Parameterized Tractability of Edge-Disjoint Paths on Directed Acyclic Graphs

• Authors: Aleksandrs Slivkins
• Venue: SIAM Journal on Discrete Mathematics
• Year: 2010

Given a graph and terminal pairs $(s_i,t_i)$, $i\in[k]$, the edge-disjoint paths problem is to determine whether there exist $s_{i}t_{i}$ paths, $i\in[k]$, 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 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 $O(m+k^{O(1)}\,k!\,n)$ 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, and (2) to an analogous special case of the unsplittable flow problem, a generalized version of disjoint paths that has capacities and demands.