Advertisement

A Hierarchy of Polynomial Kernels

  • Jouke Witteveen
  • Ralph Bottesch
  • Leen Torenvliet
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11376)

Abstract

In parameterized algorithmics the process of kernelization is defined as a polynomial time algorithm that transforms the instance of a given problem to an equivalent instance of a size that is limited by a function of the parameter. As, afterwards, this smaller instance can then be solved to find an answer to the original question, kernelization is often presented as a form of preprocessing. A natural generalization of kernelization is the process that allows for a number of smaller instances to be produced to provide an answer to the original problem, possibly also using negation. This generalization is called Turing kernelization. Immediately, questions of equivalence occur or, when is one form possible and not the other. These have been long standing open problems in parameterized complexity. In the present paper, we answer many of these. In particular we show that Turing kernelizations differ not only from regular kernelization, but also from intermediate forms as truth-table kernelizations. We achieve absolute results by diagonalizations and also results on natural problems depending on widely accepted complexity theoretic assumptions. In particular, we improve on known lower bounds for the kernel size of compositional problems using these assumptions.

Keywords

Kernelization Parameterized complexity Turing reductions Truth-table reductions Kernel lower bounds 

References

  1. 1.
    Balcázar, J.L., Díaz, J., Gabarró, J.: Structural Complexity I. Springer, Heidelberg (1995).  https://doi.org/10.1007/978-3-642-79235-9CrossRefzbMATHGoogle 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)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bodlaender, H.L., Jansen, B.M., Kratsch, S.: Kernelization lower bounds by cross-composition. SIAM J. Discret. Math. 28(1), 277–305 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cygan, M., et al.: Parameterized Algorithms. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-319-21275-3CrossRefGoogle Scholar
  5. 5.
    Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-1-4471-5559-1CrossRefzbMATHGoogle Scholar
  6. 6.
    Drucker, A.: New limits to classical and quantum instance compression. SIAM J. Comput. 44(5), 1443–1479 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. TTCSAES. Springer, Heidelberg (2006).  https://doi.org/10.1007/3-540-29953-XCrossRefzbMATHGoogle Scholar
  8. 8.
    Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct PCPs for NP. J. Comput. Syst. Sci. 77(1), 91–106 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Jansen, B.M.: Turing kernelization for finding long paths and cycles in restricted graph classes. J. Comput. Syst. Sci. 85, 18–37 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Jansen, B.M., Pilipczuk, M., Wrochna, M.: Turing kernelization for finding long paths in graphs excluding a topological minor. In: 12th International Symposium on Parameterized and Exact Computation (IPEC 2017), vol. 89, pp. 23:1–23:13. Schloss Dagstuhl-Leibniz Zentrum fuer Informatik (2018)Google Scholar
  11. 11.
    Ko, K.I.: On self-reducibility and weak P-selectivity. J. Comput. Syst. Sci. 26(2), 209–221 (1983)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kratsch, S.: Recent developments in kernelization: a survey. Bull. EATCS 2(113), 57–97 (2014)Google Scholar
  13. 13.
    Ladner, R.E., Lynch, N.A., Selman, A.L.: A comparison of polynomial time reducibilities. Theor. Comput. Sci. 1(2), 103–123 (1975)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Soare, R.I.: Turing Computability. Springer, Heidlberg (2016).  https://doi.org/10.1007/978-3-642-31933-4CrossRefzbMATHGoogle Scholar
  15. 15.
    Thomassé, S., Trotignon, N., Vušković, K.: A polynomial Turing-kernel for weighted independent set in bull-free graphs. Algorithmica 77(3), 619–641 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Trakhtenbrot, B.A.: On autoreducibility. Doklady Akademii Nauk SSSR 192(6), 1224–1227 (1970)MathSciNetGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute for Logic, Language, and ComputationUniversiteit van AmsterdamAmsterdamThe Netherlands
  2. 2.Department of Computer ScienceUniversität InnsbruckInnsbruckAustria

Personalised recommendations