algorithm, that given a graph G and an integer k, finds a graph G′ with O(k 3) vertices and integer k′≤k, such that G has a feedback vertex set of size at most k, if and only if G′ has a feedback vertex set of size at most k′. Moreover, the algorithm can be made constructive: if the reduced instance G′ has a feedback vertex set of size k′, then we can easily transform a minimum size feedback vertex set of G′ into a minimum size feedback vertex set of G. This kernelization algorithm can be used as the first step of an FPT algorithm for Feedback Vertex Set, but also as a preprocessing heuristic for Feedback Vertex Set.
We also show that the related Loop Cutset problem also has a kernel of cubic size. The kernelization algorithms are experimentally evaluated, and we report on these experiments.
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Abu-Khzam, F.N., Collins, R.L., Fellows, M.R., Langston, M.A., Suters, W.H., Symons, C.T.: Kernelization algorithms for the vertex cover problem: theory and experiments. In: Proc. 6th ACM-SIAM ALENEX, 2004, pp. 62–69. ACM-SIAM, New York, Philadelphia (2004)
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. Discrete Math. 12, 289–297 (1999)
Becker, A., Bar-Yehuda, R., Geiger, D.: Randomized algorithms for the loop cutset problem. J. Artif. Intell. Res. 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. Artif. Intell. 83, 167–188 (1996)
Bodlaender, H.L.: On disjoint cycles. Int. J. Found. Comput. Sci. 5(1), 59–68 (1994)
Bodlaender, H.L.: A linear time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25, 1305–1317 (1996)
Bodlaender, H.L.: Necessary edges in k-chordalizations of graphs. J. Comb. Optim. 7, 283–290 (2003)
Bodlaender, H.L., Penninkx, E.: A linear kernel for planar feedback vertex set. In: Grohe, M., Niedermeier, R. (eds.) Proceedings 3rd International Workshop on Parameterized and Exact Computation, IWPEC 2008. Lecture Notes in Computer Science, vol. 5018, pp. 160–171. Springer, Berlin (2008)
Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels (extended abstract). In: Proceedings 35th International Colloquium on Automata, Languages and Programming, ICALP 2008. Lecture Notes in Computer Science, vol. 5125, pp. 563–574. Springer, Berlin (2008)
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.) Proceedings 2nd International Workshop on Parameterized and Exact Computation, IWPEC 2006. Lecture Notes in Computer Science, vol. 4169, pp. 192–202. Springer, Berlin (2006)
Chen, J., Kanj, I.A., Jia, W.: Vertex cover: further observations and further improvements. J. Algorithms 41, 280–301 (2001)
Chen, J., Liu, Y., Lu, S., O’Sullivan, B., Razgon, I.: A fixed-parameter algorithm for the directed feedback vertex set problem. In: STOC ’08: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, New York, USA, 2008, pp. 177–186. ACM, New York (2008)
Clautiaux, F., Carlier, J., Moukrim, A., Négre, S.: New lower and upper bounds for graph treewidth. In: Rolim, J.D.P. (ed.) Proceedings International Workshop on Experimental and Efficient Algorithms, WEA 2003. Lecture Notes in Computer Science, vol. 2647, pp. 70–80. Springer, Berlin (2003)
Dehne, F., Fellows, M., Langston, M., Rosamond, F., Stevens, K.: An O(2O(k) n 3) FPT algorithm for the undirected feedback vertex set problem. In: Wang, L. (ed.) Proceedings 11th International Computing and Combinatorics Conference COCOON 2005. Lecture Notes in Computer Science, vol. 3595, pp. 859–869. Springer, Berlin (2005)
Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness. Congr. Numer. 87, 161–178 (1992)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Berlin (1998)
Festa, P., Pardalos, P.M., Resende, M.G.C.: Feedback set problems. In: Handbook of Combinatorial Optimization, vol. A, pp. 209–258. Kluwer, Amsterdam (1999)
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.) Proceedings 2nd International Workshop on Parameterized and Exact Computation, IWPEC 2006. Lecture Notes in Computer Science, vol. 4169, pp. 183–191. Springer, Berlin (2006)
