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Matroid-reachability-based decomposition into arborescences

Florian Hörsch, Benjamin Peyrille, Zoltán Szigeti

TL;DR

The paper generalizes matroid-based and matroid-reachability-based packing of arborescences to provide decomposition results via an $(\ell,\ell')$-limited framework, applicable to directed graphs and hypergraphs. It develops a robust polyhedral (TDI) description using biset/petal structures and OW-laminar families, including extended-graph constructions to handle packing bounds. The main contributions are necessary and sufficient conditions for existence and decomposition of such packings (with both lower and upper bounds) and their extension to hyperarborescences, supported by detailed proofs and polyhedral duality arguments. The results unify and extend prior theorems (Edmonds, Kamiyama-Katoh-Takizawa, Király, Gao-Yang) and yield polynomial-time algorithms given a matroid rank oracle, enabling practical decomposition in network design and rigidity contexts.

Abstract

The problem of matroid-reachability-based packing of arborescences was solved by Király. 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.

Matroid-reachability-based decomposition into arborescences

TL;DR

The paper generalizes matroid-based and matroid-reachability-based packing of arborescences to provide decomposition results via an -limited framework, applicable to directed graphs and hypergraphs. It develops a robust polyhedral (TDI) description using biset/petal structures and OW-laminar families, including extended-graph constructions to handle packing bounds. The main contributions are necessary and sufficient conditions for existence and decomposition of such packings (with both lower and upper bounds) and their extension to hyperarborescences, supported by detailed proofs and polyhedral duality arguments. The results unify and extend prior theorems (Edmonds, Kamiyama-Katoh-Takizawa, Király, Gao-Yang) and yield polynomial-time algorithms given a matroid rank oracle, enabling practical decomposition in network design and rigidity contexts.

Abstract

The problem of matroid-reachability-based packing of arborescences was solved by Király. 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 -limited packing of arborescences where we are given a lower bound and an upper bound on the total number of arborescences in the packing. The problem is considered for branchings and in directed hypergraphs as well.
Paper Structure (19 sections, 31 theorems, 73 equations, 3 figures)

This paper contains 19 sections, 31 theorems, 73 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. Total dual integrality
  4. Packings in directed graphs
  5. Packing arborescences
  6. Matroid-based packing of arborescences
  7. New results on matroid-based packing of arborescences
  8. Matroid-reachability-based packing of arborescences
  9. New results on matroid-reachability-based packing of arborescences
  10. Packing of branchings
  11. Packings in directed hypergraphs
  12. Proofs
  13. Proof of Theorem \ref{['bboboiboreach1']}
  14. Proof of Theorem \ref{['bboboiboreach']}
  15. Algorithmic aspects
  16. ...and 4 more sections

Key Result

Theorem 1

Let $Ax\le b$ be a TDI-system where $A$ is an integral matrix and $b$ is an integral vector. If $\max\{c^Tx: Ax\le b\}$ is finite, then it has an integral optimal solution.

Figures (3)

  • Figure 1: Our results (in bold) on packings of arborescences and their implications.
  • Figure 2: Examples of petals. The dashed areas correspond to the cores of the petals.
  • Figure 3: Instance of \ref{['bboboibo']}.

Theorems & Definitions (58)

  • Theorem 1: Edmonds, Giles TDI, Corollary 22.1b in lexbook
  • Theorem 2: Frank, Theorem 5.3 in frtdithm
  • Theorem 3: Schrijver, (41) in lexbook
  • Theorem 4: Edmonds Egy
  • Theorem 5: Kamiyama, Katoh, Takizawa japan
  • Theorem 6: Durand de Gevigney, Nguyen, Szigeti DdGNSz
  • Theorem 7: Szigeti szigrooted
  • Theorem 8
  • Theorem 9
  • Claim 1
  • ...and 48 more