On arborescence packing augmentation in hypergraphs

TL;DR

This paper investigates the minimum arc additions needed to enable specific packings of arborescences and hyperforests in hypergraphs and digraphs, generalizing many previous results in combinatorial optimization.

Contribution

It introduces new augmentation bounds for hypergraph and digraph packings, unifying various prior results in the field.

Findings

Derived bounds for hyperarborescence packings

Solved augmentation problems for hyperbranchings and hyperforests

Unified multiple previous results in combinatorial optimization

Abstract

We deepen the link between two classic areas of combinatorial optimization: augmentation and packing arborescences. We consider the following type of questions: What is the minimum number of arcs to be added to a digraph so that in the resulting digraph there exists some special kind of packing of arborescences? We answer this question for two problems: $h$-regular \textsf{M}-independent-rooted $(f,g)$-bounded $(\alpha, \beta)$-limited packing of mixed hyperarborescences and $h$-regular $(\ell, \ell')$-bordered $(\alpha, \beta)$-limited packing of $k$ hyperbranchings. We also solve the undirected counterpart of the latter, that is the augmentation problem for $h$-regular $(\ell, \ell')$-bordered $(\alpha, \beta)$-limited packing of $k$ rooted hyperforests. Our results provide a common generalization of a great number of previous results.