The disjoint paths problem: Algorithm and structure

Abstract

We consider the following well-known problem, which is called the disjoint paths problem. Input: A graph G with n vertices and m edges, k pairs of vertices (s 1,t 1),(s 2,t 2),..., (s k ,t k ) in G. Output: (Vertex- or edge-) disjoint paths P 1, P 2, ...,P k in G such that P i joins s i and t i for i = 1,2,...,k. This is certainly a central problem in algorithmic graph theory and combinatorial optimization. See the surveys [9, 31]. It has attracted attention in the contexts of transportation networks, VLSI layout and virtual circuit routing in high-speed networks or Internet. A basic technical problem is to interconnect certain prescribed "channels" on the chip such that wires belonging to different pins do not touch each other. In this simplest form, the problem mathematically amounts to finding vertex-disjoint trees or vertex-disjoint paths in a graph, each connecting a given set of vertices. © 2011 Springer-Verlag.