Routing with congestion in acyclic digraphs

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/log⁡d) for any computable function f.