  We prove that, for many parameterized problems in the class FPT, the
existence of polynomial kernels implies the collapse of the W-hierarchy (i.e.,
W[P] = FPT). The collapsing results are also extended to assumed exponential
kernels for problems in the class FPT. In particular, we establish a close
relationship between polynomial (and exponential) kernelizability and the
existence of sub-exponential time algorithms for a spectrum of circuit
satisfiability problems in FPT. To the best of our knowledge, this is the first
work that connects hardness for polynomial kernelizability of FPT problems to
parameterized intractability. Our work also offers some new insights into the
class FPT.
