The planar directed k-Vertex-Disjoint Paths problem is fixed-parameter tractable

Given a graph G and k pairs of vertices (s1,t1), ..., (sk,tk), the k-Vertex-Disjoint Paths problem asks for pairwise vertex-disjoint paths P1, ..., Pk such that Pi goes from si 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(k2)}} n^{O(1)} for the problem, that is, we show the fixed-parameter tractability of the problem.