Skip to main content
SpringerLink
Log in
Book cover

Workshop on Algorithms and Data Structures

WADS 2019: Algorithms and Data Structures pp 523–537Cite as

Wannabe Bounded Treewidth Graphs Admit a Polynomial Kernel for DFVS

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11646))

Abstract

In the Directed Feedback Vertex Set (DFVS) problem, given a digraph D and \(k\in \mathbb {N}\), the goal is to check if there exists a set of at most k vertices whose deletion from D leaves a directed acyclic graph. Resolving the existence of a polynomial kernel for DFVS parameterized by the solution size k is a central open problem in Kernelization. In this paper, we give a polynomial kernel for DFVS parameterized by k plus the size of a treewidth-\(\eta \) modulator. Our choice of parameter strictly encompasses previous positive kernelization results on DFVS. Our main result is based on a novel application of the tool of important separators embedded in state-of-the-art machinery such as protrusion decompositions.

Appeared as a Brief Announcement at ICALP 2018.

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   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
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

Notes

  1. 1.

    Throughout the paper, we do not hide constants that depend on \(\eta \) in the \(\mathcal {O}\) notation.

References

  1. Abu-Khzam, F.: A kernelization algorithm for \(d\)-HS. JCSS 76(7), 524–531 (2010)

    MATH  Google Scholar 

  2. Agrawal, A., Saurabh, S., Sharma, R., Zehavi, M.: Kernels for deletion to classes of acyclic digraphs. JCSS 92, 9–21 (2018)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Bang-Jensen, J., Maddaloni, A., Saurabh, S.: Algorithms and kernels for feedback set problems in generalizations of tournaments. Algorithmica 76, 1–24 (2015)

    MathSciNet  MATH  Google Scholar 

  5. Bar-Yehuda, R., Geiger, D., Naor, J., Roth, R.M.: Approximation algorithms for the feedback vertex set problem with applications to constraint satisfaction and bayesian inference. SIAM J. Comput. 27(4), 942–959 (1998)

    Article  MathSciNet  Google Scholar 

  6. Becker, A., Geiger, D.: Optimization of pearl’s method of conditioning and greedy-like approximation algorithms for the vertex feedback set problem. AI 83, 167–188 (1996)

    MathSciNet  Google Scholar 

  7. Bergougnoux, B., Eiben, E., Ganian, R., Ordyniak, S., Ramanujan, M.: Towards a polynomial kernel for directed feedback vertex set. In: MFCS, vol. 83 (2017)

    Google Scholar 

  8. Bonamy, M., Kowalik, Ł., Nederlof, J., Pilipczuk, M., Socała, A., Wrochna, M.: On directed feedback vertex set parameterized by treewidth. In: WG, pp. 65–78 (2018)

    Google Scholar 

  9. Cao, Y., Chen, J., Liu, Y.: On feedback vertex set: new measure and new structures. Algorithmica 73(1), 63–86 (2015)

    Article  MathSciNet  Google Scholar 

  10. Chekuri, C., Madan, V.: Constant factor approximation for subset feedback set problems via a new LP relaxation. In: SODA, pp. 808–820 (2016)

    Google Scholar 

  11. Chen, J., Fomin, F.V., Liu, Y., Lu, S., Villanger, Y.: Improved algorithms for feedback vertex set problems. JCSS 74(7), 1188–1198 (2008)

    MathSciNet  MATH  Google Scholar 

  12. Chen, J., Liu, Y., Lu, S.: An improved parameterized algorithm for the minimum node multiway cut problem. Algorithmica 55(1), 1–13 (2009)

    Article  MathSciNet  Google Scholar 

  13. 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), 21 (2008)

    Article  MathSciNet  Google Scholar 

  14. Chitnis, R., Hajiaghayi, M., Marx, D.: Fixed-parameter tractability of directed multiway cut parameterized by the size of the cutset. SIAM J. Comput. 42(4), 1674–1696 (2013)

    Article  MathSciNet  Google Scholar 

  15. Chitnis, R.H., Cygan, M., Hajiaghayi, M.T., Marx, D.: Directed subset feedback vertex set is fixed-parameter tractable. ACM TALG 11(4), 28 (2015)

