Matrices of Optimal Tree-Depth and a Row-Invariant Parameterized Algorithm for Integer Programming
| Authors | |
|---|---|
| Year of publication | 2022 |
| Type | Article in Periodical |
| Magazine / Source | SIAM JOURNAL ON COMPUTING |
| MU Faculty or unit | |
| Citation | |
| Web | https://epubs.siam.org/doi/10.1137/20M1353502 |
| Doi | http://dx.doi.org/10.1137/20M1353502 |
| Keywords | branch-depth; fixed-parameter tractability; integer programming; matroids; tree-depth |
| Description | 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 ? are solvable in time g(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 treedepth. Finally, we use these results to obtain an algorithm for integer programming running in time g(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. |
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