Parameterized Tractability of Edge-Disjoint Paths on Directed Acyclic Graphs [chapter]

Aleksandrs Slivkins
2003 Lecture Notes in Computer Science  
Given a graph and terminal pairs (s i , t i ), i ∈ [k], the edge-disjoint paths problem is to determine whether there exist s i t i paths, i ∈ [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 n O(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) n O(1) . We resolve this
more » ... estion in the negative: we show that the problem is W [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+k O(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.
doi:10.1007/978-3-540-39658-1_44 fatcat:ziapmseqyngvtbvylnvezo2cqm