Packing of mixed hyperarborescences with flexible roots via matroid intersection

TL;DR

This paper characterizes the conditions under which mixed hypergraphs can be packed with multiple flexible roots using matroid intersection, extending previous results and providing an algorithm for minimum weight solutions.

Contribution

It generalizes existing hypergraph packing results by introducing a matroid intersection approach for flexible root packings and offers an efficient algorithm.

Findings

Characterization of hypergraphs admitting flexible root packings

Extension of Edmonds' construction to mixed hypergraphs

Algorithm for minimum weight packings

Abstract

Given a mixed hypergraph $\mathcal{F}=(V,\mathcal{A}\cup \mathcal{E})$, functions $f,g:V\rightarrow \mathbb{Z}_+$ and an integer $k$, a packing of $k$ spanning mixed hyperarborescences is called $(k,f,g)$-flexible if every $v \in V$ is the root of at least $f(v)$ and at most $g(v)$ of the mixed hyperarborescences. We give a characterization of the mixed hypergraphs admitting such packings. This generalizes results of Frank and, more recently, Gao and Yang. Our approach is based on matroid intersection, generalizing a construction of Edmonds. We also obtain an algorithm for finding a minimum weight solution to the above mentioned problem.