# A Hierarchy of Polynomial Kernels

## 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.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.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.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.Cygan, M., et al.: Parameterized Algorithms. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-319-21275-3CrossRefGoogle Scholar
- 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.Drucker, A.: New limits to classical and quantum instance compression. SIAM J. Comput.
**44**(5), 1443–1479 (2015)MathSciNetCrossRefGoogle Scholar - 7.Flum, J., Grohe, M.: Parameterized Complexity Theory. TTCSAES. Springer, Heidelberg (2006). https://doi.org/10.1007/3-540-29953-XCrossRefzbMATHGoogle Scholar
- 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.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.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.Ko, K.I.: On self-reducibility and weak P-selectivity. J. Comput. Syst. Sci.
**26**(2), 209–221 (1983)MathSciNetCrossRefGoogle Scholar - 12.Kratsch, S.: Recent developments in kernelization: a survey. Bull. EATCS
**2**(113), 57–97 (2014)Google Scholar - 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.Soare, R.I.: Turing Computability. Springer, Heidlberg (2016). https://doi.org/10.1007/978-3-642-31933-4CrossRefzbMATHGoogle Scholar
- 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.Trakhtenbrot, B.A.: On autoreducibility. Doklady Akademii Nauk SSSR
**192**(6), 1224–1227 (1970)MathSciNetGoogle Scholar