### Abstract

A constructive method is described for proving the Edmonds-Giles theorem which yields a good algorithm provided that a fast subroutine is available for minimizing a submodular set function. The algorithm can be used for finding a maximum weight common independent set of two matroids, for finding a minimum weight covering of directed cuts of a digraph, and, as a new application, for finding a minimum cost k strongly connected orientation of an undirected graph. As a theoretical consequence of the algorithm, we prove a combinatorial feasibility theorem for Edmonds-Giles polyhedron and then we derive a discrete separation theorem which says, roughly, an integer valued submodular function B and an integer valued supermodular function R can be separated by an integer valued modular function provided that R ≤B.

Original language | English |
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Pages (from-to) | 97-120 |

Number of pages | 24 |

Journal | North-Holland Mathematics Studies |

Volume | 66 |

Issue number | C |

DOIs | |

Publication status | Published - 1982 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**An Algorithm for Submodular Functions on Graphs.** / Frank, A.

Research output: Contribution to journal › Article

*North-Holland Mathematics Studies*, vol. 66, no. C, pp. 97-120. https://doi.org/10.1016/S0304-0208(08)72446-0

}

TY - JOUR

T1 - An Algorithm for Submodular Functions on Graphs

AU - Frank, A.

PY - 1982

Y1 - 1982

N2 - A constructive method is described for proving the Edmonds-Giles theorem which yields a good algorithm provided that a fast subroutine is available for minimizing a submodular set function. The algorithm can be used for finding a maximum weight common independent set of two matroids, for finding a minimum weight covering of directed cuts of a digraph, and, as a new application, for finding a minimum cost k strongly connected orientation of an undirected graph. As a theoretical consequence of the algorithm, we prove a combinatorial feasibility theorem for Edmonds-Giles polyhedron and then we derive a discrete separation theorem which says, roughly, an integer valued submodular function B and an integer valued supermodular function R can be separated by an integer valued modular function provided that R ≤B.

AB - A constructive method is described for proving the Edmonds-Giles theorem which yields a good algorithm provided that a fast subroutine is available for minimizing a submodular set function. The algorithm can be used for finding a maximum weight common independent set of two matroids, for finding a minimum weight covering of directed cuts of a digraph, and, as a new application, for finding a minimum cost k strongly connected orientation of an undirected graph. As a theoretical consequence of the algorithm, we prove a combinatorial feasibility theorem for Edmonds-Giles polyhedron and then we derive a discrete separation theorem which says, roughly, an integer valued submodular function B and an integer valued supermodular function R can be separated by an integer valued modular function provided that R ≤B.

UR - http://www.scopus.com/inward/record.url?scp=77956846525&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77956846525&partnerID=8YFLogxK

U2 - 10.1016/S0304-0208(08)72446-0

DO - 10.1016/S0304-0208(08)72446-0

M3 - Article

AN - SCOPUS:77956846525

VL - 66

SP - 97

EP - 120

JO - North-Holland Mathematics Studies

JF - North-Holland Mathematics Studies

SN - 0304-0208

IS - C

ER -