  Recently a strong connection has been shown between the tractability of
integer programming (IP) with bounded coefficients on the one side and the
structure of its constraint matrix on the other side. To that end, integer
linear programming is fixed-parameter tractable with respect to the primal (or
dual) treedepth of the Gaifman graph of its constraint matrix and the largest
coefficient (in absolute value). Motivated by this, Kouteck\'y, Levin, and Onn
[ICALP 2018] asked whether it is possible to extend these result to a more
broader class of integer linear programs. More formally, is integer linear
programming fixed-parameter tractable with respect to the incidence treedepth
of its constraint matrix and the largest coefficient (in absolute value)?
  We answer this question in negative. In particular, we prove that deciding
the feasibility of a system in the standard form, ${A\mathbf{x} = \mathbf{b}},
{\mathbf{l} \le \mathbf{x} \le \mathbf{u}}$, is $\mathsf{NP}$-hard even when
the absolute value of any coefficient in $A$ is 1 and the incidence treedepth
of $A$ is 5. Consequently, it is not possible to decide feasibility in
polynomial time even if both the assumed parameters are constant, unless
$\mathsf{P}=\mathsf{NP}$. Moreover, we complement this intractability result by
showing tractability for natural and only slightly more restrictive settings,
namely: (1) treedepth with an additional bound on either the maximum arity of
constraints or the maximum number of occurrences of variables and (2) the
vertex cover number.
