Matroid Products via Submodular Coupling

2024-11-04

Symposium on the Theory of Computing

AI Summary

The classical problem of the nonexistence of tensor products of matroids—first identified by Las Vergnas in the 1970s—has remained unresolved for decades. Method: This paper introduces “submodular coupling” as a universal relaxation framework. We formally define and rigorously prove that any two submodular functions admit a monotonicity-preserving coupling; further, leveraging probabilistic coupling theory, submodular optimization, and the matroid oracle model, we construct the first algorithmically realizable matroid coupling operation. Contributions: (1) We completely overcome the four-decade-old existence bottleneck for tensor products, enabling constructive combinatorial operations for arbitrary matroid pairs; (2) We derive a new necessary condition for matroid representability; (3) We establish an intrinsic connection between tensor products and the Ingleton inequality, and confirm the existence of generic set functions under specific conditions.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

The study of matroid products traces back to the 1970s, when Lovász 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, information theory, 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. In addition, 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.

Problem

• Existence of matroid tensor products is not guaranteed
• Introduce submodular coupling as a relaxation of tensor product
• Apply coupling to matroid representability and Ingleton's inequality

Innovation

• Introduces submodular coupling for matroids
• Proves coupling always exists algorithmically
• Applies coupling to matroid representability conditions