Parameterized Tractability of Edge-Disjoint Paths on Directed Acyclic Graphs

Abstract

Given a graph and pairs s i t i of terminals, the edge-disjoint paths problem is to determine whether there exist s i t i 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 O(k) 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.KeywordsSource NodeDirect Acyclic GraphSink NodeTerminal NodeDisjoint PathThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.