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Arithmetic Circuits and Neural Networks for Regular Matroids

Christoph Hertrich, Stefan Kober, Georg Loho

TL;DR

The paper proves that regular matroids on $n$ elements admit subtraction-free $(+,\times,/)$-circuits of size $O(n^3)$ that compute the basis generating polynomial $f_M=\sum_{B\in\mathcal{B}} x^B$, and that these circuits are constructible in polynomial time from independence oracles. Through tropicalization, this yields uniform $(\max,+,-)$-circuits and ReLU networks of the same size for the weighted basis-maximization problem $\max_{B\in\mathcal{B}}\sum_{e\in B} x_e$, highlighting the neural network interpretation of tropical polynomials. A key consequence is a virtual extended formulation for matroid base polytopes of size $O(n^3)$, improving the previous $O(n^6)$ bound and raising questions about the relative power of subtractive vs. ordinary extended formulations. The proofs combine a refined Seymour decomposition for regular matroids with a generalized star-mesh transformation and Maurer’s matrix-tree theorem to express $f_M$ as a determinant, enabling an inductive, rank-reducing construction that handles the 3-connected case without delving into the hardest $3$-sum step. The results extend to MFMC matroids and provide a bridge between circuit complexity, tropical geometry, and polyhedral optimization, with potential implications for efficient representations of combinatorial polynomials and LP 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.

Arithmetic Circuits and Neural Networks for Regular Matroids

TL;DR

The paper proves that regular matroids on elements admit subtraction-free -circuits of size that compute the basis generating polynomial , and that these circuits are constructible in polynomial time from independence oracles. Through tropicalization, this yields uniform -circuits and ReLU networks of the same size for the weighted basis-maximization problem , highlighting the neural network interpretation of tropical polynomials. A key consequence is a virtual extended formulation for matroid base polytopes of size , improving the previous bound and raising questions about the relative power of subtractive vs. ordinary extended formulations. The proofs combine a refined Seymour decomposition for regular matroids with a generalized star-mesh transformation and Maurer’s matrix-tree theorem to express as a determinant, enabling an inductive, rank-reducing construction that handles the 3-connected case without delving into the hardest -sum step. The results extend to MFMC matroids and provide a bridge between circuit complexity, tropical geometry, and polyhedral optimization, with potential implications for efficient representations of combinatorial polynomials and LP formulations.

Abstract

We prove that there exist uniform -circuits of size to compute the basis generating polynomial of regular matroids on elements. By tropicalization, this implies that there exist uniform -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 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.

Paper Structure

This paper contains 32 sections, 30 theorems, 22 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Contribution.
  3. Implications on neural networks.
  4. Implications in linear programming theory.
  5. Generalization to MFMC matroids.
  6. Proof techniques.
  7. Limitations beyond regular matroids.
  8. Related work.
  9. Preliminaries
  10. Arithmetic circuits and neural networks
  11. Matroid basics
  12. Matroid basics.
  13. Representability.
  14. Regular matroids.
  15. Encoding matroids.
  16. ...and 17 more sections

Key Result

theorem thmcountertheorem

For a regular matroid $M$ with $n$ elements, there is a $(+,\times,/)$-circuit of size $O(n^3)$ computing the basis generating polynomial $f_{M}$. Given an independence oracle of $M$, this circuit can be constructed in polynomial time.

Figures (1)

  • Figure 1: Modification of the partition of a $K_{3,3}$-model, such that both endpoints of the edge $e_1$ are contained in the set $V_4$. The figure on the left shows the original model, and the figure on the right shows the modified model.

Theorems & Definitions (50)

  • theorem thmcountertheorem
  • corollary thmcountercorollary
  • corollary thmcountercorollary
  • proposition thmcounterproposition
  • proposition thmcounterproposition
  • proof
  • lemma thmcounterlemma
  • theorem thmcountertheorem: Seymour seymour1980decomposition, see truemper1992matroid
  • proposition thmcounterproposition: berczi2024reconfiguration
  • lemma thmcounterlemma
  • ...and 40 more