Matroid products via submodular coupling

TL;DR

This paper introduces a new concept called matroid coupling, inspired by probability theory, which always exists for any two submodular functions and leads to a novel operation on matroids with algorithmic and theoretical implications.

Contribution

It defines the notion of coupling for matroids and submodular functions, proving existence, constructibility, and exploring applications in matroid theory and inequalities.

Findings

A coupling always exists for any two submodular functions.

Matroid coupling can be constructed algorithmically with polynomial oracle calls.

Connections established between tensor products, Ingleton's inequality, and matroid representability.

Abstract

The study of matroid products traces back to the 1970s, when Lov\'asz and Mason studied the existence of various types of matroid products with different strengths. Among these, the tensor product is arguably the most important, which can be considered as an extension of the tensor product from linear algebra. However, Las Vergnas showed that the tensor product of two matroids does not always exist. Over the following four decades, matroid products remained surprisingly underexplored, regaining attention only in recent years due to applications in tropical geometry and the limit theory of matroids. In this paper, inspired by the concept of coupling in probability theory, we introduce the notion of coupling for matroids -- or, more generally, for submodular set functions. This operation can be viewed as a relaxation of the tensor product. Unlike the tensor product, however, we prove that a coupling always exists for any two submodular functions and can be chosen to be increasing if the original functions are increasing. As a corollary, we show that two matroids always admit a matroid coupling, leading to a novel operation on matroids. Our construction is algorithmic, providing an oracle for the coupling matroid through a polynomial number of oracle calls to the original matroids. We apply this construction to derive new necessary conditions for matroid representability and establish connection between tensor products and Ingleton's inequality. Additionally, we verify the existence of set functions that are universal with respect to a given property, meaning any set function over a finite domain with that property can be obtained as a quotient.