线性规划的灵敏度和邻近度界的紧性
Tightness of Sensitivity and Proximity Bounds for Integer Linear Programs
摘要：
我们认为ILP，其中每个变量对应于一个整数点内的多面体$\mathcal{P}$，即。例如，$\min\{c^{\top}x\mid \sum_{p\in\mathcal P\cap \mathbb Z^d} x_p p = b, x\in\mathbb Z^{|\mathcal P\cap \mathbb Z^d|}_{\ge 0}\}$形式的ILP。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。库克等人的经典结果~（数学程序。，1986）表明它至多是$Δ^{Θ(d)}$，其中$Δ$是约束矩阵中的最大系数。另一个重要的测量研究如果右侧$b$被另一个右侧$b'$替换，则最优解的变化。最优解$x$与~之间的距离$b$和一个最优解$x'$ w.r.t.~ $b'$（称为灵敏度）同样有界，即.例如，$\lVert b-b' \rVert_{1}\cdot Δ^{Θ(d)}$，也由Cook等人证明。即使在三十多年之后，这些界限基本上是这些测度的最佳已知界限。虽然这些度量的一些下界是已知的，但它们要么只适用于$Δ$的非常小的值，要么需要约束矩阵中的负项，要么具有分数右侧。因此，这些下限通常不对应于算法问题的实例。这项工作提出了每个$Δ> 0$和每个$d > 0$ ILP的上述类型的非负约束矩阵，使他们的接近度和灵敏度至少是$Δ^{Θ(d)}$。此外，这些实例与装箱问题的实例密切相关，因为它们形成配置ILP的列子集。因此，我们表明，库克等人的结果确实是紧的，即使是自然产生的组合优化问题的实例。
英文摘要：
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 $Δ^{Θ(d)}$ where $Δ$ 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., $\lVert b-b' \rVert_{1}\cdot Δ^{Θ(d)}$, 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 $Δ$, 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 $Δ> 0$ and each $d > 0$ ILPs of the above type with non-negative constraint matrices such that their proximity and sensitivity is at least $Δ^{Θ(d)}$. 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.
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