  We consider box-constrained integer programs with objective $g(Wx) + c^T x$,
where $g$ is a "complicated" function with an $m$ dimensional domain. Here we
assume we have $n \gg m$ variables and that $W \in \mathbb Z^{m \times n}$ is
an integer matrix with coefficients of absolute value at most $\Delta$. We
design an algorithm for this problem using only the mild assumption that the
objective can be optimized efficiently when all but $m$ variables are fixed,
yielding a running time of $n^m(m \Delta)^{O(m^2)}$. Moreover, we can avoid the
term $n^m$ in several special cases, in particular when $c = 0$.
  Our approach can be applied in a variety of settings, generalizing several
recent results. An important application are convex objectives of low domain
dimension, where we imply a recent result by Hunkenschr\"oder et al. [SIOPT'22]
for the 0-1-hypercube and sharp or separable convex $g$, assuming $W$ is given
explicitly. By avoiding the direct use of proximity results, which only holds
when $g$ is separable or sharp, we match their running time and generalize it
for arbitrary convex functions. In the case where the objective is only
accessible by an oracle and $W$ is unknown, we further show that their
proximity framework can be implemented in $n (m \Delta)^{O(m^2)}$-time instead
of $n (m \Delta)^{O(m^3)}$. Lastly, we extend the result by Eisenbrand and
Weismantel [SODA'17, TALG'20] for integer programs with few constraints to a
mixed-integer linear program setting where integer variables appear in only a
small number of different constraints.
