On arborescence packing augmentation in hypergraphs
Pierre Hoppenot, Zoltán Szigeti
TL;DR
The paper addresses augmentation questions for enabling packing of arborescences and their hypergraph analogues, presenting necessary and sufficient conditions for adding arcs or edges to reach target packings. It develops a unified framework based on generalized polymatroids to model feasibility via polyhedra and submodular functions, and proves augmentation lemmas that subsume and extend many prior results in Edmonds–Egy style packing theorems. The main contributions include a broad augmentation theorem for mixed hyperarborescences (with $h$-regular, $(f,g)$-bounded, and $(eta)$-limited constraints), a parallel augmentation result for hyperbranchings, and the undirected counterpart for rooted hyperforests, thereby generalizing a large family of classical results. This framework provides a versatile toolkit for network design problems where robustness requires augmenting a structure to support complex arborescence and forest packings.
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 $(α, β)$-limited packing of mixed hyperarborescences and $h$-regular $(\ell, \ell')$-bordered $(α, β)$-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 $(α, β)$-limited packing of $k$ rooted hyperforests. Our results provide a common generalization of a great number of previous results.