### Abstract

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 Bs,1+,Bs,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 xi are those used in Bs,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 uij=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 uij=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(lunate)>0 and for every k(n) with (log(n))1+ε(lunate)≤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).

Original language | English |
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Journal | Discrete Applied Mathematics |

Volume | 209 |

Issue number | C |

Pages (from-to) | 16-26 |

ISSN | 0166-218X |

DOIs | |

Publication status | Published - 2016 |

Event | 9th International Colloquium on Graph Theory and Combinatorics - Grenoble, France Duration: 30. Jun 2014 → 4. Jul 2014 http://oc.inpg.fr/conf/icgt2014/ |

### Conference

Conference | 9th International Colloquium on Graph Theory and Combinatorics |
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Country | France |

City | Grenoble |

Period | 30/06/2014 → 04/07/2014 |

Internet address |

### Fingerprint

### Keywords

- Branching flow
- Disjoint branchings
- NP-complete
- Polynomial algorithm

### Cite this

*Discrete Applied Mathematics*,

*209*(C), 16-26. https://doi.org/10.1016/j.dam.2015.10.012

}

*Discrete Applied Mathematics*, vol. 209, no. C, pp. 16-26. https://doi.org/10.1016/j.dam.2015.10.012

**The complexity of finding arc-disjoint branching flows.** / Bang-Jensen, J.; Havet, Frédéric; Yeo, Anders.

Research output: Contribution to journal › Conference article › Research › peer-review

TY - GEN

T1 - The complexity of finding arc-disjoint branching flows

AU - Bang-Jensen, J.

AU - Havet, Frédéric

AU - Yeo, Anders

PY - 2016

Y1 - 2016

N2 - 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 Bs,1+,Bs,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 xi are those used in Bs,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 uij=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 uij=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(lunate)>0 and for every k(n) with (log(n))1+ε(lunate)≤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).

AB - 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 Bs,1+,Bs,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 xi are those used in Bs,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 uij=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 uij=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(lunate)>0 and for every k(n) with (log(n))1+ε(lunate)≤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).

KW - Branching flow

KW - Disjoint branchings

KW - NP-complete

KW - Polynomial algorithm

U2 - 10.1016/j.dam.2015.10.012

DO - 10.1016/j.dam.2015.10.012

M3 - Conference article

VL - 209

SP - 16

EP - 26

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

IS - C

ER -