## The Cone of Flow Matrices: Approximation Hierarchies and Applications

- 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 $\vec{1}_P\vec{1}_P^T\in\RR^{n\times n}$, where $\vec{1}_P\in\RR^n$ is the incidence vector of the (s,t)-path P. We show that several hard flow (or path) optimization problems, that cannot be solved by using the standard arc-representation of a flow, reduce to a linear optimization problem over $\mathcal{K}$. This cone is intractable: we prove that the membership problem associated to $\mathcal{K}$ is NP-complete. However, the affine hull of this cone admits a nice description, and we give an algorithm which computes in polynomial-time the decomposition of a matrix $X\in \operatorname{span} \mathcal{K}$ as a linear combination of some $\vec{1}_P\vec{1}_P^T$'s. Then, 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 the quadratic shortest path problem, as well as a maximum flow problem with pairwise arc-capacities.

Author: | Guillaume Sagnol, Marco Blanco, Thibaut Sauvage |
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Document Type: | Article |

Parent Title (English): | Networks |

Volume: | 72 |

Issue: | 1 |

First Page: | 128 |

Last Page: | 150 |

Year of first publication: | 2018 |

Preprint: | urn:nbn:de:0297-zib-64399 |

DOI: | https://doi.org/10.1002/net.21820 |