科研之友
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摘要
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 leaf out-branching that outputs instances each of size poly. 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 -SAT , Binary NDTM Halting , Connected Vertex Cover , and Clique parameterized by . 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 ; this had been left open.
- 出版日期2015-3
