PARAMETERIZED TRACTABILITY OF EDGE-DISJOINT PATHS ON DIRECTED ACYCLIC GRAPHS.

Given a graph and terminal pairs (si, ti), i ϵ [k], the edge-disjoint paths problem is to determine whether there exist siti paths, i ϵ [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 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+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, and (2) to an analogous special case of the unsplittable flow problem, a generalized version of disjoint paths that has capacities and demands. [ABSTRACT FROM AUTHOR]