关联规划和关联树深度
Integer Programming and Incidence Treedepth
摘要：
最近，一个强有力的连接已被证明之间的易处理的整数规划（IP）与有界系数的一侧和结构的约束矩阵的另一侧。为此，整数线性规划是固定参数的易处理的，相对于其约束矩阵的Gaifman图的原始（或对偶）树深和最大系数（绝对值）。受此启发，Koutecklord，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规划类。更正式地说，整数线性规划固定参数相对于其约束矩阵的关联树深度和最大系数（绝对值）是否易于处理？我们对这个问题的回答是否定的。特别是，我们证明，决定一个系统的可行性的标准形式，${A\mathbf{x} = \mathbf{b}}, {\mathbf{l} \le \mathbf{x} \le \mathbf{u}}$，是$\mathsf{NP}$ -硬，即使在$A$中的任何系数的绝对值是1和发病树深度的$A$是5。因此，即使两个假设参数都是常数，也不可能在多项式时间内确定可行性，除非$\mathsf{P}=\mathsf{NP}$。此外，我们补充了这个棘手的结果，显示自然的，只是稍微更严格的设置，即：（1）树深度与约束的最大arity或变量的最大出现次数和（2）顶点覆盖数的额外的约束。
英文摘要：
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, Koutecký, 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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