On Polynomial Kernels for Structural Parameterizations of Odd Cycle Transversal

  • Bart M. P. Jansen
  • Stefan Kratsch
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7112)


The Odd Cycle Transversal problem (OCT) asks whether a given graph can be made bipartite (i.e., 2-colorable) by deleting at most ℓ vertices. We study structural parameterizations of OCT with respect to their polynomial kernelizability, i.e., whether instances can be efficiently reduced to a size polynomial in the chosen parameter. It is a major open problem in parameterized complexity whether Odd Cycle Transversal admits a polynomial kernel when parameterized by ℓ.

On the positive side, we show a polynomial kernel for OCT when parameterized by the vertex deletion distance to the class of bipartite graphs of treewidth at most w (for any constant w); this generalizes the parameter feedback vertex set number (i.e., the distance to a forest).

Complementing this, we exclude polynomial kernels for OCT parameterized by the distance to outerplanar graphs, conditioned on the assumption that NP \(\nsubseteq\) coNP/poly. Thus the bipartiteness requirement for the treewidth w graphs is necessary. Further lower bounds are given for parameterization by distance from cluster and co-cluster graphs respectively, as well as for Weighted OCT parameterized by the vertex cover number (i.e., the distance from an independent set).


Bipartite Graph Vertex Cover Polynomial Kernel Label Graph Outerplanar Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Agarwal, A., Charikar, M., Makarychev, K., Makarychev, Y.: O(sqrt(log n)) approximation algorithms for min uncut, min 2cnf deletion, and directed cut problems. In: Proc. 37th STOC, pp. 573–581 (2005)Google Scholar
  2. 2.
    Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci. 75(8), 423–434 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bodlaender, H.L., Fomin, F.V., Lokshtanov, D., Penninkx, E., Saurabh, S., Thilikos, D.M.: (Meta) Kernelization. In: Proc. 50th FOCS, pp. 629–638 (2009)Google Scholar
  4. 4.
    Bodlaender, H.L., Jansen, B.M.P., Kratsch, S.: Cross-composition: A new technique for kernelization lower bounds. In: Proc. 28th STACS, pp. 165–176 (2011)Google Scholar
  5. 5.
    Bodlaender, H.L., Jansen, B.M.P., Kratsch, S.: Preprocessing for Treewidth: A Combinatorial Analysis through Kernelization. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011. LNCS, vol. 6755, pp. 437–448. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  6. 6.
    Bousquet, N., Daligault, J., Thomassé, S.: Multicut is FPT. In: Proc. 43rd STOC, pp. 459–468 (2011)Google Scholar
  7. 7.
    Chen, J., Liu, Y., Lu, S., O’Sullivan, B., Razgon, I.: A fixed-parameter algorithm for the directed feedback vertex set problem. J. ACM 55(5) (2008)Google Scholar
  8. 8.
    Courcelle, B., Makowsky, J.A., Rotics, U.: Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst. 33(2), 125–150 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Downey, R., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science. Springer, New York (1999)CrossRefzbMATHGoogle Scholar
  10. 10.
    Fellows, M.R., Lokshtanov, D., Misra, N., Mnich, M., Rosamond, F.A., Saurabh, S.: The complexity ecology of parameters: An illustration using bounded max leaf number. Theory Comput. Syst. 45(4), 822–848 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fiorini, S., Hardy, N., Reed, B.A., Vetta, A.: Planar graph bipartization in linear time. Discrete Applied Mathematics 156(7), 1175–1180 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fomin, F.V., Lokshtanov, D., Misra, N., Philip, G., Saurabh, S.: Hitting forbidden minors: Approximation and kernelization. In: Proc. 28th STACS, pp. 189–200 (2011)Google Scholar
  13. 13.
    Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct PCPs for NP. J. Comput. Syst. Sci. 77(1), 91–106 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Guo, J., Gramm, J., Hüffner, F., Niedermeier, R., Wernicke, S.: Compression-based fixed-parameter algorithms for feedback vertex set and edge bipartization. J. Comput. Syst. Sci. 72(8), 1386–1396 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. SIGACT News 38(1), 31–45 (2007)CrossRefGoogle Scholar
  16. 16.
    Hüffner, F.: Algorithm engineering for optimal graph bipartization. J. Graph Algorithms Appl. 13(2), 77–98 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Jansen, B.M.P., Bodlaender, H.L.: Vertex cover kernelization revisited: Upper and lower bounds for a refined parameter. In: Proc. 28th STACS, pp. 177–188 (2011)Google Scholar
  18. 18.
    Jansen, B.M.P., Kratsch, S.: On polynomial kernels for structural parameterizations of odd cycle transversal. CoRR, abs/1107.3658 (2011)Google Scholar
  19. 19.
    Kawarabayashi, K., Reed, B.A.: An (almost) linear time algorithm for odd cyles transversal. In: Proc. 21st SODA, pp. 365–378 (2010)Google Scholar
  20. 20.
    Khot, S.: On the power of unique 2-prover 1-round games. In: Proc. 34th STOC, pp. 767–775 (2002)Google Scholar
  21. 21.
    Kratsch, S., Wahlström, M.: Compression via matroids: A randomized polynomial kernel for odd cycle transversal. CoRR, abs/1107.3068 (2011)Google Scholar
  22. 22.
    Marx, D.: Parameterized graph separation problems. Theoretical Computer Science 351(3), 394–406 (2006); Parameterized and Exact ComputationMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Marx, D., O’Sullivan, B., Razgon, I.: Treewidth reduction for constrained separation and bipartization problems. In: Proc. 27th STACS, pp. 561–572 (2010)Google Scholar
  24. 24.
    Marx, D., Razgon, I.: Fixed-parameter tractability of multicut parameterized by the size of the cutset. In: Proc. 43rd STOC, pp. 469–478 (2011)Google Scholar
  25. 25.
    Raman, V., Saurabh, S., Sikdar, S.: Improved Exact Exponential Algorithms for Vertex Bipartization and other Problems. In: Coppo, M., Lodi, E., Pinna, G.M. (eds.) ICTCS 2005. LNCS, vol. 3701, pp. 375–389. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  26. 26.
    Reed, B.A., Smith, K., Vetta, A.: Finding odd cycle transversals. Oper. Res. Lett. 32(4), 299–301 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Rizzi, R., Bafna, V., Istrail, S., Lancia, G.: Practical Algorithms and Fixed-Parameter Tractability for the Single Individual SNP Haplotyping Problem. In: Guigó, R., Gusfield, D. (eds.) WABI 2002. LNCS, vol. 2452, pp. 29–43. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  28. 28.
    Uhlmann, J., Weller, M.: Two-Layer Planarization Parameterized by Feedback Edge Set. In: Kratochvíl, J., Li, A., Fiala, J., Kolman, P. (eds.) TAMC 2010. LNCS, vol. 6108, pp. 431–442. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  29. 29.
    Wernicke, S.: On the algorithmic tractability of single nucleotide polymorphism (SNP) analysis and related problems. Master’s thesis, Wilhelm-Schickard-Institut für Informatik, Universität Tübingen (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Bart M. P. Jansen
    • 1
  • Stefan Kratsch
    • 1
  1. 1.Utrecht UniversityThe Netherlands

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