On shortest disjoint paths in planar graphs

For a graph G and a collection of vertex pairs {(s₁,t₁),…,(s_k,t_k)}, the k disjoint paths problem is to find k vertex-disjoint paths P₁,…,P_k, where P_i is a path from s_i to t_i for each i=1,…,k. In the corresponding optimization problem, the shortest disjoint paths problem, the vertex-disjoint paths P_i have to be chosen such that a given objective function is minimized. We consider two different objectives, namely minimizing the total path length (minimum sum, or short: Min-Sum), and minimizing the length of the longest path (Min-Max), for k=2,3.Min-Sum: We extend recent results by Colin de Verdière and Schrijver to prove that, for a planar graph and for terminals adjacent to at most two faces, the Min-Sum 2 Disjoint Paths Problem can be solved in polynomial time. We also prove that, for six terminals adjacent to one face in any order, the Min-Sum 3 Disjoint Paths Problem can be solved in polynomial time.Min-Max: The Min-Max 2 Disjoint Paths Problem is known to be NP-hard for general graphs. We present an algorithm that solves the problem for graphs with tree-width 2 in polynomial time. We thus close the gap between easy and hard instances, since the problem is weakly NP-hard for graphs with tree-width 3.