Efficient Algorithms for Minimal Matroid Extensions and Irreducible Decompositions of Circuit Varieties

2025-04-23

AI Summary

This work addresses the minimal irreducible decomposition of matroid circuit ideals—i.e., identifying minimal extensions covering a given matroid within the dependency poset. We provide the first combinatorial characterization that precisely links minimal extensions to cover relations in the dependency poset, thereby establishing a computable framework for circuit ideal decomposition. Our method integrates combinatorial matroid theory, poset algorithms, and techniques from computational algebraic geometry to efficiently identify minimal extensions and perform algebraic decomposition of circuit ideals. We present, for the first time, complete minimal irreducible decompositions for several classical matroids: the Vámos matroid, the Steiner system S(3,4,8), projective and affine planes, the Fano dual, and the dual of the K_{3,3} graphic matroid. These results advance the algorithmic and systematic analysis of matroid structure through algebraic decomposition.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We introduce an efficient method for decomposing the circuit variety of a given matroid $M$, based on an algorithm that identifies its minimal extensions. These extensions correspond to the smallest elements above $M$ in the poset defined by the dependency order. We apply our algorithm to several classical configurations: the V'amos matroid, the unique Steiner quadruple system $S(3,4,8)$, the 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 compute the minimal irreducible decomposition of their circuit varieties.

Problem

• Efficient decomposition of circuit varieties in matroids
• Identification of minimal matroid extensions via dependency order
• Application to classical configurations like Vámos matroid and Fano dual

Innovation

• Efficient algorithm for minimal matroid extensions
• Decomposes circuit varieties of matroids
• Applies to classical matroid configurations