Length 3 Edge-Disjoint Paths Is NP-Hard

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...

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

The Edge-Disjoint Paths Problem is NP-Complete for Partial k-Trees

Many combinatorial problems are NP-complete for general graphs, but are not NP-complete for partial k-trees (graphs of treewidth bounded by a constant k) and can be efficiently solved in polynomial time or mostly in linear time for partial k-trees. On the other hand, very few problems are known to be NP-complete for partial k-trees with bounded k. These include the subgraph isomorphism problem ...

Finding Two Edge-Disjoint Paths with Length Constraints

We consider the problem of finding, for two pairs (s1, t1) and (s2, t2) of vertices in an undirected graphs, an (s1, t1)-path P1 and an (s2, t2)-path P2 such that P1 and P2 share no edges and the length of each Pi satisfies Li, where Li ∈ {≤ ki, = ki, ≥ ki, ≤ ∞}. We regard k1 and k2 as parameters and investigate the parameterized complexity of the above problem when at least one of P1 and P2 ha...