  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.
