We are given a set $P$ of $n$ points in the plane, and a sequence of axis-aligned squares that arrive in an online fashion. The online hitting set problem consists of maintaining, by adding new points if necessary, a set $H\subseteq P$ that contains at least one point in each input square. We present an $O(\log n)$-competitive deterministic algorithm for this problem. The competitive ratio is the best possible, apart from constant factors. In fact, this is the first $O(\log n)$-competitive algorithm for the online hitting set problem that works for geometric objects of arbitrary sizes (i.e., arbitrary scaling factors) in the plane. We further generalize this result to positive homothets of a polygon with $k\geq 3$ vertices in the plane and provide an $O(k^2\log n)$-competitive algorithm.

翻译：给定平面上一个包含 $n$ 个点的集合 $P$，以及以在线方式陆续到达的轴对齐正方形序列。在线击中集问题要求通过必要时添加新点，维护一个子集 $H\subseteq P$，使得每个输入正方形中至少包含 $H$ 中的一个点。我们针对该问题提出了一种 $O(\log n)$-竞争比的确定性算法。该竞争比在常数因子范围内是最优的。事实上，这是首个针对平面上任意尺寸（即任意缩放比例）几何对象的在线击中集问题，达到 $O(\log n)$-竞争比的算法。我们进一步将该结果推广到平面上具有 $k\\geq 3$ 个顶点的多边形的正位似变换，并给出了一个 $O(k^2\\log n)$-竞争比的算法。