## Abstract

The framework of Bodlaender et al. (J Comput Sys Sci 75(8):423–434, 2009) and Fortnow and Santhanam (J Comput Sys Sci 77(1):91–106, 2011) allows us to exclude the existence of polynomial kernels for a range of problems under reasonable complexity-theoretical assumptions. However, some issues are not addressed by this framework, including the existence of Turing kernels such as the “kernelization” of leafout-branching(k) that outputs n instances each of size poly(k). Observing that Turing kernels are preserved by polynomial parametric transformations (PPTs), we define two kernelization hardness hierarchies by the PPT-closure of problems that seem fundamentally unlikely to admit efficient Turing kernelizations. This gives rise to the MK- and WK-hierarchies which are akin to the M- and W-hierarchies of parameterized complexity. We find that several previously considered problems are complete for the fundamental hardness class WK[1], including Min Ones d-Sat(k), Binary NDTM Halting(k), Connected Vertex Cover(k), and Clique parameterized by log n. We conjecture that no WK[1]-hard problem admits a polynomial Turing kernel. Our hierarchy subsumes an earlier hierarchy of Harnik and Naor that, from a parameterized perspective, is restricted to classical problems parameterized by witness size. Our results provide the first natural complete problems for, e.g., their class VC_{1}; this had been left open.

Original language | English |
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Pages (from-to) | 702-730 |

Number of pages | 29 |

Journal | Algorithmica |

Volume | 71 |

Issue number | 3 |

DOIs | |

State | Published - 1 Mar 2015 |

## Keywords

- Complexity hierarchies
- Kernelization
- Parameterized complexity
- Turing kernelization

## ASJC Scopus subject areas

- General Computer Science
- Computer Science Applications
- Applied Mathematics