Approximation Hierarchies for the cone of flow matrices
- 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: | In Proceedings |
Parent Title (English): | INOC 2017 – 8th International Network Optimization Conference |
Volume: | 64 |
First Page: | 275 |
Last Page: | 284 |
Series: | Electronic Notes in Discrete Mathematics |
Year of first publication: | 2018 |
Preprint: | urn:nbn:de:0297-zib-68424 |
DOI: | https://doi.org/10.1016/j.endm.2018.02.002 |