The min knapsack problem appears as a major component in the structure of capacitated covering problems. Its polyhedral relaxations have been extensively studied, leading to strong relaxations for networking, scheduling and facility location problems.

A valid inequality ${\alpha }^{T}x\ge {\alpha }_0$ with $\alpha \ge 0$ for a min knapsack instance is said to have pitch $\le \pi$ ($\pi$ a positive integer) if the $\pi$ smallest strictly positive ${\alpha }_j$ sum to at least ${\alpha }_0$. An inequality with coefficients and right-hand side in $\{0,1,\dots ,\pi \}$ has pitch $\le \pi$. The notion of pitch has been used for measuring the complexity of valid inequalities for the min knapsack polytope. Separating inequalities of pitch-1 is already NP-Hard. In this paper, we show an algorithm for efficiently separating inequalities with coefficients in $\{0,1,\dots ,\pi \}$ for any fixed $\pi$ up to an arbitrarily small additive error. As a special case, this allows for approximate separation of inequalities with pitch at most 2. We moreover investigate the integrality gap of minimum knapsack instances when bounded pitch inequalities (possibly in conjunction with other inequalities) are added. Among other results, we show that the CG closure of minimum knapsack has unbounded integrality gap even after a constant number of rounds.

最小背包问题似乎是带电容覆盖问题结构的主要组成部分。其多面体松弛已被广泛研究，导致网络，调度和设施位置问题的强烈松弛。

有效的不平等 ${\alpha }^{Ť}X\ge {\alpha }_0$ 与 $\alpha \ge 0$ 最小背包实例据说具有音高 $\le \pi$ （$\pi$ 一个正整数），如果 $\pi$ 最小严格正 ${\alpha }_{Ĵ}$ 总和至少 ${\alpha }_0$。带有系数和右手边的不等式$\{0，\mathrm{1个}，\dots ，\pi \}$ 有音高 $\le \pi$。间距的概念已被用于测量最小背包多面体的有效不等式的复杂性。间距1的不等式已经是NP-Hard。在本文中，我们展示了一种有效分离系数不等式的算法$\{0，\mathrm{1个}，\dots ，\pi \}$ 对于任何固定 $\pi$直至任意小的加法误差。作为一种特殊情况，这允许不等距与间距最大为2的近似分离。此外，当添加有界间距不等式（可能与其他不等式结合）时，我们研究了最小背包实例的完整性间隙。在其他结果中，我们表明，即使经过一定数量的回合，最小背包的CG闭合也具有无穷大的完整性缺口。