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.

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整数编程和关联树深度

最近，在一侧具有有界系数的整数编程（IP）的可处理性与另一侧具有约束矩阵的结构之间，已显示出很强的联系。为此，相对于其约束矩阵和最大系数（绝对值）的盖夫曼图的原始（或对偶）树深度，整数线性规划是固定参数可控的。受此启发，Kouteck'y，Levin和Onn [ICALP 2018]询问是否有可能将这些结果扩展到更广泛的整数线性程序类别。更正式地说，就其约束矩阵的入射树深度和最大系数（绝对值）而言，整数线性规划固定参数是否易于处理？我们以否定回答这个问题。尤其是，我们证明以标准形式$ {A \ mathbf {x} = \ mathbf {b}}，{\ mathbf {l} \ le \ mathbf {x} \ le \ mathbf {u} } $是$ \ mathsf {NP} $-hard，即使$ A $中任何系数的绝对值为1且关联树深度$ A $为5。因此，无法确定多项式时间的可行性即使两个假定参数都是常数，除非$ \ mathsf {P} = \ mathsf {NP} $。此外，我们通过显示自然且仅是限制性稍强的设置的易处理性来补充这种难处理性结果，即：（1）树深度，在最大约束约束或最大变量出现次数上附加界限，以及（2）顶点封面号码。即使$ A $中任何系数的绝对值为1且$ A $的关联树深度为5，也为$ \ mathsf {NP} $-hard。因此，即使两个参数都不能确定多项式时间的可行性除非$ \ mathsf {P} = \ mathsf {NP} $，否则假定的参数是恒定的。此外，我们通过显示自然且仅是限制性稍强的设置的易处理性来补充这种难处理性结果，即：（1）树深度，在最大约束约束或最大变量出现次数上附加界限，以及（2）顶点封面号码。即使$ A $中任何系数的绝对值为1且$ A $的关联树深度为5，也为$ \ mathsf {NP} $-hard。因此，即使两个参数都不能确定多项式时间的可行性除非$ \ mathsf {P} = \ mathsf {NP} $，否则假定的参数是恒定的。此外，我们通过显示自然且仅是限制性稍强的设置的易处理性来补充这种难处理性结果，即：（1）树深度，在最大约束约束或最大变量出现次数上附加界限，以及（2）顶点封面号码。