### Abstract

The problem of computing a minimum cost subgraph D^{′} = (V, A^{′}) of a directed graph D = (V, A) so as to contain k edge-disjoint paths from a specified root r_{0} ∈ V to every other node in V was solved by Edmonds [J. Edmonds, Submodular functions, matroids, and certain polyhedra, in: R. Guy, H. Hanani, N. Sauer, J. Schönheim (Eds.), Combinatorial Structures and their Applications, Gordon and Breach, New York, 1970, pp. 69-87] by an elegant reduction to weighted matroid intersection. A corresponding problem when openly disjoint paths are requested rather than edge-disjoint ones was solved in [A. Frank, É. Tardos, An application of submodular flows, Linear Algebra Appl. 114-115 (1989) 329-348] with the help of submodular flows. Here we show that the use of submodular flows is actually avoidable and even a common generalization of the two rooted k-connection problems reduces to matroid intersection. The approach is based on a new matroid construction extending what Whiteley [W. Whiteley, Some matroids from discrete applied geometry, in: J.E. Bonin, J.G. Oxley, B. Servatius (Eds.), Matroid Theory, in: Contemp. Math., vol. 197, Amer. Math. Soc, Providence, RI, 1996, pp. 171-311] calls count matroids. We also provide a polyhedral description using supermodular functions on bi-sets and this approach enables us to handle more general rooted k-connection problems. For example, with the help of a submodular flow algorithm the following restricted version of the generalized Steiner-network problem is solvable in polynomial time: given a digraph D = (V, A) with a root-node r_{0}, a terminal set T, and a cost function c : A → R_{+} so that each edge of positive cost has its head in T, find a subgraph D^{′} = (V, A^{′}) of D of minimum cost so that there are k openly disjoint paths in D^{′} from r_{0} to every node in T.

Original language | English |
---|---|

Pages (from-to) | 1242-1254 |

Number of pages | 13 |

Journal | Discrete Applied Mathematics |

Volume | 157 |

Issue number | 6 |

DOIs | |

Publication status | Published - Mar 28 2009 |

### Fingerprint

### Keywords

- Arborescences
- Connectivity of digraphs
- Matroid intersection

### ASJC Scopus subject areas

- Applied Mathematics
- Discrete Mathematics and Combinatorics

### Cite this

*Discrete Applied Mathematics*,

*157*(6), 1242-1254. https://doi.org/10.1016/j.dam.2008.03.040

**Rooted k-connections in digraphs.** / Frank, A.

Research output: Contribution to journal › Article

*Discrete Applied Mathematics*, vol. 157, no. 6, pp. 1242-1254. https://doi.org/10.1016/j.dam.2008.03.040

}

TY - JOUR

T1 - Rooted k-connections in digraphs

AU - Frank, A.

PY - 2009/3/28

Y1 - 2009/3/28

N2 - The problem of computing a minimum cost subgraph D′ = (V, A′) of a directed graph D = (V, A) so as to contain k edge-disjoint paths from a specified root r0 ∈ V to every other node in V was solved by Edmonds [J. Edmonds, Submodular functions, matroids, and certain polyhedra, in: R. Guy, H. Hanani, N. Sauer, J. Schönheim (Eds.), Combinatorial Structures and their Applications, Gordon and Breach, New York, 1970, pp. 69-87] by an elegant reduction to weighted matroid intersection. A corresponding problem when openly disjoint paths are requested rather than edge-disjoint ones was solved in [A. Frank, É. Tardos, An application of submodular flows, Linear Algebra Appl. 114-115 (1989) 329-348] with the help of submodular flows. Here we show that the use of submodular flows is actually avoidable and even a common generalization of the two rooted k-connection problems reduces to matroid intersection. The approach is based on a new matroid construction extending what Whiteley [W. Whiteley, Some matroids from discrete applied geometry, in: J.E. Bonin, J.G. Oxley, B. Servatius (Eds.), Matroid Theory, in: Contemp. Math., vol. 197, Amer. Math. Soc, Providence, RI, 1996, pp. 171-311] calls count matroids. We also provide a polyhedral description using supermodular functions on bi-sets and this approach enables us to handle more general rooted k-connection problems. For example, with the help of a submodular flow algorithm the following restricted version of the generalized Steiner-network problem is solvable in polynomial time: given a digraph D = (V, A) with a root-node r0, a terminal set T, and a cost function c : A → R+ so that each edge of positive cost has its head in T, find a subgraph D′ = (V, A′) of D of minimum cost so that there are k openly disjoint paths in D′ from r0 to every node in T.

AB - The problem of computing a minimum cost subgraph D′ = (V, A′) of a directed graph D = (V, A) so as to contain k edge-disjoint paths from a specified root r0 ∈ V to every other node in V was solved by Edmonds [J. Edmonds, Submodular functions, matroids, and certain polyhedra, in: R. Guy, H. Hanani, N. Sauer, J. Schönheim (Eds.), Combinatorial Structures and their Applications, Gordon and Breach, New York, 1970, pp. 69-87] by an elegant reduction to weighted matroid intersection. A corresponding problem when openly disjoint paths are requested rather than edge-disjoint ones was solved in [A. Frank, É. Tardos, An application of submodular flows, Linear Algebra Appl. 114-115 (1989) 329-348] with the help of submodular flows. Here we show that the use of submodular flows is actually avoidable and even a common generalization of the two rooted k-connection problems reduces to matroid intersection. The approach is based on a new matroid construction extending what Whiteley [W. Whiteley, Some matroids from discrete applied geometry, in: J.E. Bonin, J.G. Oxley, B. Servatius (Eds.), Matroid Theory, in: Contemp. Math., vol. 197, Amer. Math. Soc, Providence, RI, 1996, pp. 171-311] calls count matroids. We also provide a polyhedral description using supermodular functions on bi-sets and this approach enables us to handle more general rooted k-connection problems. For example, with the help of a submodular flow algorithm the following restricted version of the generalized Steiner-network problem is solvable in polynomial time: given a digraph D = (V, A) with a root-node r0, a terminal set T, and a cost function c : A → R+ so that each edge of positive cost has its head in T, find a subgraph D′ = (V, A′) of D of minimum cost so that there are k openly disjoint paths in D′ from r0 to every node in T.

KW - Arborescences

KW - Connectivity of digraphs

KW - Matroid intersection

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U2 - 10.1016/j.dam.2008.03.040

DO - 10.1016/j.dam.2008.03.040

M3 - Article

VL - 157

SP - 1242

EP - 1254

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

IS - 6

ER -