On complete classes of valuated matroids

TL;DR

This paper characterizes a broad class of valuated matroids called R-minor valuated matroids, demonstrates their properties, and refutes a conjecture linking all gross substitute valuations to matroid-based functions.

Contribution

It introduces R-minor valuated matroids, shows their closure properties, and provides counterexamples to the Matroid Based Valuation Conjecture.

Findings

R-minor valuated matroids include indicator functions of matroids

Not all valuated matroids are R-minor based, as shown by sparse paving matroids

The Matroid Based Valuation Conjecture is refuted

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.