Finding k partially disjoint paths in a directed planar graph

The partially disjoint paths problem is: given: a directed graph, vertices \(r_1,s_1,\ldots ,r_k,s_k\), and a set F of pairs \(\{i,j\}\) from \(\{1,\ldots ,k\}\), find: for each \(i=1,\ldots ,k\) a directed \(r_i-s_i\) path \(P_i\) such that if \(\{i,j\}\in F\) then \(P_i\) and \(P_j\) are disjoint. We show that for fixed k, this problem is solvable in polynomial time if the directed graph is planar. More generally, the problem is solvable in polynomial time for directed graphs embedded on a fixed compact surface. Moreover, one may specify for each edge a subset of \(\{1,\ldots ,k\}\) prescribing which of the \(r_i-s_i\) paths are allowed to traverse this edge.