  The central conjecture of parameterized complexity states that FPT is not
equal to W[1], and is generally regarded as the parameterized counterpart to P
!= NP. We revisit the issue of the plausibility of FPT != W[1], focusing on two
aspects: the difficulty of proving the conjecture (assuming it holds), and how
the relation between the two classes might differ from the one between P and
NP.
  Regarding the first aspect, we give new evidence that separating FPT from
W[1] would be considerably harder than doing the same for P and NP. Our main
result regarding the relation between FPT and W[1] states that the closure of
W[1] under relativization with FPT-oracles is precisely the class W[P],
implying that either FPT is not low for W[1], or the W-Hierarchy collapses.
This theorem also has consequences for the A-Hierarchy (a parameterized version
of the Polynomial Hierarchy), namely that unless W[P] is a subset of some level
A[t], there are structural differences between the A-Hierarchy and the
Polynomial Hierarchy. We also prove that under the unlikely assumption that
W[P] collapses to W[1] in a specific way, the collapse of any two consecutive
levels of the A-Hierarchy implies the collapse of the entire hierarchy to a
finite level; this extends a result of Chen, Flum, and Grohe (2005).
  Finally, we give weak (oracle-based) evidence that the inclusion of W[t] in
A[t] is strict for t>1, and that the W-Hierarchy is proper. The latter result
answers a question of Downey and Fellows (1993).
