On complete classes of valuated matroids

TL;DR

A rich class of valuated matroids is characterized, called R-minor valuatedMatroids, that includes the indicator functions of matroIDS, and is closed under operations such as taking minors, duality, and induction by network.

Abstract

We characterize a rich class of valuated matroids, called R-minor valuated matroids that includes the indicator functions of matroids, and is closed under operations such as taking minors, duality, and induction by network. We exhibit a family of valuated matroids that are not R-minor based on sparse paving matroids. Valuated matroids are inherently related to gross substitute valuations in mathematical economics. By the same token we refute the Matroid Based Valuation Conjecture by Ostrovsky and Paes Leme (Theoretical Economics 2015) asserting that every gross substitute valuation arises from weighted matroid rank functions by repeated applications of merge and endowment operations. Our result also has implications in the context of Lorentzian polynomials: it reveals the limitations of known construction operations.

Figures (11)

• Figure 4: A bipartite graph $G = (V\cup W, U;E)$ with edge weights $c\in {{\mathbb{R}}}^E$ and matroid ${\mathcal{M}}$ on vertex set $U$. This gives rise to an R-minor valuated matroid on $V$, as described in Definition \ref{['def:intro-R-minor']}.
• Figure 5: The bipartite graph realising the transversally valuated matroid from Example \ref{['ex:transversal+VM']}. The dashed edges have weight zero and the solid edges have weight one.
• Figure 6: Given a valuated matroid $g$ on $U$ and $w \in ({{\mathbb{R}}} \cup \{-\infty\})^U$, the principal extension $g^w$ is realized as the induction of $g$ via the above bipartite graph, as given in Remark \ref{['rem:principle+extension']}. The dashed edges are weighted zero, while the solid edges $(p,u)$ are weighted $w_u$.
• Figure 7: The inclusion relationship between classes of valuated matroids.
• Figure 8: Two representations of the Snowflake, defined in Example \ref{['ex:snowflake']}. The left is a valuated gammoid representation, where the element $7$ is contracted. The right is an R-induced representation with induced matroid $U_{2,3}$. All edges are weighted zero.

Theorems & Definitions (107)

• definition 1.1: R-minor, R-induced
• definition 1.1
• Theorem 1.2: Main
• definition 2.1
• remark 2.2
• Example 2.3
• Example 2.4
• Example 2.5
• definition 2.6: Valuated matroid union
• Example 2.7