The Planar Directed K-Vertex-Disjoint Paths Problem Is Fixed-Parameter Tractable

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 pair wise vertex-disjoint paths P
₁
, ..., Pk such that Pi goes from si to ti. Schrijver [SICOMP'94] proved that the k-Vertex-Disjoint Paths problem on planar directed graphs can be solved in time
^nO(k)
. We give an algorithm with running time 2
^2O(k2)
* n
^O(1)
for the problem, that is, we show the fixed-parameter tractability of the problem.