Abstract
The Subset Feedback Vertex Set problem takes as input a weighted graph G and a vertex subset S of G, and the task is to find a set of vertices of total minimum weight to be removed from G such that in the remaining graph no cycle contains a vertex of S. This problem is a generalization of two classical NP-complete problems: Feedback Vertex Set and Multiway Cut. We show that it can be solved in time O(1.8638n) for input graphs on n vertices. To the best of our knowledge, no exact algorithm breaking the trivial 2n n O(1)-time barrier has been known for Subset Feedback Vertex Set, even in the case of unweighted graphs. The mentioned running time is a consequence of the more general main result of this paper: we show that all minimal subset feedback vertex sets of a graph can be enumerated in O(1.8638n) time.
This work is supported by the Research Council of Norway.
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References
Open problems from Dagstuhl seminar 09511. Dagstuhl Seminar 09511 (2009)
Calinescu, G.: Multiway cut. In: Encyclopedia of Algorithms, Springer, Heidelberg (2008)
Chen, J., Fomin, F.V., Liu, Y., Lu, S., Villanger, Y.: Improved algorithms for feedback vertex set problems. J. Comput. Syst. Sci. 74(7), 1188–1198 (2008)
Cygan, M., Pilipczuk, M., Pilipczuk, M., Wojtaszczyk, J.O.: Subset feedback vertex set is fixed parameter tractable. In: CoRR, abs/1004.2972 (2010)
Even, G., Naor, J., Zosin, L.: An 8-approximation algorithm for the subset feedback vertex set problem. SIAM J. Comput. 30(4), 1231–1252 (2000)
Fomin, F.V., Gaspers, S., Pyatkin, A.V., Razgon, I.: On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52(2), 293–307 (2008)
Fomin, F.V., Grandoni, F., Kratsch, D.: A measure & conquer approach for the analysis of exact algorithms. J. ACM 56, Article 25(5) (2009)
Fomin, F.V., Kratsch, D.: Exact Exponential Algorithms. In: Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin (2010)
Fomin, F.V., Villanger, Y.: Finding induced subgraphs via minimal triangulations. In: Proceedings of STACS 2010, vol. 5, pp. 383–394. Schloss Dagstuhl—Leibniz-Zentrum fuer Informatik (2010)
Garey, M.R., Johnson, D.S.: Computers and Intractability. W. H. Freeman and Co., New York (1978)
Garg, N., Vazirani, V.V., Yannakakis, M.: Multiway cuts in node weighted graphs. J. Algorithms 50(1), 49–61 (2004)
Raman, V., Saurabh, S., Subramanian, C.R.: Faster fixed parameter tractable algorithms for finding feedback vertex sets. ACM Transactions on Algorithms 2(3), 403–415 (2006)
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)
Thomassé, S.: A 4k2 kernel for feedback vertex set. ACM Transactions on Algorithms 6, Article 32(2) (2010)
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Fomin, F.V., Heggernes, P., Kratsch, D., Papadopoulos, C., Villanger, Y. (2011). Enumerating Minimal Subset Feedback Vertex Sets. In: Dehne, F., Iacono, J., Sack, JR. (eds) Algorithms and Data Structures. WADS 2011. Lecture Notes in Computer Science, vol 6844. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22300-6_34
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DOI: https://doi.org/10.1007/978-3-642-22300-6_34
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