Set cover and hitting set are fundamental problems in combinatorial optimization which are well-studied in the offline, online, and dynamic settings. We study the geometric versions of these problems and present new online and dynamic algorithms for them. In the online version of set cover (resp. hitting set), $m$ sets (resp.~$n$ points) are give $n$ points (resp.~$m$ sets) arrive online, one-by-one. In the dynamic versions, points (resp. sets) can arrive as well as depart. Our goal is to maintain a set cover (resp. hitting set), minimizing the size of the computed solution. For online set cover for (axis-parallel) squares of arbitrary sizes, we present a tight $O(\log n)$-competitive algorithm. In the same setting for hitting set, we provide a tight $O(\log N)$-competitive algorithm, assuming that all points have integral coordinates in $[0,N)^{2}$. No online algorithm had been known for either of these settings, not even for unit squares (apart from the known online algorithms for arbitrary set systems). For both dynamic set cover and hitting set with $d$-dimensional hyperrectangles, we obtain $(\log m)^{O(d)}$-approximation algorithms with $(\log m)^{O(d)}$ worst-case update time. This partially answers an open question posed by Chan et al. [SODA'22]. Previously, no dynamic algorithms with polylogarithmic update time were known even in the setting of squares (for either of these problems). Our main technical contributions are an \emph{extended quad-tree }approach and a \emph{frequency reduction} technique that reduces geometric set cover instances to instances of general set cover with bounded frequency.

翻译：设置封面和点击是组合优化中的根本问题, 可在离线、 在线和动态设置中进行仔细研究 。 我们的目标是保持一个设置的封面( 重新点击), 并展示这些问题的新的在线和动态算法 。 在设置覆盖的在线版本( 重试 ), $set (resp. ~ 美元点) 给美元点( resp.~ $ 美元 ), 单次到达 。 在动态版本中, 点( resp. 数) 可以到达并且离开 。 我们的目标是保持一个设置的套套套( 重击), 最大限度地缩小计算解决方案的大小。 对于任意大小的( 轴- parallel) 方格的在线覆盖, 我们展示了一个紧凑的 $( log N) 数( log N) 标准, 我们提供一个直线的算算算法, 假设所有点都有 $0, N) 和 d=2美元 。 我们没有知道这些设置的在线算法, 甚至不是单位主数 格式 。</s>