# Matrices of optimal tree-depth and a row-invariant parameterized algorithm for integer programming

## TL;DR

This paper establishes a connection between the dual tree-depth of integer programming matrices and matroid branch-depth, providing algorithms to optimize matrix structure for more efficient problem solving.

## Contribution

It proves the equivalence of minimum dual tree-depth and matroid branch-depth, and develops algorithms to compute these parameters and optimize matrix structure for integer programming.

## Key findings

- Dual tree-depth equals matroid branch-depth.
- Algorithms for computing branch-depth and row-equivalent matrices.
- Improved integer programming solving time based on branch-depth.

## Abstract

A long line of research on fixed parameter tractability of integer programming culminated with showing that integer programs with n variables and a constraint matrix with dual tree-depth d and largest entry D are solvable in time g(d,D)poly(n) for some function g. However, the dual tree-depth of a constraint matrix is not preserved by row operations, i.e., a given integer program can be equivalent to another with a smaller dual tree-depth, and thus does not reflect its geometric structure. We prove that the minimum dual tree-depth of a row-equivalent matrix is equal to the branch-depth of the matroid defined by the columns of the matrix. We design a fixed parameter algorithm for computing branch-depth of matroids represented over a finite field and a fixed parameter algorithm for computing a row-equivalent matrix with minimum dual tree-depth. Finally, we use these results to obtain an algorithm for integer programming running in time g(d*,D)poly(n) where d* is the branch-depth of the constraint matrix; the branch-depth cannot be replaced by the more permissive notion of branch-width.