  We consider 4-block $n$-fold integer programming, which can be written as
$\max\{w\cdot x: H x=b, l\le x\le u, x\in \mathbb{Z}^{N} \}$ where the
constraint matrix $H$ is composed of small submatrices $A,B,C,D$ such that the
first row of $H$ is $(C,D,D,\cdots,D)$, the first column of $H$ is
$(C,B,B,\cdots,B)$, the main diagonal of $H$ is $(C,A,A,\cdots,A)$, and all the
other entries are $0$. The special case where $B=C=0$ is known as $n$-fold
integer programming.
  Prior algorithmic results for 4-block $n$-fold integer programming and its
special cases usually take $\Delta$, the largest absolute value among entries
of $H$ as part of the parameters. In this paper, we explore the possibility of
getting rid of $\Delta$ from parameters, i.e., we are looking for algorithms
that runs polynomially in $\log\Delta$. We show that, assuming $\text{P}\neq
\text{NP}$, this is not possible even if $A=(1,1,\Delta)$ and $B=C=0$. However,
this becomes possible if $A=(1,1,\cdots,1)$ or $A\in \mathbb{Z}^{1\times 2}$,
or more generally if $A\in\mathbb{Z}^{s_A\times t_A} $ where $t_A=s_A+1$ and
the rank of matrix $A$ satisfies that $\text{rank}(A)=s_A$. More precisely,
  1. If $A=(1,\ldots,1)\in \mathbb{Z}^{1\times t_A} $, then 4-block $n$-fold IP
can be solved in $(t_A+t_B)^{O(t_A+t_B)}\cdot poly(n,\log\Delta)$ time.
  2. If $A\in\mathbb{Z}^{s_A\times t_A} $, $t_A=s_A+1$ and
$\text{rank}(A)=s_A$, then 4-block $n$-fold IP can be solved in
$(t_A+t_B)^{O(t_A+t_B)}\cdot n^{O(t_A)}\cdot poly(\log\Delta)$ time;
Specifically, if in addition we have $B=C=0$ (i.e., $n$-fold integer
programming), then it can be solved in linear time $n\cdot poly(t_A,\log
\Delta)$.
