  The {\sc weak 2-linkage} problem for digraphs asks for a given digraph and
vertices $s_1,s_2,t_1,t_2$ whether $D$ contains a pair of arc-disjoint paths
$P_1,P_2$ such that $P_i$ is an $(s_i,t_i)$-path. This problem is NP-complete
for general digraphs but polynomially solvable for acyclic digraphs
\cite{fortuneTCS10}. Recently it was shown \cite{bercziESA17} that if $D$ is
equipped with a weight function $w$ on the arcs
  which satisfies that all edges have positive weight, then there is a
polynomial algorithm for the variant of the weak-2-linkage problem when both
paths have to be shortest paths in $D$. In this paper we consider the unit
weight case and prove that for every pair constants $k_1,k_2$, there is a
polynomial algorithm which decides whether the input digraph $D$ has a pair of
arc-disjoint paths $P_1,P_2$ such that $P_i$ is an $(s_i,t_i)$-path and the
length of $P_i$ is no more than $d(s_i,t_i)+k_i$, for $i=1,2$, where
$d(s_i,t_i)$ denotes the length of the shortest $(s_i,t_i)$-path. We prove
that, unless the exponential time hypothesis (ETH) fails, there is no
polynomial algorithm for deciding the existence of a solution $P_1,P_2$ to the
{\sc weak 2-linkage} problem where each path $P_i$ has length at most
$d(s_i,t_i)+ c\log^{1+\epsilon}{}n$ for some constant $c$.
  We also prove that the {\sc weak 2-linkage} problem remains NP-complete if we
require one of the two paths to be a shortest path while the other path has no
restriction on the length.