    MathSciNet  MATH  Google Scholar 

  16. Cygan, M., et al.: Parameterized Algorithms. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-21275-3

    Book  MATH  Google Scholar 

  17. Cygan, M., Lokshtanov, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: On the hardness of losing width. TOCS 54(1), 73–82 (2014)

    MathSciNet  MATH  Google Scholar 

  18. Cygan, M., Nederlof, J., Pilipczuk, M., Pilipczuk, M., van Rooij, J.M.M., Wojtaszczyk, J.O.: Solving connectivity problems parameterized by treewidth in single exponential time. In: FOCS, pp. 150–159 (2011)

    Google Scholar 

  19. Cygan, M., Pilipczuk, M., Pilipczuk, M., Wojtaszczyk, J.O.: Subset feedback vertex set is fixed-parameter tractable. SIDMA 27(1), 290–309 (2013)

    Article  MathSciNet  Google Scholar 

  20. Dom, M., Guo, J., Hüffner, F., Niedermeier, R., Truß, A.: Fixed-parameter tractability results for feedback set problems in tournaments. JDA 8(1), 76–86 (2010)

    MathSciNet  MATH  Google Scholar 

  21. Downey, R.G., Fellows, M.R.: Fixed-parameter intractability. In: CCC, pp. 36–49 (1992)

    Google Scholar 

  22. Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness I: basic results. SIAM J. Comput. 24(4), 873–921 (1995)

    Article  MathSciNet  Google Scholar 

  23. Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. TCS. Springer, London (2013). https://doi.org/10.1007/978-1-4471-5559-1

    Book  MATH  Google Scholar 

  24. Erdős, P., Pósa, L.: On independent circuits contained in a graph. Can. J. Math. 17, 347–352 (1965)

    Article  MathSciNet  Google Scholar 

  25. Even, G., Naor, J., Schieber, B., Sudan, M.: Approximating minimum feedback sets and multicuts in directed graphs. Algorithmica 20(2), 151–174 (1998)

    Article  MathSciNet  Google Scholar 

  26. Flum, J., Grohe, M.: Parameterized Complexity Theory. TTCSAES. Springer, Heidelberg (2006). https://doi.org/10.1007/3-540-29953-X

    Book  MATH  Google Scholar 

  27. Fomin, F.V., Lokshtanov, D., Misra, N., Saurabh, S.: Planar F-deletion: approximation, kernelization and optimal FPT algorithms. In: FOCS, pp. 470–479 (2012). http://www.ii.uib.no/~daniello/papers/PFDFullV1.pdf

  28. Fomin, F.V., Lokshtanov, D., Saurabh, S., Zehavi, M.: Kernelization: Theory of Parameterized Preprocessing. Cambridge University Press, Cambridge (2018)

    Book  Google Scholar 

  29. Fomin, F.V., Saurabh, S.: Kernelization methods for fixed-parameter tractability. In: Bordeaux, L., Hamadi, Y., Kohli, P. (eds.) Tractability, pp. 260–282. Cambridge University Press, Cambridge (2014)

    Google Scholar 

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

    Article  Google Scholar 

  31. Guruswami, V., Lee, E.: Inapproximability of H-transversal/packing. In: APPROX/RANDOM. LIPIcs, vol. 40, pp. 284–304 (2015)

    Google Scholar 

  32. Kakimura, N., Kawarabayashi, K., Kobayashi, Y.: Erdös-Pósa property and its algorithmic applications: parity constraints, subset feedback set, and subset packing. In: SODA, pp. 1726–1736 (2012)

    Google Scholar 

  33. Kakimura, N., ichi Kawarabayashi, K., Marx, D.: Packing cycles through prescribed vertices. JCTB 101(5), 378–381 (2011)

    Article  MathSciNet  Google Scholar 

  34. Kawarabayashi, K., Kobayashi, Y.: Fixed-parameter tractability for the subset feedback fet problem and the S-cycle packing problem. JCTB 102, 1020–1034 (2012)

