Packing branchings under cardinality constraints on their root sets

TL;DR

This paper extends Edmonds' theorem to characterize the packing of branchings in digraphs with specific cardinality constraints on their root sets, using arborescence augmentation.

Contribution

It provides a new characterization for completing arc-disjoint arborescences to meet prescribed root set size constraints.

Findings

Characterization of when branchings can be completed under constraints

Conditions for augmenting existing arborescences to spanning ones

Extension of Edmonds' theorem to constrained root sets

Abstract

Edmonds' fundamental theorem on arborescences characterizes the existence of $k$ pairwise arc-disjoint spanning arborescences with prescribed root sets in a digraph. In this paper, we study the problem of packing branchings in digraphs under cardinality constraints on their root sets by arborescence augmentation. Let $D=(V+x,A)$ be a digraph, $\mathcal{P}=$ $\{I_{1}, \ldots, I_{l} \}$ be a partition of $[k]$, $c_{1}, \ldots, c_{l}, c'_{1}, \ldots, c'_{l}$ be nonnegative integers such that $c_{\alpha} \leq c'_{\alpha}$ for $\alpha \in [l]$, $F_{1}, \ldots, F_{k}$ be $k$ arc-disjoint $x$-arborescences in $D$ such that $\sum_{i \in I_{\alpha}}d_{F_{i}}^{+}(x)$ $\leq c'_{\alpha}$ for $\alpha \in [l]$. We give a characterization on when $F_{1}, \ldots, F_{k}$ can be completed to arc-disjoint spanning $x$-arborescences $F^{*}_{1}, \ldots, F^{*}_{k}$ such that for any $\alpha \in [l]$, $ c_{\alpha} \leq \sum_{i \in I_{\alpha}}d^{+}_{F^{*}_{i}}(x)$ $ \leq c'_{\alpha}$.