## Abstract

The framework of Bodlaender et al. (ICALP 2008, JCSS 2009) and Fortnow and Santhanam (STOC 2008, JCSS 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 Leaf Out Branching(k) into a disjunction over 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 ordinary 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 k logn. We conjecture that no WK[1]-hard problem admits a polynomial Turing kernel. Our hierarchy subsumes an earlier hierarchy of Harnik and Naor (FOCS 2006, SICOMP 2010) 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 GB |
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Pages (from-to) | 202-215 |

Number of pages | 14 |

Journal | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

DOIs | |

State | Published - 1 Dec 2013 |

Event | 8th International Symposium on Parameterized and Exact Computation, IPEC 2013 - Sophia Antipolis, France Duration: 4 Sep 2013 → 6 Sep 2013 |

## ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science (all)