TY - JOUR

T1 - A Completeness Theory for Polynomial (Turing) Kernelization

AU - Hermelin, Danny

AU - Kratsch, Stefan

AU - Sołtys, Karolina

AU - Wahlström, Magnus

AU - Wu, Xi

N1 - Funding Information:
Main work done while Stefan Kratsch was supported by the Netherlands Organization for Scientific Research (NWO), Project “KERNELS: Combinatorial Analysis of Data Reduction”.
Publisher Copyright:
© 2014, Springer Science+Business Media New York.

PY - 2015/3/1

Y1 - 2015/3/1

N2 - 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 VC1; this had been left open.

AB - 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 VC1; this had been left open.

KW - Complexity hierarchies

KW - Kernelization

KW - Parameterized complexity

KW - Turing kernelization

UR - http://www.scopus.com/inward/record.url?scp=84923674126&partnerID=8YFLogxK

U2 - 10.1007/s00453-014-9910-8

DO - 10.1007/s00453-014-9910-8

M3 - Article

AN - SCOPUS:84923674126

SN - 0178-4617

VL - 71

SP - 702

EP - 730

JO - Algorithmica

JF - Algorithmica

IS - 3

ER -