A Shortcut to (Sun)Flowers: Kernels in Logarithmic Space or Linear Time

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9235)


We investigate whether kernelization results can be obtained if we restrict kernelization algorithms to run in logarithmic space. This restriction for kernelization is motivated by the question of what results are attainable for preprocessing via simple and/or local reduction rules. We find kernelizations for \(d\)-hitting set( k ), \(d\)-set packing( k ), edge dominating set( k ), and a number of hitting and packing problems in graphs, each running in logspace. Additionally, we return to the question of linear-time kernelization. For \(d\)-hitting set( k ) a linear-time kernel was given by van Bevern [Algorithmica (2014)]. We give a simpler procedure and save a large constant factor in the size bound. Furthermore, we show that we can obtain a linear-time kernel for \(d\)-set packing( k ).


  1. 1.
    Cai, L., Chen, J., Downey, R.G., Fellows, M.R.: Advice classes of parameterized tractability. Ann. Pure Appl. Logic 84(1), 119–138 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Dell, H., Marx, D.: Kernelization of packing problems. In: Proceedings of the Twenty-third Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 68–81. SIAM (2012)Google Scholar
  3. 3.
    Erdős, P., Rado, R.: Intersection theorems for systems of sets. J. Lond. Math. Soc. 1(1), 85–90 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Fafianie, S., Kratsch, S.: A shortcut to (sun) flowers: Kernels in logarithmic space or linear time. arXiv report 1504.08235 (2015)Google Scholar
  5. 5.
    Flum, J., Grohe, M.: Parameterized complexity theory, volume XIV of texts in theoretical computer science. In: An EATCS Series (2006)Google Scholar
  6. 6.
    Fomin, F.V., Lokshtanov, D., Saurabh, S.: Efficient computation of representative sets with applications in parameterized and exact algorithms. In: Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 142–151. SIAM (2014)Google Scholar
  7. 7.
    Håstad, J., Jukna, S., Pudlák, P.: Top-down lower bounds for depth-three circuits. Comput. Complex. 5(2), 99–112 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Jukna, S.: Extremal combinatorics: with applications in computer science. Springer Science & Business Media, Heidelberg (2011)CrossRefzbMATHGoogle Scholar
  9. 9.
    Kratsch, S., et al.: Recent developments in kernelization: a survey. Bull. EATCS 2(113), 58–97 (2014)MathSciNetGoogle Scholar
  10. 10.
    Lokshtanov, D.: New methods in parameterized algorithms and complexity. Ph.D thesis, Citeseer (2009)Google Scholar
  11. 11.
    Niedermeier, R., Rossmanith, P.: An efficient fixed-parameter algorithm for 3-hitting set. J. Discrete Algorithms 1(1), 89–102 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    van Bevern, R.: Towards optimal and expressive kernelization for d-hitting set. Algorithmica 70(1), 129–147 (2014)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.University of BonnBonnGermany

Personalised recommendations