Abstrakt
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 Δ are solvable in time g(d, Δ )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 treedepth.
Finally, we use these results to obtain an algorithm for integer programming running in time g(dASTERISK OPERATOR, Δ )poly(n) where dASTERISK OPERATOR is the branch-depth of the constraint matrix; the branch-depth cannot be replaced by the more permissive notion of branch-width.