  Fixed-parameter algorithms and kernelization are two powerful methods to
solve $\mathsf{NP}$-hard problems. Yet, so far those algorithms have been
largely restricted to static inputs.
  In this paper we provide fixed-parameter algorithms and kernelizations for
fundamental $\mathsf{NP}$-hard problems with dynamic inputs. We consider a
variety of parameterized graph and hitting set problems which are known to have
$f(k)n^{1+o(1)}$ time algorithms on inputs of size $n$, and we consider the
question of whether there is a data structure that supports small updates (such
as edge/vertex/set/element insertions and deletions) with an update time of
$g(k)n^{o(1)}$; such an update time would be essentially optimal. Update and
query times independent of $n$ are particularly desirable. Among many other
results, we show that Feedback Vertex Set and $k$-Path admit dynamic algorithms
with $f(k)\log^{O(1)}n$ update and query times for some function $f$ depending
on the solution size $k$ only.
  We complement our positive results by several conditional and unconditional
lower bounds. For example, we show that unlike their undirected counterparts,
Directed Feedback Vertex Set and Directed $k$-Path do not admit dynamic
algorithms with $n^{o(1)}$ update and query times even for constant solution
sizes $k\leq 3$, assuming popular hardness hypotheses. We also show that
unconditionally, in the cell probe model, Directed Feedback Vertex Set cannot
be solved with update time that is purely a function of $k$.
