Parameterized Tractability of Edge-Disjoint Paths on Directed Acyclic Graphs

Given a graph and pairs siti of terminals, the edge-disjoint paths problem is to determine whether there exist siti paths that do not share any edges. We consider this problem on acyclic digraphs. It is known to be NP-complete and solvable in time n where k is the number of paths. It has been a long-standing open question whether it is fixed-parameter tractable in k. We resolve this question in the negative: we show that the problem is W [1]-hard. In fact it remains W [1]-hard even if the demand graph consists of two sets of parallel edges. On a positive side, we give an O(m + k! n) algorithm for the special case when G is acyclic and G+H is Eulerian, where H is the demand graph. We generalize this result (1) to the case when G+H is “nearly” Eulerian, (2) to an analogous special case of the unsplittable flow problem. Finally, we consider a related NP-complete routing problem when only the first edge of each path cannot be shared, and prove that it is fixed-parameter tractable on directed graphs.