Characterization of Matrices with Bounded Graver Bases and Depth Parameters and Applications to Integer Programming
| Authors | |
|---|---|
| Year of publication | 2022 |
| Type | Article in Proceedings |
| Conference | Proceedings of the 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022) |
| MU Faculty or unit | |
| Citation | |
| Doi | http://dx.doi.org/10.4230/LIPIcs.ICALP.2022.29 |
| 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 an equivalent sparse matrix if it exists and provide structural results characterizing the existence of a sparse equivalent matrix in terms of the structural properties of the associated column matroid. In particular, our results imply that the ??1-norm of the Graver basis is bounded by a function of the maximum ??1-norm of a circuit of A. We use our results to design a parameterized algorithm that constructs a matrix equivalent to an input matrix A that has small primal/dual tree-depth and entry complexity if such an equivalent matrix exists. Our results yield parameterized algorithms for integer programming when parameterized by the ??1-norm of the Graver basis of the constraint matrix, when parameterized by the ??1-norm of the circuits of the constraint matrix, when parameterized by the smallest primal tree-depth and entry complexity of a matrix equivalent to the constraint matrix, and when parameterized by the smallest dual tree-depth and entry complexity of a matrix equivalent to the constraint matrix. |
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