Length-bounded disjoint paths in planar graphs

The following problem is considered: given: an undirected planar graph G=(V,E) embedded in , distinct pairs of vertices {r1,s1},…,{rk,sk} of G adjacent to the unbounded face, positive integers b1,…,bk and a function ; find: pairwise vertex-disjoint paths P1,…,Pk such that for each i=1,…,k, Pi is a ri-si-path and the sum of the l-length of all edges in Pi is at most bi. It is shown that the problem is NP-hard in the strong sense. A pseudo-polynomial-time algorithm is given for the case of k=2