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The Undirected Feedback Vertex Set Problem Has a Poly(k) Kernel

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Parameterized and Exact Computation (IWPEC 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4169))

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Abstract

Resolving a noted open problem, we show that the Undirected Feedback Vertex Set problem, parameterized by the size of the solution set of vertices, is in the parameterized complexity class Poly(k), that is, polynomial-time pre-processing is sufficient to reduce an initial problem instance (G,k) to a decision-equivalent simplified instance (G′,k′) where k′ ≤k, and the number of vertices of G′ is bounded by a polynomial function of k. Our main result shows an O(k 11) kernelization bound.

This research has been supported in part by the U.S. National Science Foundation under grant CCR–0075792, by the U.S. Office of Naval Research under grant N00014–01–1–0608, by the U.S. Department of Energy under contract DE–AC05–00OR22725, and by the Australian Research Council under the auspices of the Australian Centre for Bioinformatics, through Federation Fellowship support of the first author, and through Discovery Project support of the second and third authors.

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References

  1. 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: Arge, L., Italiano, G., Sedgewick, R. (eds.) Proceedings of the 6th Workshop on Algorithm Engineering and Experiments (ALENEX) Proc. Applied Mathematics 115, New Orleans, January 2004. ACM/SIAM (2004)

    Google Scholar 

  2. Alber, J., Fellows, M., Niedermeier, R.: Polynomial time data reduction for dominating set. Journal of the ACM 51, 363–384 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bafna, V., Berman, P., Fujito, T.: A 2-approximation algorithm for the undirected feedback vertex set problem. SIAM Journal on Discrete Mathematics 12, 289–297 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  4. Becker, A., Bar-Yehuda, R., Geiger, D.: Random algorithms for the loop cutset problem. Journal of Artificial Intelligence Research 12, 219–234 (2000)

    MATH  MathSciNet  Google Scholar 

  5. Bodlaender, H., Koster, A., van den Eijkhof, F.: Preprocessing rules for triangulation of probabilistic networks. Computational Intelligence 21, 286–305 (2005)

    Article  MathSciNet  Google Scholar 

  6. Bodlaender, H.: On disjoint cycles. International Journal of Foundations of Computer Science 5, 59–68 (1994)

    Article  MATH  Google Scholar 

  7. Chor, B., Fellows, M., Juedes, D.W.: Linear kernels in linear time, or how to save k colors in O(n 2) steps. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 257–269. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  8. Downey, R., Fellows, M.: Fixed-parameter tractability and completeness. Congressus Numerantium 87, 161–187 (1992)

    MathSciNet  Google Scholar 

  9. Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)

    Google Scholar 

  10. Dehne, F., Fellows, M.R., Langston, M.A., Rosamond, F.A., Stevens, K.: An O *(2O(k)) FPT algorithm for the undirected feedback vertex set problem. In: Wang, L. (ed.) COCOON 2005. LNCS, vol. 3595, pp. 859–869. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  11. Dehne, F., Fellows, M., Rosamond, F.A., Shaw, P.: Greedy localization, iterative compression and modeled crown reductions: new FPT techniques, an improved algorithm for set splitting and a novel 2k kernelization for vertex cover. In: Downey, R.G., Fellows, M.R., Dehne, F. (eds.) IWPEC 2004. LNCS, vol. 3162, pp. 271–280. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  12. Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)

    Google Scholar 

  13. Flum, J., Grohe, M., Weyer, M.: Bounded fixed-parameter tractability and log2 n nondeterministic bits. In: Díaz, J., et al. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 555–567. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  14. Festa, P., Pardalos, P.M., Resende, M.G.C.: Feedback set problems. In: Du, D.Z., Pardalos, P.M. (eds.) Handbook of Combinatorial Optimization, vol. A, pp. 209–258. Kluwer, Dordrecht (1999)

    Google Scholar 

  15. Guo, J., Gramm, J., Hüffner, F., Niedermeier, R., Wernicke, S.: Improved fixed-parameter algorithms for two feedback set problems. In: Dehne, F., López-Ortiz, A., Sack, J.-R. (eds.) WADS 2005. LNCS, vol. 3608, pp. 158–169. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  16. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, New York (1979)

