We study the version of the k-disjoint paths problem where k demand pairs
(s1,t1), …, (sk,tk) are specified in the input and the paths in
the solution are allowed to intersect, but such that no vertex is on more than
c paths. We show that on directed acyclic graphs the problem is solvable in
time nO(d) if we allow congestion k−d for k paths. Furthermore, we
show that, under a suitable complexity theoretic assumption, the problem cannot
be solved in time f(k)no(d/logd) for any computable function f