## Approximation Hierarchies for the cone of flow matrices

Please always quote using this URN: urn:nbn:de:0297-zib-68424

- Let $G$ be a directed acyclic graph with $n$ arcs, a source $s$ and a sink $t$. We introduce the cone $K$ of flow matrices, which is a polyhedral cone generated by the matrices $1_P 1_P^T \in R^{n\times n}$, where $1_P\in R^n$ is the incidence vector of the $(s,t)$-path $P$. Several combinatorial problems reduce to a linear optimization problem over $K$. This cone is intractable, but we provide two convergent approximation hierarchies, one of them based on a completely positive representation of $K$. We illustrate this approach by computing bounds for a maximum flow problem with pairwise arc-capacities.

Author: | Guillaume Sagnol, Marco Blanco, Thibaut Sauvage |
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Document Type: | ZIB-Report |

Tag: | Approximation hierarchies; Copositive programming; Flows in graphs |

MSC-Classification: | 90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING / 90Cxx Mathematical programming [See also 49Mxx, 65Kxx] / 90C25 Convex programming |

90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING / 90Cxx Mathematical programming [See also 49Mxx, 65Kxx] / 90C35 Programming involving graphs or networks [See also 90C27] | |

Date of first Publication: | 2018/04/23 |

Series (Serial Number): | ZIB-Report (18-20) |

ISSN: | 1438-0064 |

Published in: | Appeared in: Electronic Notes in Discrete Mathematics Volume 64, February 2018, Pages 275–284 INOC 2017 – 8th International Network Optimization Conference |

DOI: | https://doi.org/10.1016/j.endm.2018.02.002 |