Gerards, A.M.H.: Matching. In: Ball, M.O. et al. (ed.) Network Models. Handbooks in Operations Research and Management Sciences, vol. 7, Chap. 3, pp. 135–224. Elsevier Science, Amsterdam (1995)
Gogate, V., Dechter, R.: A complete anytime algorithm for treewidth. In: Proceedings of the 20th Annual Conference on Uncertainty in Artificial Intelligence UAI-04. Arlington, Virginia, USA, 2004, pp. 201–208. AUAI Press, Berkeley (2004)
Goldberg, A.V., Karzanov, A.V.: Maximum skew-symmetric flows and matchings. Math. Program. 100, 537–568 (2004)
Guo, J.: A more effective linear kernelization for cluster editing. In: Chen, B., Paterson, M., Zhang, G. (eds.) Proceedings First International Symposium on Combinatorics, Algorithms, Probabilistic and Experimental Methodologies ESCAPE 2007. Lecture Notes in Computer Science, vol. 4614, pp. 36–47. Springer, Berlin (2007)
Guo, J., Gramm, J., Hüffner, F., Niedermeier, R., Wernicke, S.: Improved fixed-parameter algorithms for two feedback set problems. In: Proc. 9th Int. Workshop on Algorithms and Data Structures WADS 2004. Lecture Notes in Computer Science, vol. 3608, pp. 158–168. Springer, Berlin (2004)
Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. ACM SIGACT News 38, 31–45 (2007)
Kanj, I.A., Pelsmajer, M.J., Schaefer, M.: Parameterized algorithms for feedback vertex set. In: Downey, R.G., Fellows, M.R. (eds.) Proceedings 1st International Workshop on Parameterized and Exact Computation, IWPEC 2004. Lecture Notes in Computer Science, vol. 3162, pp. 235–248. Springer, Berlin (2004)
Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–104. Plenum Press, New York (1972)
Mehlhorn, K., Näher, S.: LEDA: A Platform for Combinatorial and Geometric Computing. Cambridge University Press, Cambridge (1995)
Nagamochi, H., Ibaraki, T.: A linear-time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph. Algorithmica 7, 583–596 (1992)
Nemhauser, G.L., Trotter, L.E.: Vertex packing: structural properties and algorithms. Math. Program. 8, 232–248 (1975)
Niedermeier, R.: Invitation to fixed-parameter algorithms. Universität Tübingen. Habilitation Thesis (2002)
Pearl, J.: Probablistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, Palo Alto (1988)
Raman, V., Saurabh, S., Subramanian, C.R.: Faster fixed parameter tractable algorithms for undirected feedback vertex set. In: Proceedings 13th International Symposium on Algorithms and Computation, ISAAC 2002. Lecture Notes in Computer Science, vol. 2518, pp. 241–248. Springer, Berlin (2002)
Raman, V., Saurabh, S., Subramanian, C.R.: Faster algorithms for feedback vertex set. Electronic Notes in Discrete Mathematics 19, 273–279 (2005). Proceedings 2nd Brazilian Symposium on Graphs, Algorithms, and Combinatorics, GRACO 2005
Razgon, I.: Exact computation of maximum induced forest. In: Arge, L., Freivalds, R. (eds.) Proceedings of the 10th Scandinavian Workshop on Algorithm Theory, SWAT 2006. Lecture Notes in Computer Science, vol. 4059, pp. 160–171. Springer, Berlin (2006)
Schrijver, A.: Combinatorial Optimization. Polyhedra and Efficiency. Springer, Berlin (2003)
Thomassé, S.: A quadratic kernel for feedback vertex set. In: SODA ’09: Proceedings of the Nineteenth Annual ACM–SIAM Symposium on Discrete Algorithms, Philadelphia, PA, USA, 2009. pp. 115–119. Society for Industrial and Applied Mathematics, Philadelphia (2009)
Treewidthlib. http://www.cs.uu.nl/people/hansb/treewidthlib (2004)
Tutte, W.T.: A short proof of the factor theorem for finite graphs. Can. J. Math. 6, 347–352 (1954)
van Dijk, T.C.: Fixed parameter complexity of feedback problems. Master’s thesis, Utrecht University (2007)
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Bodlaender, H.L., van Dijk, T.C. A Cubic Kernel for Feedback Vertex Set and Loop Cutset. Theory Comput Syst 46, 566–597 (2010). https://doi.org/10.1007/s00224-009-9234-2
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DOI: https://doi.org/10.1007/s00224-009-9234-2