Arithmetic Circuits and Neural Networks for Regular Matroids

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 for the base polytope of regular matroids 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.

Combinatorics (math.CO); Computational Complexity (cs.CC); Discrete Mathematics (cs.DM); Machine Learning (cs.LG); Optimization and Control (math.OC)

arXiv:2511.02406 [math.CO]

(or arXiv:2511.02406v2 [math.CO] for this version)

https://doi.org/10.48550/arXiv.2511.02406 Focus to learn more arXiv-issued DOI via DataCite

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