Solving integer programs of the form $\min \{\mathbf{x} \mid A\mathbf{x} = \mathbf{b}, \mathbf{l} \leq \mathbf{x} \leq \mathbf{u}, \mathbf{x} \in \mathbb{Z}^n \}$ is, in general, $\mathsf{NP}$-hard. Hence, great effort has been put into identifying subclasses of integer programs that are solvable in polynomial or $\mathsf{FPT}$ time. A common scheme for many of these integer programs is a star-like structure of the constraint matrix. The arguably simplest form that is not a star is a path. We study integer programs where the constraint matrix $A$ has such a path-like structure: every non-zero coefficient appears in at most two consecutive constraints. We prove that even if all coefficients of $A$ are bounded by 8, deciding the feasibility of such integer programs is $\mathsf{NP}$-hard via a reduction from 3-SAT. Given the existence of efficient algorithms for integer programs with star-like structures and a closely related pattern where the sum of absolute values is column-wise bounded by 2 (hence, there are at most two non-zero entries per column of size at most 2), this hardness result is surprising.

翻译：求解形如 $\min \{\mathbf{x} \mid A\mathbf{x} = \mathbf{b}, \mathbf{l} \leq \mathbf{x} \leq \mathbf{u}, \mathbf{x} \in \mathbb{Z}^n \}$ 的整数规划问题通常是 $\mathsf{NP}$-困难的。因此，大量研究致力于识别可在多项式时间或 $\mathsf{FPT}$ 时间内求解的整数规划子类。许多此类整数规划的一个共同特点是约束矩阵具有星形结构。而非星形结构中最简单的形式可视为路径。本文研究约束矩阵 $A$ 具有此类路径结构的整数规划：每个非零系数最多出现在两个相邻约束中。我们证明，即使 $A$ 的所有系数均以 8 为界，通过从 3-SAT 问题归约，判定此类整数规划的可行性仍是 $\mathsf{NP}$-困难的。考虑到星形结构整数规划及其密切相关模式（即每列绝对值之和以 2 为界，故每列最多有两个非零元素且其大小不超过 2）存在高效算法，这一困难性结果令人意外。