A Completeness Theory for Polynomial (Turing) Kernelization
Algorithmica(2014)
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 (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 k 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.
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关键词
Parameterized complexity,Kernelization,Turing kernelization,Complexity hierarchies
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