Recently a strong connection has been shown between the tractability of integer programming (IP) with bounded coefficients on the one side and the structure of its constraint matrix on the other side. To that end, integer linear programming is fixed-parameter tractable with respect to the primal (or dual) treedepth of the Gaifman graph of its constraint matrix and the largest coefficient (in absolute value). Motivated by this, Kouteck\'y, Levin, and Onn [ICALP 2018] asked whether it is possible to extend these result to a more broader class of integer linear programs. More formally, is integer linear programming fixed-parameter tractable with respect to the incidence treedepth of its constraint matrix and the largest coefficient (in absolute value)? We answer this question in negative. In particular, we prove that deciding the feasibility of a system in the standard form, ${A\mathbf{x} = \mathbf{b}}, {\mathbf{l} \le \mathbf{x} \le \mathbf{u}}$, is $\mathsf{NP}$-hard even when the absolute value of any coefficient in $A$ is 1 and the incidence treedepth of $A$ is 5. Consequently, it is not possible to decide feasibility in polynomial time even if both the assumed parameters are constant, unless $\mathsf{P}=\mathsf{NP}$. Moreover, we complement this intractability result by showing tractability for natural and only slightly more restrictive settings, namely: (1) treedepth with an additional bound on either the maximum arity of constraints or the maximum number of occurrences of variables and (2) the vertex cover number.

翻译：最近显示,在整数编程(IP)的可移动性(IP)和其约束系数的制约矩阵结构之间,最近显示了一个强有力的连接。为此,整数线线性编程对于其约束矩阵和最大系数(绝对值)的原始(或双)树深度(Gaifman)图的制约矩阵和最大系数(绝对值)而言,是固定的参数。为此,Kouteck\'y,Levin和On[CricalP 2018]询问是否有可能将这些结果扩大到一个范围更广的整数线性程序类别。更正式地说,整数线性编程固定参数的可移动性固定参数,相对于其约束矩阵和最大系数(绝对值)的发生率,我们回答这个问题是否定的。我们证明,在标准格式中确定一个系统的可行性,${A\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\1\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\