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

TL;DR

The paper studies the circuit imbalance measure $\kappa_{\mathbf{A}}$ for real matrices $\mathbf{A} \in \mathbb{R}^{d\times n}$ with non-collinear columns, proving a polynomial bound on the number of columns $n$ in terms of the rank $d$ and $\kappa_{\mathbf{A}}$, namely $n \le \pi d^4 \kappa_{\mathbf{A}}$ (and $O(d^4\kappa_{\mathbf{A}})$ up to constants). The authors extend results for $\Delta$-modular integer matrices to the real setting by showing the class of $\kappa$-bounded real matroids is minor-closed and excludes a rank-2 uniform minor $U_{2,l}$ with $l = O(\kappa)$, reducing the problem to simple rank-2 minors and a geometric packing argument on lines. A central technical contribution gives a polynomial bound $n \le O(d^4 l)$ for simple rank-$d$ complex representable matroids excluding a line of length $l$, which, via matroid reductions, yields the main real bound. The paper also develops a stronger rank bound for design matrices: for a $(q,k,t)$-design matrix, $\operatorname{rank}(\mathbf{A}) \ge n - \frac{nt(q-1)}{k}$, derived through matrix scaling and an analysis of $M = \mathbf{B}^*\mathbf{B}$, with consequences for proximity bounds to integer solutions (via Graver bases) and refined incidence-plane results. Overall, these results generalize prior $O(d^4\Delta)$ bounds to all parameter ranges for real matrices and link matroid structure, design matrices, and LP/IP proximity in a unified framework.

Abstract

For a real matrix $A \in \mathbb{R}^{d \times n}$ 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 $(l-3)\binom{d}{2} + d$ 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 $l$).

Figures (3)

• Figure 1.1: Left: all the lines defined by some points $S \subseteq \mathbb{R}^2$. Right: $\mathop{\mathrm{maxlines}}\nolimits(S)=4$ attained by $u$.
• Figure 3.1: Visualization of many $(d-1)$-dimensional flats containing $G$ for $d=3$.
• Figure 6.1: Special vs. ordinary lines

Theorems & Definitions (43)

• Theorem 1.1
• Lemma 1.2
• Theorem 1.3: geelen2024excluding
• Theorem 1.4: geelen2021excluding
• Theorem 1.5
• Corollary 1.6
• Definition 1.7
• Lemma 1.8
• Theorem 1.9
• Theorem 1.10: dvir2014improved