We improve the running times of $O(1)$-approximation algorithms for the set cover problem in geometric settings, specifically, covering points by disks in the plane, or covering points by halfspaces in three dimensions. In the unweighted case, Agarwal and Pan [SoCG 2014] gave a randomized $O(n\log^4 n)$-time, $O(1)$-approximation algorithm, by using variants of the multiplicative weight update (MWU) method combined with geometric data structures. We simplify the data structure requirement in one of their methods and obtain a deterministic $O(n\log^3 n\log\log n)$-time algorithm. With further new ideas, we obtain a still faster randomized $O(n\log n(\log\log n)^{O(1)})$-time algorithm. For the weighted problem, we also give a randomized $O(n\log^4n\log\log n)$-time, $O(1)$-approximation algorithm, by simple modifications to the MWU method and the quasi-uniform sampling technique.

中文翻译：

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几何集覆盖的更快逼近算法

我们针对几何设置中的集合覆盖问题改进了 $O(1)$-近似算法的运行时间，特别是通过平面上的圆盘覆盖点，或在三个维度上通过半空间覆盖点。在未加权的情况下，Agarwal 和 Pan [SoCG 2014] 通过使用乘法权重更新 (MWU) 的变体，给出了随机的 $O(n\log^4 n)$-time、$O(1)$-近似算法结合几何数据结构的方法。我们在他们的一种方法中简化了数据结构要求，并获得了确定性的 $O(n\log^3 n\log\log n)$-time 算法。有了更多的新想法，我们获得了一个更快的随机 $O(n\log n(\log\log n)^{O(1)})$-time 算法。对于加权问题，我们也给出了一个随机的$O(n\log^4n\log\log n)$-time，$O(1)$-近似算法，