  The problem DFA-Intersection-Nonemptiness asks if a given number of
deterministic automata accept a common word. In general, this problem is
PSPACE-complete. Here, we investigate this problem for the subclasses of
commutative automata and automata recognizing sparse languages. We show that in
both cases DFA-Intersection-Nonemptiness is complete for NP and for the
parameterized class $W[1]$, where the number of input automata is the
parameter, when the alphabet is fixed. Additionally, we establish the same
result for Tables Non-Empty Join, a problem that asks if the join of several
tables (possibly containing null values) in a database is non-empty. Lastly, we
show that Bounded NFA-Intersection-Nonemptiness, parameterized by the length
bound, is $\mbox{co-}W[2]$-hard with a variable input alphabet and for
nondeterministic automata recognizing finite strictly bounded languages,
yielding a variant leaving the realm of $W[1]$.
