Parameterized Tractability of Edge-Disjoint Paths on Directed Acyclic Graphs

11th Annual European Symp. on Algorithms (ESA) |

Journal version appeared in SIAM J. on Discrete Math 24(1), pp. 146-157, 2010.

Given a graph and terminal pairs (si; ti), i 2 [k], the edge-disjoint paths problem is to determine whether there exist siti paths, i 2 [k], that do not share any edges. We consider this problem on acyclic digraphs. It is known to be NP-complete and solvable in time nO(k) where n is the number of nodes. It has been a long-standing open question whether it is fixed-parameter tractable in k, i.e. whether it admits an algorithm with running time of the form f(k) nO(1). We resolve this question in the negative: we show that the problem isW[1]-hard, hence unlikely to be fixed-parameter tractable. 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+kO(1) 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, a generalized version of disjoint paths that has capacities and demands.