Arithmetic Circuits and Neural Networks for Regular Matroids

TL;DR

This paper demonstrates that uniform arithmetic circuits and neural networks can efficiently compute basis generating polynomials of regular matroids, leading to new insights in optimization and extended formulations.

Contribution

It introduces the first polynomial-size uniform circuits and neural networks for regular matroids and applies this to improve linear programming formulations.

Findings

Existence of $O(n^3)$ size uniform circuits for regular matroids

Neural networks of the same size for weighted basis maximization

Difference of extended formulations can outperform individual formulations

Abstract

We prove that there exist uniform $(+,\times,/)$-circuits of size $O(n^3)$ to compute the basis generating polynomial of regular matroids on $n$ elements. By tropicalization, this implies that there exist uniform $(\max,+,-)$-circuits and ReLU neural networks of the same size for weighted basis maximization of regular matroids. As a consequence in linear programming theory, we obtain a first example where taking the difference of two extended formulations can be more efficient than the best known individual extended formulation of size $O(n^6)$ by Aprile and Fiorini. Such differences have recently been introduced as virtual extended formulations. The proof of our main result relies on a fine-tuned version of Seymour's decomposition of regular matroids which allows us to identify and maintain graphic substructures to which we can apply a local version of the star-mesh transformation.