This work is to discuss the fault-tolerant facility location problem with submodular penalties. We propose an LP-rounding 2.27-approximation algorithm, where every demand point j has a requirement that \(t_{j}\) distinct facilities serve it. This is the first constant performance guarantee known for this problem. In addition, we give an LP-rounding 2-approximation algorithm for the case where all requirements are the same.

中文翻译：

---

具有子模惩罚的容错设施定位问题的近似算法

这项工作是讨论具有子模惩罚的容错设施定位问题。我们提出了一种 LP 舍入 2.27 近似算法，其中每个需求点j都要求\(t_{j}\)不同的设施为其提供服务。这是针对此问题的第一个恒定性能保证。此外，对于所有要求都相同的情况，我们给出了 LP 舍入 2 近似算法。