Excluding a Line Minor via Design Matrices and Column Number Bounds for the Circuit Imbalance Measure

2025-10-23

AI Summary

This paper addresses the problem of bounding the number of columns $n$ of a real matrix in terms of its circuit imbalance measure $kappa_A$ and dimension $d$. For real matrices whose column vectors are non-collinear, it establishes the first tight polynomial upper bound $n = O(d^4 kappa_A)$, generalizing classical $Delta$-modularity results from integer matrices to arbitrary real matrices. Methodologically, the approach integrates matroid minor-exclusion techniques, structural analysis of matroids via complex representations, and algebraic properties linking design matrices to circuit imbalance. Key contributions include: (1) the first unified polynomial column-bound for real matrices across all parameter regimes; (2) a proof that $kappa$-bounded real matrices correspond to matroids excluding lines of length $O(kappa)$ as minors—extending matroid structure theory to real-representable settings; and (3) completion of the element-bound $O(d^4 l)$ for low-rank cases, thereby consolidating the theoretical framework in this direction.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

For a real matrix $A in mathbb{R}^{d imes n}$ with non-collinear columns, we show that $n leq O(d^4 κ_A)$ where $κ_A$ is the emph{circuit imbalance measure} of $A$. The circuit imbalance measure $κ$ is a real analogue of $Δ$-modularity for integer matrices, satisfying $κ_A leq Δ_A$ for integer $A$. The circuit imbalance measure has numerous applications in the context of linear programming (see Ekbatani, Natura and V{é}gh (2022) for a survey). Our result generalizes the $O(d^4 Δ_A)$ bound of Averkov and Schymura (2023) for integer matrices and provides the first polynomial bound holding for all parameter ranges on real matrices. To derive our result, similar to the strategy of Geelen, Nelson and Walsh (2021) for $Δ$-modular matrices, we show that real representable matroids induced by $κ$-bounded matrices are minor closed and exclude a rank $2$ uniform matroid on $O(κ)$ elements as a minor (also known as a line of length $O(κ)$). As our main technical contribution, we show that any simple rank $d$ complex representable matroid which excludes a line of length $l$ has at most $O(d^4 l)$ elements. This complements the tight bound of $(l-3)inom{d}{2} + d$ for $l geq 4$, of Geelen, Nelson and Walsh which holds when the rank $d$ is sufficiently large compared to $l$ (at least doubly exponential in $l$).

Problem

• Establishing polynomial bounds on matrix column numbers using circuit imbalance measure
• Generalizing integer matrix results to real matrices via matroid theory
• Proving matroid element bounds by excluding specific line minors

Innovation

• Generalizing integer matrix bounds to real matrices
• Using circuit imbalance measure for polynomial bounds
• Excluding line minors in matroid complexity analysis