  In the Directed Disjoint Paths problem, we are given a digraph $D$ and a set
of requests $\{(s_1, t_1), \ldots, (s_k, t_k)\}$, and the task is to find a
collection of pairwise vertex-disjoint paths $\{P_1, \ldots, P_k\}$ such that
each $P_i$ is a path from $s_i$ to $t_i$ in $D$. This problem is NP-complete
for fixed $k=2$ and W[1]-hard with parameter $k$ in DAGs. A few positive
results are known under restrictions on the input digraph, such as being planar
or having bounded directed tree-width, or under relaxations of the problem,
such as allowing for vertex congestion. Positive results are scarce, however,
for general digraphs. In this article we propose a novel global congestion
metric for the problem: we only require the paths to be "disjoint enough", in
the sense that they must behave properly not in the whole graph, but in an
unspecified part of size prescribed by a parameter. Namely, in the Disjoint
Enough Directed Paths problem, given an $n$-vertex digraph $D$, a set of $k$
requests, and non-negative integers $d$ and $s$, the task is to find a
collection of paths connecting the requests such that at least $d$ vertices of
$D$ occur in at most $s$ paths of the collection. We study the parameterized
complexity of this problem for a number of choices of the parameter, including
the directed tree-width of $D$. Among other results, we show that the problem
is W[1]-hard in DAGs with parameter $d$ and, on the positive side, we give an
algorithm in time $\mathcal{O}(n^{d+2} \cdot k^{d\cdot s})$ and a kernel of
size $d \cdot 2^{k-s}\cdot \binom{k}{s} + 2k$ in general digraphs. This latter
result has consequences for the Steiner Network problem: we show that it is FPT
parameterized by the number $k$ of terminals and $p$, where $p = n - q$ and $q$
is the size of the solution.
