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 \(\ell _1\)-norm of the Graver basis is bounded by a function of the maximum \(\ell _1\)-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 \(\ell _1\)-norm of the Graver basis of the constraint matrix, when parameterized by the \(\ell _1\)-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.

对整数规划的固定参数易处理性的深入研究集中于利用约束矩阵A的稀疏性与其 Graver 基元素范数之间的关系。特别地，当由原始树深度和A的条目复杂性参数化时，以及当由对偶树深度和A的条目复杂性参数化时，整数规划是固定参数易于处理的；这两个参数化都意味着A是稀疏的，特别是其非零条目的数量分别与列数或行数呈线性关系。我们研究了将给定矩阵转换为行等效稀疏矩阵（如果存在）的预处理器，并提供了根据相关列拟阵的结构属性来表征稀疏行等效矩阵存在性的结构结果。特别是，我们的结果意味着Graver 基的\(\ell _1\) -范数受到A电路的最大\(\ell _1\) -范数的函数的限制。我们使用我们的结果设计一个参数化算法，该算法构造一个与输入矩阵A等效的矩阵行，如果存在这样的行等效矩阵，则该矩阵具有较小的原始/对偶树深度和条目复杂度。当由约束矩阵的 Graver 基的\(\ell _1\) -范数参数化时，当由约束矩阵电路的\(\ell _1\) -范数参数化时，我们的结果产生整数规划的参数化算法，当通过与约束矩阵等效的矩阵行的最小原始树深度和条目复杂度参数化时，以及当通过与约束矩阵等效的矩阵行的最小对偶树深度和条目复杂度参数化时。