  We consider integer programs whose constraint matrix has a special block
structure: $\min\{f(x):H_{com}x=b, l\le x\le u,x\in\mathbb{Z}^{t_B+nt_A}\}$,
where the objective function $f$ is separable convex and the constraint matrix
$H_ {com}$ is composed of small submatrices $A_i,B,C,D_i$ such that the first
row is $(C,D_1,D_2,\ldots,D_n)$, the first column is $(C,B,B,\ldots,B)^{\top}$,
the main diagonal is $(C,A_1,A_2,\ldots,A_n)$, and the rest entries are 0.
Furthermore, $\text{rank}(B)$=1. We study fixed parameter tractable (FPT)
algorithms by taking as parameters the number of rows and columns of small
submatrices, together with the largest absolute value over their entries.
  We call the IP (almost) combinatorial 4-block n-fold IP. It generalizes the
generalized n-fold IP and is a special case of the generalized 4-block n-fold
IP. The existence of FPT algorithms for the generalized 4-block n-fold IP is a
major open problem, which motivates us to study special cases of the
generalized 4-block n-fold IP to find structural insights.
  We show the $\ell_{\infty}$-norm of Graver basis elements of combinatorial
4-block n-fold IP is $\Omega(n)$. There is some FPT-value $\lambda$ such that
for any nonzero element $g\in\{x: H_{com} x= 0\}$, $\lambda g$ can always be
decomposed into Graver basis elements in the same orthant whose
$\ell_{\infty}$-norm is FPT-bounded (while g might not admit such a
decomposition). Then we can bound the $\ell_{\infty}$-norm of Graver basis
elements by $O_{FPT}(n)$ and develop $O_{FPT}({n^4\hat{L}^2})$-time algorithms
($O_{FPT}$ hides a multiplicative FPT-term, and $\hat{L}$ denotes the logarithm
of the largest number occurring in the input).
  As applications, combinatorial 4-block n-fold IP can be used to model some
classical problems, including scheduling with rejection and bicriteria
scheduling.
