We consider column‐sparse covering integer programs, a generalization of set cover. We develop a new rounding scheme based on the partial resampling variant of the Lovász Local Lemma developed by Harris and Srinivasan. This achieves an approximation ratio of , where a_min is the minimum covering constraint and is the maximum ℓ₁‐norm of any column of the covering matrix A (whose entries are scaled to lie in [0, 1]). With additional constraints on the variable sizes, we get an approximation ratio of (where is the maximum number of nonzero entries in any column of A). These results improve asymptotically over results of Srinivasan and results of Kolliopoulos and Young. We show nearly‐matching lower bounds. We also show that the rounding process leads to negative correlation among the variables.

中文翻译：

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部分重采样以近似覆盖整数程序†

我们考虑列稀疏覆盖整数程序，它是集合覆盖的一般化。我们根据Harris和Srinivasan开发的LovászLocal Lemma的部分重采样变体开发了一种新的取整方案。这实现的近似比，其中一个_分是最小覆盖约束和为最大ℓ ₁所述覆盖矩阵的任何列的范数甲（其条目被缩放成位于[0,1]）。在变量大小受到其他限制的情况下，我们得到的近似比为（其中是A的任何列中非零项的最大数量）。这些结果比Srinivasan的结果以及Kolliopoulos和Young的结果渐近改善。我们展示了几乎匹配的下限。我们还表明，舍入过程导致变量之间的负相关。