Given a graph G and k pairs of vertices (s_1,t_1), ..., (s_k,t_k), the k-Vertex-Disjoint Paths problem asks for pairwise vertex-disjoint paths P_1, ..., P_k such that P_i goes from s_i to t_i. Schrijver [SICOMP'94] proved that the k-Vertex-Disjoint Paths problem on planar directed graphs can be solved in time n^{ O(k)}. We give an algorithm with running time 2^{ 2^{ O(k^2)}} n^{ O(1)} for the problem, that is, we show the fixed-parameter tractability of the problem.
Categories: cs.DM cs.DS math.CO