  It is a notorious open question whether integer programs (IPs), with an
integer coefficient matrix $M$ whose subdeterminants are all bounded by a
constant $\Delta$ in absolute value, can be solved in polynomial time. We
answer this question in the affirmative if we further require that, by removing
a constant number of rows and columns from $M$, one obtains a submatrix $A$
that is the transpose of a network matrix.
  Our approach focuses on the case where $A$ arises from $M$ after removing $k$
rows only, where $k$ is a constant. We achieve our result in two main steps,
the first related to the theory of IPs and the second related to graph minor
theory.
  First, we derive a strong proximity result for the case where $A$ is a
general totally unimodular matrix: Given an optimal solution of the linear
programming relaxation, an optimal solution to the IP can be obtained by
finding a constant number of augmentations by circuits of $[A\; I]$.
  Second, for the case where $A$ is transpose of a network matrix, we
reformulate the problem as a maximum constrained integer potential problem on a
graph $G$. We observe that if $G$ is $2$-connected, then it has no rooted
$K_{2,t}$-minor for $t = \Omega(k \Delta)$. We leverage this to obtain a
tree-decomposition of $G$ into highly structured graphs for which we can solve
the problem locally. This allows us to solve the global problem via dynamic
programming.
