The complexity of finding arc-disjoint branching flows

The concept of arc-disjoint flows in networks was recently introduced in Bang-Jensen and Bessy (2014). This is a very general framework within which many well-known and important problems can be formulated. In particular, the existence of arc-disjoint branching flows, that is, flows which send one unit of flow from a given source s to all other vertices, generalizes the concept of arc-disjoint out-branchings (spanning out-trees) in a digraph. A pair of out-branchings B_s,1⁺,B_s,2⁺ from a root s in a digraph D=(V,A) on n vertices corresponds to arc-disjoint branching flows x₁,x₂ (the arcs carrying flow in x_i are those used in B_s,i⁺, i=1,2) in the network that we obtain from D by giving all arcs capacity n−1. It is then a natural question to ask how much we can lower the capacities on the arcs and still have, say, two arc-disjoint branching flows from the given root s. We prove that for every fixed integer k≥2 it is • an NP-complete problem to decide whether a network N=(V,A,u) where u_ij=k for every arc ij has two arc-disjoint branching flows rooted at s.• a polynomial problem to decide whether a network N=(V,A,u) on n vertices and u_ij=n−k for every arc ij has two arc-disjoint branching flows rooted at s.The algorithm for the later result generalizes the polynomial algorithm, due to Lovász, for deciding whether a given input digraph has two arc-disjoint out-branchings rooted at a given vertex. Finally we prove that under the so-called Exponential Time Hypothesis (ETH), for every ϵ>0 and for every k(n) with (log(n))^1+ϵ≤k(n)≤n2 (and for every large i we have k(n)=i for some n) there is no polynomial algorithm for deciding whether a given digraph contains two arc-disjoint branching flows from the same root so that no arc carries flow larger than n−k(n).