We consider 4-block $n$-fold integer programming, which can be written as $max\{\mathbf{w}\cdot \mathbf{x}:H\mathbf{x}=\mathbf{b},\mathbf{l}\le \mathbf{x}\le \mathbf{u},\mathbf{x}\in {\mathbb{Z}}^N\}$, where the constraint matrix $H$ is composed of small matrices $A,B,C,D$ such that the first row of $H$ is $(C,D,D,\dots ,D)$, the first column of $H$ is $(C,B,B,\dots ,B)$, the main diagonal of $H$ is $(C,A,A,\dots ,A)$, and all the other entries are 0. There are $n$ copies of $D$, $B$, and $A$. 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 solving the problems polynomially when the number of rows and columns of the small matrices are constant. We show that, assuming $\text{P}\ne \text{NP}$, this is not possible even if $A=(1,1,\Delta )$ and $B=C=0$. However, this becomes possible if $A=(1,\dots ,1)$ or $A\in {\mathbb{Z}}^{1\times 2}$, or more generally if $A\in {\mathbb{Z}}^{s_A\times t_A}$, $t_A=s_A+1$ and the rank of matrix $A$ satisfies that $\text{rank}\left(A\right)=s_A$.

我们考虑 4-block$n$-fold 整数规划，可以写成$最大限度\{\mathbf{w}\cdot \mathbf{X}：H\mathbf{X}=\mathbf{b},\mathbf{l}\le \mathbf{X}\le \mathbf{你},\mathbf{X}\in {\mathbb{Z}}^{ñ}\}$, 其中约束矩阵$H$由小矩阵组成$\mathrm{一个},乙,C,D$这样第一行$H$是$(C,D,D,\dots ,D)$，第一列$H$是$(C,乙,乙,\dots ,乙)$, 的主对角线$H$是$(C,\mathrm{一个},\mathrm{一个},\dots ,\mathrm{一个})$，所有其他条目为0。有$n$的副本$D$,$乙$， 和$\mathrm{一个}$. 特殊情况$乙=C=0$被称为$n$-折叠整数规划。

4块的先验算法结果$n$-fold 整数规划及其特殊情况通常需要$\Delta$, 的条目中最大的绝对值$H$，作为参数的一部分。在本文中，我们探讨了当小矩阵的行数和列数恒定时，多项式求解问题的可能性。我们证明，假设$\text{磷}\ne \text{NP}$, 这也是不可能的，即使$\mathrm{一个}=(1,1,\Delta )$和$乙=C=0$. 然而，这成为可能，如果$\mathrm{一个}=(1,\dots ,1)$或者$\mathrm{一个}\in {\mathbb{Z}}^{1\times 2}$，或更一般地，如果$\mathrm{一个}\in {\mathbb{Z}}^{s_{\mathrm{一个}}\times {吨}_{\mathrm{一个}}}$,${吨}_{\mathrm{一个}}=s_{\mathrm{一个}}+1$和矩阵的秩$\mathrm{一个}$满足$\text{秩}\left(\mathrm{一个}\right)=s_{\mathrm{一个}}$.