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Efficient Algorithms for Maximal Matroid Degenerations and Irreducible Decompositions of Circuit Varieties

Emiliano Liwski, Fatemeh Mohammadi, Rémi Prébet

TL;DR

This work develops an efficient framework for decomposing circuit varieties of matroids by reducing the problem to maximal matroid degenerations under the weak order, encoded via labeled hypergraphs and rank-submodularity. It then builds a rank-four optimized algorithm and stratifies the search to manage combinatorial explosion, delivering explicit irreducible decompositions for classical rank-four configurations such as the Vámos matroid, the Steiner system S(3,4,8), the dual Fano, and the dual of K_{3,3}. The results yield non-redundant decompositions and connect to conditional independence models and rigidity theory, with open conjectures for Steiner-system–based families and symmetry-driven simplifications. An open-source implementation accompanies the work, enabling practical computation of circuit-variety decompositions beyond current symbolic-algebra capabilities.

Abstract

Matroid theory provides a unifying framework for studying dependence across combinatorics, geometry, and applications ranging from rigidity to statistics. In this work, we study circuit varieties of matroids, defined by their minimal dependencies, which play a central role in modeling determinantal varieties, rigidity problems, and conditional independence relations. We introduce an efficient computational strategy for decomposing the circuit variety of a given matroid $M$, based on an algorithm that identifies its maximal degenerations. These degenerations correspond to the largest matroids lying below $M$ in the weak order. Our framework yields explicit and computable decompositions of circuit varieties that were previously out of reach for symbolic or numerical algebra systems. We apply our strategy to several classical configurations, including the Vámos matroid, the unique Steiner quadruple system $S(3,4,8)$, projective and affine planes, the dual of the Fano matroid, and the dual of the graphic matroid of $K_{3,3}$. In each case, we successfully compute the minimal irreducible decomposition of their circuit varieties.

Efficient Algorithms for Maximal Matroid Degenerations and Irreducible Decompositions of Circuit Varieties

TL;DR

This work develops an efficient framework for decomposing circuit varieties of matroids by reducing the problem to maximal matroid degenerations under the weak order, encoded via labeled hypergraphs and rank-submodularity. It then builds a rank-four optimized algorithm and stratifies the search to manage combinatorial explosion, delivering explicit irreducible decompositions for classical rank-four configurations such as the Vámos matroid, the Steiner system S(3,4,8), the dual Fano, and the dual of K_{3,3}. The results yield non-redundant decompositions and connect to conditional independence models and rigidity theory, with open conjectures for Steiner-system–based families and symmetry-driven simplifications. An open-source implementation accompanies the work, enabling practical computation of circuit-variety decompositions beyond current symbolic-algebra capabilities.

Abstract

Matroid theory provides a unifying framework for studying dependence across combinatorics, geometry, and applications ranging from rigidity to statistics. In this work, we study circuit varieties of matroids, defined by their minimal dependencies, which play a central role in modeling determinantal varieties, rigidity problems, and conditional independence relations. We introduce an efficient computational strategy for decomposing the circuit variety of a given matroid , based on an algorithm that identifies its maximal degenerations. These degenerations correspond to the largest matroids lying below in the weak order. Our framework yields explicit and computable decompositions of circuit varieties that were previously out of reach for symbolic or numerical algebra systems. We apply our strategy to several classical configurations, including the Vámos matroid, the unique Steiner quadruple system , projective and affine planes, the dual of the Fano matroid, and the dual of the graphic matroid of . In each case, we successfully compute the minimal irreducible decomposition of their circuit varieties.

Paper Structure

This paper contains 31 sections, 24 theorems, 94 equations, 11 figures, 6 algorithms.

Table of Contents

  1. Introduction
  2. Motivation
  3. Outline and our results
  4. Preliminaries
  5. Matroids
  6. Paving, nilpotent and inductively connected matroids
  7. Decomposing using maximal matroid degenerations
  8. Reduction to the maximal matroid degenerations
  9. Reduction to labeled hypergraphs
  10. Reduction to simple matroids by removing loops and double points
  11. Algorithm for identifying maximal matroid degenerations
  12. Comparing matroids
  13. Algorithm for identifying $\textup{max}_{ \hbox{$\,\preceq$}}\!\left(\Delta\right)$
  14. Termination of Algorithm \ref{['algo 21']}.
  15. Correctness of Algorithm \ref{['algo 21']}.
  16. ...and 16 more sections

Key Result

Proposition 2.3

The rank function of a matroid is submodular, meaning that for any subsets $A$ and $B$ of the ground set, the following inequality holds:

Figures (11)

  • Figure 1: Uniform matroid $U_{2,7}$ and Fano plane.
  • Figure 2: Examples of rank 3 matroids.
  • Figure 3: (a) Uniform matroid $U_{2,7}$; (b) Matroid $M_{\textup{Fano}}(6)$; (c) $A_{1}'$; (d) $B_{1}'$.
  • Figure 4: Matroids in $\textup{max}_{ \hbox{$\,\preceq$}}\!\left(\Delta\right)$ for the hypergraph $\Delta$ from Example \ref{['example: hypergraph delta']}.
  • Figure 5: Matroids in $\textup{max}_{ \hbox{$\,\preceq$}}\!\left(\Lambda\right)$ for the hypergraph $\Lambda$ from Example \ref{['ex: hypergraph lambda']}: (a) $M_1$ (Left); (b) $M_2$ (Center); (c) $M_3$ (Right).
  • ...and 6 more figures

Theorems & Definitions (103)

  • Definition 1.1
  • Example 1.2
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3: Oxley
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Example 2.7
  • Definition 2.8
  • ...and 93 more