    MATH  Google Scholar 

  17. Guo, J.: Algorithm design techniques for parameterized problems. Ph.D. Thesis, Friedrich-Schiller-Universität, Jena (2006)

    Google Scholar 

  18. 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. 235–247. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  19. Lokshtanov, D., Sloper, C.: Fixed-parameter set-splitting, linear kernel and improved running time. In: Algorithms and Complexity in Durham 2005: Proceedings of the First ACiD Workshop. Texts in Algorithmics, vol. 4, pp. 105–113. King’s College Press (2005)

    Google Scholar 

  20. Niedermeier, R.: Invitation to fixed-parameter algorithms, Habilitationschrift, University of Tubingen (2002) (Electronic file available from R. Niedermeier)

    Google Scholar 

  21. Niedermeier, R.: Invitation to Fixed Parameter Algorithms. Oxford University Press, Oxford (2006)

    Book  MATH  Google Scholar 

  22. Nemhauser, G.L., Trotter, L.E.: Vertex packings: structural properties and algorithms. Mathematical Programming 8, 232–248 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  23. Prieto-Rodriguez, E.: Systematic kernelization in FPT algorithm design. Ph.D. Thesis, School of EE&CS, University of Newcastle, Australia (2005)

    Google Scholar 

  24. 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)

    Chapter  Google Scholar 

  25. Raman, V., Saurabh, S., Subramanian, C.R.: Faster algorithms for feedback vertex set. In: Proceedings of the 2nd Brazilian Symposium on Graphs, Algorithms and Combinatorics, GRACO 2005, Angra dos Reis (Rio de Janeiro), Brazil, April 27-29, 2005. Electronic Notes in Discrete Mathematics. Elsevier, Amsterdam (2005)

    Google Scholar 

  26. Sloper, C.: Techniques in parameterized algorithm design. Ph.D. Thesis, Department of Informatics, University of Bergen, Norway (2005)

    Google Scholar 

  27. Weihe, K.: Covering trains by stations, or the power of data reduction. In: Proc. ALEX 1998, pp. 1–8 (1998)

    Google Scholar 

  28. Weihe, K.: On the Differences Between ‘Practical’ and ‘Applied’ (invited paper). In: Näher, S., Wagner, D. (eds.) WAE 2000. LNCS, vol. 1982, pp. 1–10. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  29. Weyer, M.: Bounded fixed-parameter tractability: the case of 2poly(k). In: Downey, R.G., Fellows, M.R., Dehne, F. (eds.) IWPEC 2004. LNCS, vol. 3162, pp. 49–60. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Queensland, Brisbane, QLD, 4072

    Kevin Burrage & Shev Mac

  2. Griffith University, Brisbane, QLD, 4111, Australia

    Vladimir Estivill-Castro

  3. School of EE & CS, University of Newcastle, Callaghan, NSW, 2308, Australia

    Michael Fellows

  4. Department of Computer Science, University of Tennessee, Knoxville, TN, 37996-3450, USA

    Michael Langston

  5. Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN, 37831-6164, USA

    Michael Langston

  6. The Retreat for the Arts and Sciences, Newcastle, Australia

    Frances Rosamond

Authors
  1. Kevin Burrage
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  2. Vladimir Estivill-Castro
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  3. Michael Fellows
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  4. Michael Langston
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  5. Shev Mac
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  6. Frances Rosamond
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Editor information

Editors and Affiliations

  1. Department of Information and Computing Sciences, Utrecht University, The Netherlands

    Hans L. Bodlaender

  2. University of Tennessee, 37996-3450, Knoxville, Tennessee, U.S.A.

    Michael A. Langston

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Burrage, K., Estivill-Castro, V., Fellows, M., Langston, M., Mac, S., Rosamond, F. (2006). The Undirected Feedback Vertex Set Problem Has a Poly(k) Kernel. In: Bodlaender, H.L., Langston, M.A. (eds) Parameterized and Exact Computation. IWPEC 2006. Lecture Notes in Computer Science, vol 4169. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11847250_18

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  • DOI: https://doi.org/10.1007/11847250_18

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-39098-5

  • Online ISBN: 978-3-540-39101-2

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