Characterization of matrices with bounded Graver bases and depth parameters and applications to integer programming

TL;DR

This paper investigates the structural properties of matrices with bounded Graver bases and depth parameters, providing new algorithms for transforming matrices into sparse, structured forms to improve fixed parameter tractability in integer programming.

Contribution

It characterizes when a matrix can be transformed into a sparse, structured form based on matroid properties and bounds the Graver basis norm using circuit norms, leading to new parameterized algorithms.

Findings

Bounded the $ ext{l}_1$-norm of the Graver basis by circuit norms.

Provided structural characterizations for the existence of sparse row-equivalent matrices.

Developed algorithms to find row-equivalent matrices with small depth and complexity.

Abstract

An intensive line of research on fixed parameter tractability of integer programming is focused on exploiting the relation between the sparsity of a constraint matrix $A$ and the norm of the elements of its Graver basis. In particular, integer programming is fixed parameter tractable when parameterized by the primal tree-depth and the entry complexity of $A$, and when parameterized by the dual tree-depth and the entry complexity of $A$; both these parameterization imply that $A$ is sparse, in particular, the number of its non-zero entries is linear in the number of columns or rows, respectively. We study preconditioners transforming a given matrix to a row-equivalent sparse matrix if it exists and provide structural results characterizing the existence of a sparse row-equivalent matrix in terms of the structural properties of the associated column matroid. In particular, our results imply that the $\ell_1$-norm of the Graver basis is bounded by a function of the maximum $\ell_1$-norm of a circuit of $A$. We use our results to design a parameterized algorithm that constructs a matrix row-equivalent to an input matrix $A$ that has small primal/dual tree-depth and entry complexity if such a row-equivalent matrix exists. Our results yield parameterized algorithms for integer programming when parameterized by the $\ell_1$-norm of the Graver basis of the constraint matrix, when parameterized by the $\ell_1$-norm of the circuits of the constraint matrix, when parameterized by the smallest primal tree-depth and entry complexity of a matrix row-equivalent to the constraint matrix, and when parameterized by the smallest dual tree-depth and entry complexity of a matrix row-equivalent to the constraint matrix.