Matrices of Optimal Tree-Depth and a Row-Invariant Parameterized Algorithm for Integer Programming
Timothy F. N. Chan, Jacob W. Cooper, Martin Koutecký, Daniel Král͏̌, Kristýna Pekárková · SIAM Journal on Computing · 2022
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 $\Delta$ are solvable in time $g(d,\Delta){poly}(n)$ 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 $g(d^*,\Delta){poly}(n)$ where $d^*$ is the branch-depth of the constraint matrix; the branch-depth cannot be replaced by the more permissive notion of branch-width.