Reachability in arborescence packings

TL;DR

This paper unifies and extends key results on packing arborescences and reachability in directed graphs using a simple proof technique, and develops algorithms for finding such packings efficiently.

Contribution

It introduces a unified proof approach linking various arborescence packing theorems and presents algorithms for constructing these packings with optimization capabilities.

Findings

A strong version of Edmonds' theorem implies Kamiyama et al.'s result.

Matroid-based packing theorems imply reachability packing results.

New algorithms efficiently find and optimize arborescence packings.

Abstract

Fortier et al. proposed several research problems on packing arborescences. Some of them were settled in that article and others were solved later by Matsuoka and Tanigawa and by Gao and Yang. The last open problem is settled in this article. We show how to turn an inductive idea used in the latter two articles into a simple proof technique that allows to relate previous results on arborescence packings. We show how a strong version of Edmonds' theorem on packing spanning arborescences implies Kamiyama, Katoh and Takizawa's result on packing reachability arborescences and how Durand de Gevigney, Nguyen and Szigeti's theorem on matroid-based packing of arborescences implies Kir\'aly's result on matroid-reachability-based packing of arborescences. Finally, we deduce a new result on matroid-reachability-based packing of mixed hyperarborescences from a theorem on matroid-based packing of mixed hyperarborescences due to Fortier et al.. In the last part of the article, we deal with the algorithmic aspects of the problems considered. We first obtain algorithms to find the desired packings of arborescences in all settings and then apply Edmonds' weighted matroid intersection algorithm to also find solutions minimizing a given weight function.