Matrices of optimal tree-depth and a row-invariant parameterized algorithm for integer programming
| Authors | |
|---|---|
| Year of publication | 2020 |
| Type | Article in Proceedings |
| Conference | Proceedings of the 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020), p. "26:1"-"26:19", 19 pp. 2020. |
| MU Faculty or unit | |
| Citation | |
| Doi | http://dx.doi.org/10.4230/LIPIcs.ICALP.2020.26 |
| Keywords | matroid algorithms; width parameters; integer programming; fixed parameter tractability; branch-width; branch-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 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. |
| Related projects: |