Time complexity of two disjoint simple paths

Shortest Two Disjoint Paths in Polynomial Time

Given an undirected graph and two pairs of vertices (si, ti) for i ∈ {1, 2} we show that there is a polynomial time Monte Carlo algorithm that finds disjoint paths of smallest total length joining si and ti for i ∈ {1, 2} respectively, or concludes that there most likely are no such paths at all. Our algorithm applies to both the vertexand edge-disjoint versions of the problem. Our algorithm is...

Finding Two Disjoint Simple Paths on Two Sets of Points is NP-Complete

Finding two disjoint simple paths on two given sets of points is a geometric problem introduced by Jeff Erickson. This problem has various applications in computational geometry, like robot motion planning, generating polygon etc. We will present a reduction from planar Hamiltonian path to this problem, and prove that it is NP-Complete. To the best of our knowledge, no study has considered its ...

Two Arc Disjoint Paths in Eulerian Diagraphs

Let G be an Eulerian digraph, and fx 1 ; x 2 g; fy 1 ; y 2 g be two pairs of vertices in G. A directed path from a vertex s to a vertex t is called a st-path. An instance (G; fx 1 ; x 2 g; fy 1 ; y 2 g) is called feasible if there is a choice of h; i; j; k with fh; ig = fj; kg = f1; 2g such that G has two arc-disjoint x h x i and y j y k -paths. In this paper, we characterize the structure of m...

The disjoint paths problem in quadratic time

We consider the following well-known problem, which is called the disjoint paths problem. Input: A graph G, k pairs of vertices (s1, t1), (s2, t2), . . . , (sk, tk) in G (which are sometimes called terminals). Output : Vertex disjoint paths P1, . . . , Pk in G such that Pi joins si and ti for i = 1, 2, . . . , k. We present an O(n) time algorithm for this problem for fixed k. This improves the ...

Finding non-dominated bicriteria shortest pairs of disjoint simple paths