Characterization of Matrices with Bounded Graver Bases and Depth Parameters and Applications to Integer Programming
| Authors | |
|---|---|
| Year of publication | 2024 |
| Type | Article in Periodical |
| Magazine / Source | MATHEMATICAL PROGRAMMING |
| MU Faculty or unit | |
| Citation | |
| Web | https://link.springer.com/article/10.1007/s10107-023-02048-x |
| Doi | http://dx.doi.org/10.1007/s10107-023-02048-x |
| Keywords | Integer programming; width parameters; matroids; Graver basis; tree-depth; fixed-parameter tractability |
| Description | An intensive line of research on fixed parameter tractability of integer programming is focused on exploiting the relation between the sparsity of a constraint matrix A and the norm of the elements of its Graver basis. In particular, integer programming is fixed parameter tractable when parameterized by the primal tree-depth and the entry complexity of A, and when parameterized by the dual tree-depth and the entry complexity of A; both these parameterization imply that A is sparse, in particular, the number of its non-zero entries is linear in the number of columns or rows, respectively. We study preconditioners transforming a given matrix to a row-equivalent sparse matrix if it exists and provide structural results characterizing the existence of a sparse row-equivalent matrix in terms of the structural properties of the associated column matroid. In particular, our results imply that the l_1-norm of the Graver basis is bounded by a function of the maximum l_1-norm of a circuit of A. We use our results to design a parameterized algorithm that constructs a matrix row-equivalent to an input matrix A that has small primal/dual tree-depth and entry complexity if such a row-equivalent matrix exists. Our results yield parameterized algorithms for integer programming when parameterized by the l_1-norm of the Graver basis of the constraint matrix, when parameterized by the l_1-norm of the circuits of the constraint matrix, when parameterized by the smallest primal tree-depth and entry complexity of a matrix row-equivalent to the constraint matrix, and when parameterized by the smallest dual tree-depth and entry complexity of a matrix row-equivalent to the constraint matrix. |
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