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On generic $Δ$-modular integer matrices with two rows

Björn Kriepke, Matthias Schymura

TL;DR

This work resolves the column-number problem for Δ-modular matrices with two rows by reducing the general case to a structured family $M(a,b)$ of type $m$ via lattice-width considerations, and then bounding the column count through a tailored linear program tied to Euler totients. For large Δ (and for a concrete range up to 1{,}550), the maximal number of columns is shown to be $g(Δ,2) = 2\\left\\lfloor\\frac{Δ+5}{6}\\right\\;2 + 2\\left\\lfloor\\frac{Δ+1}{3}\\right\\;2 + 2$, i.e. $g(Δ,2) = Δ + c(Δ)$ with $c(Δ)$ determined by $Δ mod 6$, and the function is eventualy even and non-decreasing. The analysis yields upper bounds for the size of $M(a,b)$ in small types and demonstrates that the bound is tight in several congruence classes; it also yields a complete characterization of the unique rank-two excluded minor for the class of Δ-submodular matroids in the studied regimes. While the approach provides strong asymptotics and structural insight, the authors note substantial computational limitations and discuss avenues for refining the constants and expanding the range where the exact formula applies. Overall, the results advance understanding of Δ-modular matrices, with implications for matroid theory and related algorithmic questions in integer programming.

Abstract

The column number question asks for the maximal number of columns of an integer matrix with the property that all its rank size minors are bounded by a fixed parameter $Δ$ in absolute value. Polynomial upper bounds have been proved in various settings in recent years, with consequences for algorithmic questions in integer linear programming and matroid theory. In this paper, we focus on the exact determination of the maximal column number of such matrices with two rows and no vanishing $2$-minors. We prove that for large enough $Δ$, this number is a quasi-linear function, non-decreasing and always even. Such basic structural properties of column number functions are barely known, but expected to hold in other settings as well. Moreover, our results identify the unique excluded (co)rank two minors for the class of matroids that are representable as a $Δ$-submodular matrix.

On generic $Δ$-modular integer matrices with two rows

TL;DR

This work resolves the column-number problem for Δ-modular matrices with two rows by reducing the general case to a structured family of type via lattice-width considerations, and then bounding the column count through a tailored linear program tied to Euler totients. For large Δ (and for a concrete range up to 1{,}550), the maximal number of columns is shown to be , i.e. with determined by , and the function is eventualy even and non-decreasing. The analysis yields upper bounds for the size of in small types and demonstrates that the bound is tight in several congruence classes; it also yields a complete characterization of the unique rank-two excluded minor for the class of Δ-submodular matroids in the studied regimes. While the approach provides strong asymptotics and structural insight, the authors note substantial computational limitations and discuss avenues for refining the constants and expanding the range where the exact formula applies. Overall, the results advance understanding of Δ-modular matrices, with implications for matroid theory and related algorithmic questions in integer programming.

Abstract

The column number question asks for the maximal number of columns of an integer matrix with the property that all its rank size minors are bounded by a fixed parameter in absolute value. Polynomial upper bounds have been proved in various settings in recent years, with consequences for algorithmic questions in integer linear programming and matroid theory. In this paper, we focus on the exact determination of the maximal column number of such matrices with two rows and no vanishing -minors. We prove that for large enough , this number is a quasi-linear function, non-decreasing and always even. Such basic structural properties of column number functions are barely known, but expected to hold in other settings as well. Moreover, our results identify the unique excluded (co)rank two minors for the class of matroids that are representable as a -submodular matrix.

Paper Structure

This paper contains 9 sections, 12 theorems, 92 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Auxiliary estimates on sums involving the totient function
  3. A reduction to a class of well-structured matrices
  4. The maximal size of the matrices M(a,b) for small m
  5. Matrices M(a,b) of type 2
  6. Matrices M(a,b) of type 3
  7. An upper bound on the value of (\ref{['lp:primal_version']}) for large dimensions
  8. Closing the gap and proof of Theorem \ref{['thm:upper_bound_on_g_Delta2']}
  9. Improvements and limitations of our approach

Key Result

Theorem 1.1

For every $385 \leq \Delta \leq 1\,550$ and every $\Delta\geq 10^8$, we have

Figures (1)

  • Figure 1: The shape of (near) optimal solutions $Y$ to the dual problem \ref{['lp:dual_version']}.

Theorems & Definitions (28)

  • Theorem 1.1
  • Corollary 1.3
  • Lemma 2.1
  • proof
  • Definition 3.1
  • Example 3.2
  • Lemma 3.3: cf. kriepkeSizeIntegerPrograms2025
  • proof
  • Proposition 3.4
  • proof
  • ...and 18 more