Matroid-reachability-based decomposition into arborescences

TL;DR

This paper addresses the complex problem of decomposing matroid-reachability-based packings of arborescences, extending previous work to include $( ext{ell}, ext{ell}')$-limited packings and broader graph structures.

Contribution

It introduces a solution to the decomposition problem for matroid-reachability-based packings, generalizing to limited packings and directed hypergraphs.

Findings

Solved the decomposition problem for matroid-reachability-based packings.

Extended the framework to include $( ext{ell}, ext{ell}')$-limited packings.

Applied results to branchings and directed hypergraphs.

Abstract

The problem of matroid-reachability-based packing of arborescences was solved by Kir\'aly. Here we solve the corresponding decomposition problem that turns out to be more complicated. The result is obtained from the solution of the more general problem of matroid-reachability-based $(\ell,\ell')$-limited packing of arborescences where we are given a lower bound $\ell$ and an upper bound $\ell'$ on the total number of arborescences in the packing. The problem is considered for branchings and in directed hypergraphs as well.