    Article  Google Scholar 

  35. Kawarabayashi, K., Král, D., Krcál, M., Kreutzer, S.: Packing directed cycles through a specified vertex set. In: SODA, pp. 365–377 (2013)

    Google Scholar 

  36. Kociumaka, T., Pilipczuk, M.: Faster deterministic feedback vertex set. IPL 114(10), 556–560 (2014)

    Article  MathSciNet  Google Scholar 

  37. Kratsch, S.: Recent developments in kernelization. Bull. EATCS 113, 58–97 (2014)

    MathSciNet  MATH  Google Scholar 

  38. Le, T., Lokshtanov, D., Saurabh, S., Thomassé, S., Zehavi, M.: Subquadratic kernels for implicit 3-hitting set and 3-set packing problems. In: SODA, pp. 331–342 (2018)

    Google Scholar 

  39. Lokshtanov, D., Misra, N., Saurabh, S.: Kernelization – preprocessing with a guarantee. In: Bodlaender, H.L., Downey, R., Fomin, F.V., Marx, D. (eds.) The Multivariate Algorithmic Revolution and Beyond. LNCS, vol. 7370, pp. 129–161. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-30891-8_10

    Chapter  Google Scholar 

  40. Lokshtanov, D., Ramanujan, M.S., Saurabh, S.: When recursion is better than iteration. In: SODA, pp. 1916–1933 (2018)

    Google Scholar 

  41. Mnich, M., van Leeuwen, E.J.: Polynomial kernels for deletion to classes of acyclic digraphs. Discrete Optim. 25, 48–76 (2017)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  43. Pontecorvi, M., Wollan, P.: Disjoint cycles intersecting a set of vertices. JCTB 102(5), 1134–1141 (2012)

    Article  MathSciNet  Google Scholar 

  44. Raman, V., Saurabh, S., Subramanian, C.R.: Faster fixed parameter tractable algorithms for finding feedback vertex sets. ACM TALG 2(3), 403–415 (2006)

    Article  MathSciNet  Google Scholar 

  45. Reed, B.A., Robertson, N., Seymour, P., Thomas, R.: Packing directed circuits. Combinatorica 16(4), 535–554 (1996)

    Article  MathSciNet  Google Scholar 

  46. Seymour, P.: Packing directed circuits fractionally. Combinatorica 15(2), 281–288 (1995)

    Article  MathSciNet  Google Scholar 

  47. Seymour, P.: Packing circuits in eulerian digraphs. Combinatorica 16(2), 223–231 (1996)

    Article  MathSciNet  Google Scholar 

  48. Wahlström, M.: Half-integrality, LP-branching and FPT algorithms. In: SODA, pp. 1762–1781 (2014)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of California, Santa Barbara, USA

    Daniel Lokshtanov

  2. University of Warwick, Coventry, UK

    M. S. Ramanujan

  3. Institute of Mathematical Sciences, HBNI, Chennai, India

    Saket Saurabh & Roohani Sharma

  4. Ben-Gurion University of the Negev, Beersheba, Israel

    Meirav Zehavi

Authors
  1. Daniel Lokshtanov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. S. Ramanujan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Saket Saurabh
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Roohani Sharma
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Meirav Zehavi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roohani Sharma .

Editor information

Editors and Affiliations

  1. Department of Computing Science, University of Alberta, Edmonton, AB, Canada

    Zachary Friggstad

  2. School of Computer Science, Carleton University, Manotick, ON, Canada

    Jörg-Rüdiger Sack

  3. Department of Computing Science, University of Alberta, Edmonton, AB, Canada

    Mohammad R Salavatipour

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Lokshtanov, D., Ramanujan, M.S., Saurabh, S., Sharma, R., Zehavi, M. (2019). Wannabe Bounded Treewidth Graphs Admit a Polynomial Kernel for DFVS. In: Friggstad, Z., Sack, JR., Salavatipour, M. (eds) Algorithms and Data Structures. WADS 2019. Lecture Notes in Computer Science(), vol 11646. Springer, Cham. https://doi.org/10.1007/978-3-030-24766-9_38

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-24766-9_38

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-24765-2

  • Online ISBN: 978-3-030-24766-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation