到背包多面体格点的距离
Distances to Lattice Points in Knapsack Polyhedra
摘要：
给出了背包多面体顶点到最近可行格点的最大范数距离的最优上界。在一个随机设置，我们表明，平均上界可以显着提高。作为一个推论，我们得到了一个最佳的上界的整数背包问题的添加剂的完整性差距，并表明，一个“典型”的背包问题的完整性差距是大大小于的完整性差距，发生在最坏的情况下。我们还证明，在一般情况下，整数规划差距承认一个自然的最佳下界。
英文摘要：
We give an optimal upper bound for the maximum-norm distance from a vertex of a knapsack polyhedron to its nearest feasible lattice point. In a randomised setting, we show that the upper bound can be significantly improved on average. As a corollary, we obtain an optimal upper bound for the additive integrality gap of integer knapsack problems and show that the integrality gap of a "typical" knapsack problem is drastically smaller than the integrality gap that occurs in a worst case scenario. We also prove that, in a generic case, the integer programming gap admits a natural optimal lower bound.
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