Block-structured integer programming: Can we parameterize without the largest coefficient?

We consider 4-block $n$-fold integer programming, which can be written as $max\{w\cdot x:Hx=b,l\le x\le u,x\in Z^N\}$, 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 $A\in Z^{1\times 2}$, or more generally if $A\in Z^{s_A\times t_A}$, $t_A=s_A+1$ and the rank of matrix $A$ satisfies that $\text{rank}\left(A\right)=s_A$.