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

For a real matrix A ∈ R d×n with non-collinear columns, we show that n ≤ O(d 4 κ A ) where κ A is the circuit imbalance measure of A. The circuit imbalance measure κ is a real analogue of ∆-modularity for integer matrices, satisfying κ A ≤ ∆ A for integer A. The circuit imbalance measure has numerous applications in the context of linear programming (see Ekbatani, Natura and Végh (2022) for a survey). Our result generalizes the O(d 4 ∆ 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 ∆-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) d 2 + d for l ≥ 4, of Geelen, Nelson and Walsh which holds when the rank d is sufficiently large compared to l (at least doubly exponential in l).

Our proof of the above relies on an improvement of a Sylvester-Gallai type theorem of Dvir, Saraf and Wigderson (2014). Refining their design matrix technique, we show that for any full dimensional set of n points in C d there always exists a point that lies on at least (1 -4 d )n many distinct lines (the constant 4 is improved from 12). The excluded minor bound follows by inductively applying this result to find good elements to contract in the matroid, where the improved constant reduces the dependence on d from d 12 to d 4 . Interestingly, by relying on geometric techniques, our proof avoids the use of any difficult matroid machinery.