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：集合覆盖与击中问题是组合优化中的基本问题，在离线、在线及动态场景下均有深入研究。本文研究这些问题的几何版本，并提出新的在线与动态算法。在集合覆盖（或击中问题）的在线版本中，m个集合（或n个点）预先给出，n个点（或m个集合）逐个在线到达。在动态版本中，点（或集合）既可到达也可离开。我们的目标是维护一个集合覆盖（或击中问题），并最小化所求解的大小。针对任意大小的（轴平行）方形的在线集合覆盖问题，我们提出了一个紧的O(log n)-竞争算法。在相同设定下，我们为击中问题提供了一个紧的O(log N)-竞争算法，假设所有点的坐标均为[0,N)²内的整数。此前这两种设定下均无已知的在线算法（甚至对于单位方形，除已知的针对任意集合系统的在线算法外）。对于d维超矩形的动态集合覆盖与击中问题，我们获得了近似比为(log m)^{O(d)}且最坏情况更新时间为(log m)^{O(d)}的算法，部分回答了Chan等人[SODA'22]提出的开放问题。此前，即使对于方形（针对这两个问题中的任何一个），也未知具有多对数更新时间的动态算法。我们的主要技术贡献包括一种扩展四叉树方法与一种频率约简技术，该技术将几何集合覆盖实例化为具有有界频率的普通集合覆盖实例。