论文标题
整数线性程序的灵敏度和接近性界限的紧密度
Tightness of Sensitivity and Proximity Bounds for Integer Linear Programs
论文作者
论文摘要
我们考虑ILP，其中每个变量对应于polytope $ \ Mathcal {p} $，i中的一个积分点。 e。，形式的$ \ min \ {c^{\ top} x \ mid \ sum_ {p \ in \ Mathcal p \ cap \ cap \ cap \ mathbb z^d} x_p p = b，x \ in \ in \ in \ mathb z^{| \ mathc p \ cap p \ cap \ cap \ cap \ cap \ cap \ mathb z^$} 最佳分数解决方案与最佳积分解决方案（称为接近度）之间的距离是重要的措施。 Cook等人的经典结果（Math。program。，1986）表明它最多是$δ^{θ（d）} $，其中$δ$是约束矩阵中最大的系数。 另一个重要的措施研究了最佳解决方案中的变化，如果右侧$ b $由另一个右侧$ b'$取代。最佳解决方案$ x $ w.r.t. e。，$ \ lvert b-b'\ rvert_ {1} \cdotΔ^{θ（d）} $，也由Cook等人显示。 即使经过三十多年，这些界限本质上也是这些措施最著名的界限。 尽管这些度量已知一些下限，但它们要么仅适用于非常小的$δ$的值，要么在约束矩阵中需要负条目，要么具有分数右侧。 因此，这些下限通常与算法问题中的实例不符。 这项工作为每个$δ> 0 $以及上述类型的每个$ d> 0 $ ilps提供，具有非负约束矩阵，使它们的接近性和敏感性至少为$δ^{θ（d）} $。 此外，这些实例与垃圾箱问题的实例密切相关，因为它们形成了配置ILP的列的子集。 因此，我们表明库克等人的结果。确实很紧，即使是由于组合优化的问题自然引起的情况。
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.
