  The propositional planning problem is a notoriously difficult computational
problem. Downey et al. (1999) initiated the parameterized analysis of planning
(with plan length as the parameter) and B\"ackstr\"om et al. (2012) picked up
this line of research and provided an extensive parameterized analysis under
various restrictions, leaving open only one stubborn case. We continue this
work and provide a full classification. In particular, we show that the case
when actions have no preconditions and at most $e$ postconditions is
fixed-parameter tractable if $e\leq 2$ and W[1]-complete otherwise. We show
fixed-parameter tractability by a reduction to a variant of the Steiner Tree
problem; this problem has been shown fixed-parameter tractable by Guo et al.
(2007). If a problem is fixed-parameter tractable, then it admits a
polynomial-time self-reduction to instances whose input size is bounded by a
function of the parameter, called the kernel. For some problems, this function
is even polynomial which has desirable computational implications. Recent
research in parameterized complexity has focused on classifying fixed-parameter
tractable problems on whether they admit polynomial kernels or not. We revisit
all the previously obtained restrictions of planning that are fixed-parameter
tractable and show that none of them admits a polynomial kernel unless the
polynomial hierarchy collapses to its third level.
