Augmenting a hypergraph to have a matroid-based $(f,g)$-bounded $(α,β)$-limited packing of rooted hypertrees
Pierre Hoppenot, Zoltán Szigeti
TL;DR
This work advances the theory of packing rooted trees and hypertrees under matroid constraints by developing a partition-based submodular/supermodular framework, including trimming techniques to move between hypergraphs and graphs. It provides a unified characterization for ${\sf M}$-based $(f,g)$-bounded and $(\alpha,\beta)$-limited packings of rooted trees and hypertrees, and extends these results to hypergraph augmentations. Building on Nash-Williams–Tutte and Katoh–Tanigawa, the authors deploy matroid intersections and generalized partition matroids to obtain necessary and sufficient conditions, with broad applicability to hypergraph rigidity-like settings and related combinatorial packing problems. The augmentation results offer practical guidance for adding edges to achieve desired packings in graphs and hypergraphs, highlighting the versatility of the approach for future combinatorial optimization applications. Key concepts include $e_E(\mathcal{P})$ bounds, partition uncrossing, the trimming lemma, and matroid-based rank relaxations.$
Abstract
The aim of this paper is to further develop the theory of packing trees in a graph. We first prove the classic result of Nash-Williams \cite{NW} and Tutte \cite{Tu} on packing spanning trees by adapting Lovász' proof \cite{Lov} of the seminal result of Edmonds \cite{Egy} on packing spanning arborescences in a digraph. Our main result on graphs extends the theorem of Katoh and Tanigawa \cite{KT} on matroid-based packing of rooted trees by characterizing the existence of such a packing satisfying the following further conditions: for every vertex $v$, there are a lower bound $f(v)$ and an upper bound $g(v)$ on the number of trees rooted at $v$ and there are a lower bound $α$ and an upper bound $β$ on the total number of roots. We also answer the hypergraphic version of the problem. Furthermore, we are able to solve the augmentation version of the latter problem, where the goal is to add a minimum number of edges to have such a packing. The methods developed in this paper to solve these problems may have other applications in the future.