We present a new generalization of the bin covering problem that is known to be a strongly NP-hard problem. In our generalization there is a positive constant $\Delta$, and we are given a set of items each of which has a positive size. We would like to find a partition of the items into bins. We say that a bin is near exact covered if the total size of items packed into the bin is between $1$ and $1+\Delta$. Our goal is to maximize the number of near exact covered bins. If $\Delta=0$ or $\Delta>0$ is given as part of the input, our problem is shown here to have no approximation algorithm with a bounded asymptotic approximation ratio (assuming that $P\neq NP$). However, for the case where $\Delta>0$ is seen as a constant, we present an asymptotic fully polynomial time approximation scheme (AFPTAS) that is our main contribution.

翻译：我们展示了一个新的垃圾箱的概略化, 这个问题已知是一个强烈的NP- 硬性问题。 在我们的概括化中, 我们有一个正常数$\ Delta$, 我们得到的是每件都有正数大小的物品。 我们想要找到将物品分割到垃圾箱中的方法。 我们说, 如果装入垃圾箱的物品总大小在$和$$+Delta$之间, 垃圾箱就近被覆盖了。 我们的目标是最大限度地增加近似已覆盖的垃圾箱的数量。 如果输入的一部分是$\ Delta=0 或$\ Delta>0, 我们的问题在这里显示没有带有捆绑的微量接近率的近似算法( 假设是$P\ neq NP$ ) 。 但是, 如果将$\ Delta>0美元看成是常数, 我们提出一个微量的完全多元时间接近计划( AFPTAS ), 也就是我们的主要贡献 。