Skip to main content
SpringerLink
Log in
Book cover

International Symposium on Parameterized and Exact Computation

IPEC 2011: Parameterized and Exact Computation pp 132–144Cite as

On Polynomial Kernels for Structural Parameterizations of Odd Cycle Transversal

  • Conference paper

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7112))

Abstract

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).

This work was supported by the Netherlands Organization for Scientific Research (NWO), project “KERNELS: Combinatorial Analysis of Data Reduction”.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   54.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. 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)

    Chapter  Google Scholar 

  6. Bousquet, N., Daligault, J., Thomassé, S.: Multicut is FPT. In: Proc. 43rd STOC, pp. 459–468 (2011)

    Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Downey, R., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science. Springer, New York (1999)

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fiorini, S., Hardy, N., Reed, B.A., Vetta, A.: Planar graph bipartization in linear time. Discrete Applied Mathematics 156(7), 1175–1180 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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. Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct PCPs for NP. J. Comput. Syst. Sci. 77(1), 91–106 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  15. Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. SIGACT News 38(1), 31–45 (2007)

    Article  Google Scholar 

  16. Hüffner, F.: Algorithm engineering for optimal graph bipartization. J. Graph Algorithms Appl. 13(2), 77–98 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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. Jansen, B.M.P., Kratsch, S.: On polynomial kernels for structural parameterizations of odd cycle transversal. CoRR, abs/1107.3658 (2011)

    Google Scholar 

  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. Khot, S.: On the power of unique 2-prover 1-round games. In: Proc. 34th STOC, pp. 767–775 (2002)

    Google Scholar 

  21. Kratsch, S., Wahlström, M.: Compression via matroids: A randomized polynomial kernel for odd cycle transversal. CoRR, abs/1107.3068 (2011)

    Google Scholar 

  22. Marx, D.: Parameterized graph separation problems. Theoretical Computer Science 351(3), 394–406 (2006); Parameterized and Exact Computation

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. 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)

    Chapter  Google Scholar 

  26. Reed, B.A., Smith, K., Vetta, A.: Finding odd cycle transversals. Oper. Res. Lett. 32(4), 299–301 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Chapter  Google Scholar 

  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 

Download references

Author information

Authors and Affiliations

  1. Utrecht University, The Netherlands

    Bart M. P. Jansen & Stefan Kratsch

Authors
  1. Bart M. P. Jansen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stefan Kratsch
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. MTA SZTAKI, Lágymányosi u. 11, 1111, Budapest, Hungary

    Dániel Marx

  2. Lehrgebiet Theoretische Informatik, RWTH Aachen, Ahornstraße 55, 52056, Aachen, Germany

    Peter Rossmanith

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Jansen, B.M.P., Kratsch, S. (2012). On Polynomial Kernels for Structural Parameterizations of Odd Cycle Transversal. In: Marx, D., Rossmanith, P. (eds) Parameterized and Exact Computation. IPEC 2011. Lecture Notes in Computer Science, vol 7112. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28050-4_11

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-28050-4_11

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-28049-8

  • Online ISBN: 978-3-642-28050-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation