The Induced Disjoint Paths Problem

Abstract

For a graph G and a collection of vertex pairs {(s ₁, t ₁), ...,(s _k , t _k )}, the 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. This problem is one of the classic problems in combinatorial optimization and algorithmic graph theory, and has many applications, for example in transportation networks, VLSI layout, and recently, virtual circuits routing in high-speed networks.

As an extension of the disjoint paths problem, we introduce a new problem which we call the induced disjoint paths problem. In this problem we are given a graph G and a collection of vertex pairs {(s ₁, t ₁), ...,(s _k , t _k )}. The objective is to find k paths P ₁, ..., P _k such that P _i is a path from s _i to t _i and P _i and P _j have neither common vertices nor adjacent vertices for any distinct i, j.

This problem setting is a generalization of the disjoint paths problem, since if we subdivide each edge, then desired disjoint paths would be induced disjoint paths. The problem is motivated by not only the disjoint paths problem but also the recognition of an induced subgraph. The latter has been developed in the recent years, and this is actually connected to the Strong Perfect Graph Theorem [4], and the recognition of the perfect graphs [2].

In this paper, we shall investigate the complexity issues of this problem. The induced disjoint paths problem has several variants depending on whether k is a fixed constant or a part of the input, whether the graph is directed or undirected, and whether the graph is planar or not. We show that the induced disjoint paths problem is

(i) solvable in polynomial time when k is fixed and G is a directed planar graph,

(ii) solvable in linear time when k is fixed and G is an undirected planar graph,

(iii) NP-hard when k = 2 and G is an acyclic directed graph,

(iv) NP-hard when k = 2 and G is an undirected general graph.

(i) generalizes the result by Schrijver [22], while (ii) generalizes the result by Reed, Robertson, Schrijver and Seymour [17].