A completeness theory for polynomial (Turing) kernelization

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.