Basic Packing of Arborescences

TL;DR

This paper characterizes digraphs that can be packed with arborescences under matroid constraints, extending classical results and providing polynomial-time algorithms for minimum cost packings.

Contribution

It extends Edmonds' and Katoh-Tanigawa's results to directed graphs with matroid constraints, offering a complete characterization and efficient algorithms.

Findings

Characterization of digraphs with arborescence packings under matroid constraints

Complete description of the convex hull of incidence vectors of basic packings

Polynomial-time algorithm for minimum cost arborescence packings

Abstract

We provide the directed counterpart of a slight extension of Katoh and Tanigawa's result on rooted-tree decompositions with matroid constraints. Our result characterises digraphs having a packing of arborescences with matroid constraints. It is a proper extension of Edmonds' result on packing of spanning arborescences and implies - using a general orientation result of Frank - the above result of Katoh and Tanigawa. We also give a complete description of the convex hull of the incidence vectors of the basic packings of arborescences and prove that the mimimum cost version of the problem can be solved in polynomial time.