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A characterization of positive spanning sets with ties to strongly connected digraphs

Denis Cornaz, Sébastien Kerleau, Clément W. Royer

TL;DR

This paper develops a comprehensive decomposition framework linking positive spanning sets (PSSs) to digraph theory by extending ear-decomposition concepts from graphs to matrices. It shows that a network matrix is a PSS for $\mathbb{R}^n$ if and only if the associated digraph is strongly edge-connected, and then generalizes this to a matrix-theoretic ear decomposition using acyclic and circuit matrices. The central result establishes a structural equivalence: a nonzero matrix $\mathbf{M}$ is a PSS of $\mathbb{R}^n$ if and only if it is structurally equivalent to an IN matrix with $\ell=n$ and $k>0$; if not, it is structurally equivalent to an INA matrix. This leads to a powerful certificate-based approach for recognizing PSSs and yields a characterization of positive bases via critical structures, with detailed descriptions for near-extreme sizes and the connection to minimally strongly connected digraphs. The findings open paths for applying these certificates in optimization and for further exploration of orthogonalized or k-spanning variants of positive bases.

Abstract

Positive spanning sets (PSSs) are families of vectors that span a given linear space through non-negative linear combinations. Despite certain classes of PSSs being well understood, a complete characterization of PSSs remains elusive. In this paper, we explore a relatively understudied relationship between positive spanning sets and strongly edge-connected digraphs, in that the former can be viewed as a generalization of the latter. We leverage this connection to define a decomposition structure for positive spanning sets inspired by the ear decomposition from digraph theory.

A characterization of positive spanning sets with ties to strongly connected digraphs

TL;DR

This paper develops a comprehensive decomposition framework linking positive spanning sets (PSSs) to digraph theory by extending ear-decomposition concepts from graphs to matrices. It shows that a network matrix is a PSS for if and only if the associated digraph is strongly edge-connected, and then generalizes this to a matrix-theoretic ear decomposition using acyclic and circuit matrices. The central result establishes a structural equivalence: a nonzero matrix is a PSS of if and only if it is structurally equivalent to an IN matrix with and ; if not, it is structurally equivalent to an INA matrix. This leads to a powerful certificate-based approach for recognizing PSSs and yields a characterization of positive bases via critical structures, with detailed descriptions for near-extreme sizes and the connection to minimally strongly connected digraphs. The findings open paths for applying these certificates in optimization and for further exploration of orthogonalized or k-spanning variants of positive bases.

Abstract

Positive spanning sets (PSSs) are families of vectors that span a given linear space through non-negative linear combinations. Despite certain classes of PSSs being well understood, a complete characterization of PSSs remains elusive. In this paper, we explore a relatively understudied relationship between positive spanning sets and strongly edge-connected digraphs, in that the former can be viewed as a generalization of the latter. We leverage this connection to define a decomposition structure for positive spanning sets inspired by the ear decomposition from digraph theory.

Paper Structure

This paper contains 19 sections, 24 theorems, 31 equations, 3 figures.

Table of Contents

  1. Keywords
  2. 2020 Mathematics Subject Classification
  3. Introduction
  4. Notations
  5. Digraphs and ear decomposition
  6. Positive spanning sets
  7. Definition and characterization
  8. Connection with digraphs
  9. Ear decomposition of positive spanning sets
  10. Acyclic and circuit matrices
  11. A new certificate for positive spanning sets
  12. On the one hand, suppose that $\mathbf{M}$ is a PSS of $\mathbb{R}^n$.
  13. On the other hand, suppose that $\mathbf{M}$ is a not a PSS of $\mathbb{R}^n$.
  14. Applications to positive bases
  15. Positive bases and digraphs
  16. ...and 4 more sections

Key Result

Proposition 2.1

A connected digraph $G=(V,A)$ is not strongly edge-connected if and only if there exists an oriented cut of $G$, i.e. a set $\tilde{A}\subset A$ and two vertex-disjoint subgraphs $G_1=(V_1,A_1)$ and $G_2=(V_2,A_2)$ of $G$ with $V_1\neq \emptyset$, $V_2 \neq \emptyset$, such that

Figures (3)

  • Figure 1: An ear decomposition $(G_1,G_2,G_3)$ where $G_1=(\{v_1\},\emptyset)$, $G_2$ is obtained by adding the blue arcs and vertices and $G_3$ is obtained by adding the red arcs and vertices.
  • Figure 2: Digraph, spanning tree (plain edges) and network matrix. The arcs are numbered to match the column ordering
  • Figure :

Theorems & Definitions (40)

  • Proposition 2.1
  • Definition 2.1: Ear and ear decomposition
  • Theorem 2.1
  • Definition 3.1: Positive span and positive spanning set
  • Proposition 3.1
  • Proposition 3.2
  • Definition 3.2: Structural equivalence
  • Proposition 3.3
  • Definition 3.3: Network matrix
  • Proposition 3.4
  • ...and 30 more