We present an improved approximation algorithm for k-level facility location problem with submodular penalties, the new approximation ratio is 2.9444 for any constant k, which improves the current best approximation ratio 3.314. The central ideas in our results are as follows: first, we restructure the problem as an uncapacitated facility location problem, then we use the primal-dual scheme with greedy augmentation. The key technique of our result is that we change the way of last opening facility set in primal-dual approximation algorithm to get much more tight result for k-level facility location problem with submodular penalties.

中文翻译：

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具有子模惩罚的 k 级设施选址问题的组合逼近算法

我们针对具有子模惩罚的k级设施选址问题提出了一种改进的近似算法，对于任何常数k，新的近似比为 2.9444 ，这改进了当前的最佳近似比 3.314。我们的结果的中心思想如下：首先，我们将问题重构为无能力的设施位置问题，然后我们使用带有贪婪增强的原始对偶方案。我们的结果的关键技术是我们改变了原对偶近似算法中最后打开设施集的方式，以获得具有子模惩罚的k级设施位置问题的更严格的结果。