The 2 edge-disjoint 3-paths polyhedron

4.1 Edge Disjoint Paths

Problem Statement: Given a directed graphG and a set of terminal pairs {(s1, t1), (s2, t2), · · · , (sk, tk)}, our goal is to connect as many pairs as possible using non edge intersecting paths. Edge disjoint paths problem is NP-Complete and is closely related to the multicommodity flow problem. In fact integer multicommodity flow is a generalization of this problem. We describe a greedy approx...

Reconstructing edge-disjoint paths

From edge-disjoint paths to independent paths

Let f(k) denote the maximum such that every simple undirected graph containing two vertices s, t and k edge-disjoint s–t paths, also contains two vertices u, v and f(k) independent u–v paths. Here, a set of paths is independent if none of them contains an interior vertex of another. We prove that f(k) = ( k if k ≤ 2, and 3 otherwise. Since independent paths are edge-disjoint, it is clear that f...

Reconstructing edge-disjoint paths faster

For a simple undirected graph with n vertices and m edges, we consider a data structure that given a query of a pair of vertices u, v and an integer k ≥ 1, it returns k edge-disjoint uv-paths. The data structure takes Õ(n) time to build, using O( √ mn log n) space, and each query takes O( √ kn) time, which is optimal and beats the previous query time of O(knα(n)).

Length 3 Edge-Disjoint Paths and Partial Orientation

In 2003, it was claimed that the following problem was solvable in polynomial time: do there exist k edge-disjoint paths of length exactly 3 between vertices s and t in a given graph? The proof was flawed, and we show that this problem is NP-hard even if we disallow multiple edges. We use a reduction from Partial Orientation, a problem recently shown by Pálvölgyi to be NP-hard. In [2], Bley dis...