On Shortest Disjoint Paths in Planar Graphs

Abstract

For a graph and a set of vertex pairs {(s 1, t 1), ..., (s k , t k )}, the k disjoint paths problem is to find k vertex-disjoint paths P 1, ..., 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 Verdiere 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 at least 3.