arXiv subjects

Ingo Stallknecht

Publications and source records attributed to Ingo Stallknecht.

4 recordsLinked to original sources

arXiv2022

On the column number and forbidden submatrices for $Δ$-modular matrices

An integer matrix $\mathbf{A}$ is $Δ$-modular if the determinant of each $\text{rank}(\mathbf{A}) \times \text{rank}(\mathbf{A})$ submatrix of $\mathbf{A}$ has absolute value at most $Δ$. The study of $Δ$-modular matrices appears in the theory of integer programming, where an open conjecture is whether integer programs defined by $Δ$-modular constraint matrices can be solved in polynomial time if $Δ$ is considered constant. The conjecture is only known to hold true when $Δ\in \{1,2\}$. In light of this conjecture, a natural question is to understand structural properties of $Δ$-modular matrices. We consider the column number question -- how many nonzero, pairwise non-parallel columns can a rank-$r$ $Δ$-modular matrix have? We prove that for each positive integer $Δ$ and sufficiently large integer $r$, every rank-$r$ $Δ$-modular matrix has at most $\binom{r+1}{2} + 80Δ^7 \cdot r$ nonzero, pairwise non-parallel columns, which is tight up to the term $80Δ^7$. This is the first upper bound of the form $\binom{r+1}{2} + f(Δ)\cdot r$ with $f$ a polynomial function. Underlying our results is a partial list of matrices that cannot exist in a $Δ$-modular matrix. We believe this partial list may be of independent interest in future studies of $Δ$-modular matrices.

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arXiv2022

Polynomial upper bounds on the number of differing columns of $Δ$-modular integer programs

We study integer-valued matrices with bounded determinants. Such matrices appear in the theory of integer programs (IP) with bounded determinants. For example, Artmann et al. showed that an IP can be solved in strongly polynomial time if the constraint matrix is bimodular, that is, the determinants are bounded in absolute value by two. Determinants are also used to bound the $\ell_1$-distance between IP solutions and solutions of its linear relaxation. One of the first works to quantify the complexity of IPs with bounded determinants was that of Heller, who identified the maximum number of differing columns in a totally unimodular matrix. Each extension of Heller's bound to general determinants has been super-polynomial in the determinants or the number of equations. We provide the first column bound that is polynomial in both values. For integer programs with box constraints, our result gives the first $\ell_1$-distance bound that is polynomial in the determinants and the number of equations. Our result can also be used to derive a bound on the height of Graver basis elements that is polynomial in the determinants and the number of equations. Furthermore, we show a tight bound on the number of differing columns in a bimodular matrix; this is the first tight bound since Heller. Our analysis reveals combinatorial properties of bimodular IPs that may be of independent interest.

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arXiv2021

Notes on $\{a,b,c\}$-Modular Matrices

Let $A \in \mathbb{Z}^{m \times n}$ be an integral matrix and $a$, $b$, $c \in \mathbb{Z}$ satisfy $a \geq b \geq c \geq 0$. The question is to recognize whether $A$ is $\{a,b,c\}$-modular, i.e., whether the set of $n \times n$ subdeterminants of $A$ in absolute value is $\{a,b,c\}$. We will succeed in solving this problem in polynomial time unless $A$ possesses a duplicative relation, that is, $A$ has nonzero $n \times n$ subdeterminants $k_1$ and $k_2$ satisfying $2 \cdot |k_1| = |k_2|$. This is an extension of the well-known recognition algorithm for totally unimodular matrices. As a consequence of our analysis, we present a polynomial time algorithm to solve integer programs in standard form over $\{a,b,c\}$-modular constraint matrices for any constants $a$, $b$ and $c$.

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arXiv2020

Improving proximity bounds using sparsity

We refer to the distance between optimal solutions of integer programs and their linear relaxations as proximity. In 2018, Eisenbrand and Weismantel proved that proximity is independent of the dimension for programs in standard form. We improve their bounds using existing and novel results on the sparsity of integer solutions. We first bound proximity in terms of the largest absolute value of any full-dimensional minor in the constraint matrix, and this bound is tight up to a polynomial factor in the number of constraints. We also give an improved bound in terms of the largest absolute entry in the constraint matrix, after efficiently transforming the program into an equivalent one. Our results are stated in terms of general sparsity bounds, so any new results on sparse solutions immediately improves our work. Generalizations to mixed integer programs are also discussed.

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