TY - GEN

T1 - A completeness theory for polynomial (Turing) kernelization

AU - Hermelin, Danny

AU - Kratsch, Stefan

AU - Sołtys, Karolina

AU - Wahlström, Magnus

AU - Wu, Xi

PY - 2013/12/1

Y1 - 2013/12/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84893034217&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-03898-8_18

DO - 10.1007/978-3-319-03898-8_18

M3 - Conference contribution

AN - SCOPUS:84893034217

SN - 9783319038971

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 202

EP - 215

BT - Parameterized and Exact Computation - 8th International Symposium, IPEC 2013, Revised Selected Papers

T2 - 8th International Symposium on Parameterized and Exact Computation, IPEC 2013

Y2 - 4 September 2013 through 6 September 2013

ER -