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.

翻译：整数规划固定参数可处理性的研究主线聚焦于约束矩阵$A$的稀疏性与其Graver基元素范数之间的关系。具体而言，当以原始树深度和$A$的项复杂度为参数时，以及以对偶树深度和$A$的项复杂度为参数时，整数规划是固定参数可处理的；这两种参数化均意味着$A$是稀疏的，特别地，其非零元数目分别与列数或行数呈线性关系。我们研究预条件子（若存在）将给定矩阵变换为行等价稀疏矩阵的方法，并通过关联列拟阵的结构性质给出了行等价稀疏矩阵存在性的结构性刻画。特别地，我们的结果表明Graver基的$\ell_1$范数受$A$的电路最大$\ell_1$范数的函数所界。我们利用这些结果设计了一个参数化算法，该算法能够构造与输入矩阵$A$行等价的矩阵（若存在），使其具有较小的原始/对偶树深度和项复杂度。我们的结果产生了整数规划的参数化算法，这些算法分别以约束矩阵Graver基的$\ell_1$范数、约束矩阵电路的$\ell_1$范数、与约束矩阵行等价的最小原始树深度和项复杂度，以及与约束矩阵行等价的最小对偶树深度和项复杂度为参数。