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We consider integer programs whose constraint matrix has a special blockstructure: $\min\{f(x):H_{com}x=b, l\le x\le u,x\in\mathbb{Z}^{t_B+nt_A}\}$,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 firstrow 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 smallsubmatrices, together with the largest absolute value over their entries.  We call the IP (almost) combinatorial 4-block n-fold IP. It generalizes thegeneralized n-fold IP and is a special case of the generalized 4-block n-foldIP. The existence of FPT algorithms for the generalized 4-block n-fold IP is amajor open problem, which motivates us to study special cases of thegeneralized 4-block n-fold IP to find structural insights.  We show the $\ell_{\infty}$-norm of Graver basis elements of combinatorial4-block n-fold IP is $\Omega(n)$. There is some FPT-value $\lambda$ such thatfor any nonzero element $g\in\{x: H_{com} x= 0\}$, $\lambda g$ can always bedecomposed into Graver basis elements in the same orthant whose$\ell_{\infty}$-norm is FPT-bounded (while g might not admit such adecomposition). Then we can bound the $\ell_{\infty}$-norm of Graver basiselements 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 logarithmof the largest number occurring in the input).  As applications, combinatorial 4-block n-fold IP can be used to model someclassical problems, including scheduling with rejection and bicriteriascheduling.

FPT Algorithms for a Special Block-structured Integer Program with Applications in Scheduling