On shortest disjoint paths in planar graphs

  • Authors:

  • Yusuke Kobayashi;Christian Sommer

  • Affiliations:

  • Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, Tokyo 113-8656, Japan;Department of Computer Science, Graduate School of Information Science and Technology, University of Tokyo, Tokyo 113-0033, Japan

  • Venue:
  • Discrete Optimization
  • Year:
  • 2010

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Abstract

For a graph G and a collection 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 3.