We consider ILPs, where each variable corresponds to an integral point within a polytope $\mathcal{P}$, i. e., ILPs of the form $\min\{c^{\top}x\mid \sum_{p\in\mathcal P\cap \mathbb Z^d} x_p p = b, x\in\mathbb Z^{|\mathcal P\cap \mathbb Z^d|}_{\ge 0}\}$. The distance between an optimal fractional solution and an optimal integral solution (called proximity) is an important measure. A classical result by Cook et al.~(Math. Program., 1986) shows that it is at most $\Delta^{\Theta(d)}$ where $\Delta$ is the largest coefficient in the constraint matrix. Another important measure studies the change in an optimal solution if the right-hand side $b$ is replaced by another right-hand side $b'$. The distance between an optimal solution $x$ w.r.t.~$b$ and an optimal solution $x'$ w.r.t.~$b'$ (called sensitivity) is similarly bounded, i. e., $\lVert b-b' \rVert_{1}\cdot \Delta^{\Theta(d)}$, also shown by Cook et al. Even after more than thirty years, these bounds are essentially the best known bounds for these measures. While some lower bounds are known for these measures, they either only work for very small values of $\Delta$, require negative entries in the constraint matrix, or have fractional right-hand sides. Hence, these lower bounds often do not correspond to instances from algorithmic problems. This work presents for each $\Delta > 0$ and each $d > 0$ ILPs of the above type with non-negative constraint matrices such that their proximity and sensitivity is at least $\Delta^{\Theta(d)}$. Furthermore, these instances are closely related to instances of the Bin Packing problem as they form a subset of columns of the configuration ILP. We thereby show that the results of Cook et al. are indeed tight, even for instances arising naturally from problems in combinatorial optimization.

中文翻译：

---

整数线性规划的灵敏度和邻近边界的严格性

我们考虑 ILP，其中每个变量对应于多面体 $\mathcal{P}$ 中的一个积分点，即 $\min\{c^{\top}x\mid \sum_{p\in\ mathcal P\cap \mathbb Z^d} x_p p = b, x\in\mathbb Z^{|\mathcal P\cap \mathbb Z^d|}_{\ge 0}\}$。最优分数解和最优积分解之间的距离（称为接近度）是一个重要的度量。Cook et al.~(Math. Program., 1986) 的经典结果表明它至多是 $\Delta^{\Theta(d)}$，其中 $\Delta$ 是约束矩阵中的最大系数。如果右侧 $b$ 被另一个右侧 $b'$ 替换，则另一个重要的度量研究最优解的变化。最优解 $x$wrt~$b$ 和最优解 $x'$wrt~$b'$ 之间的距离（称为灵敏度）同样有界，即，$\lVert bb' \rVert_{1}\cdot \Delta^{\Theta(d)}$，也由 Cook 等人展示。即使在 30 多年之后，这些界限本质上仍然是这些度量的最广为人知的界限。虽然这些度量的一些下限是已知的，但它们要么仅适用于 $\Delta$ 的非常小的值，需要约束矩阵中的负项，或者具有小数右侧。因此，这些下限通常与算法问题的实例不对应。这项工作为上述类型的每个 $\Delta > 0$ 和每个 $d > 0$ ILP 提供了非负约束矩阵，使得它们的接近度和灵敏度至少为 $\Delta^{\Theta(d)}$ . 此外，这些实例与装箱问题的实例密切相关，因为它们构成了配置 ILP 的列的子集。我们从而表明库克等人的结果。确实很紧，即使对于组合优化中的问题自然产生的实例也是如此。