Matrices of Optimal Tree-Depth and a Row-Invariant Parameterized Algorithm for Integer Programming

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