On Shortest Disjoint Paths in Planar Graphs

For a graph G and a collection of vertex pairs {(s1, t1), . . . , (sk, tk)}, the k disjoint paths problem is to find k vertex-disjoint paths P1, . . . , Pk, where Pi is a path from si to ti for each i = 1, . . . , k. In the corresponding optimization problem, the shortest disjoint paths problem, the vertex-disjoint paths Pi 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 Verdi`ere 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 kn...

• Type: Conference
• Year: 2009
• Where: ISAAC
• Authors: Yusuke Kobayashi, Christian Sommer 0002
• Keywords: Disjoint Paths Problem, ISAAC 2009, Polynomial Time, Vertex-disjoint Paths