We consider the online version of the piercing set problem, where geometric objects arrive one by one, and the online algorithm must maintain a valid piercing set for the already arrived objects by making irrevocable decisions. It is easy to observe that any deterministic algorithm solving this problem for intervals in $\mathbb{R}$ has a competitive ratio of at least $\Omega(n)$. This paper considers the piercing set problem for similarly sized objects. We propose a deterministic online algorithm for similarly sized fat objects in $\mathbb{R}^d$. For homothetic hypercubes in $\mathbb{R}^d$ with side length in the range $[1,k]$, we propose a deterministic algorithm having a competitive ratio of at most~$3^d\lceil\log_2 k\rceil+2^d$. In the end, we show deterministic lower bounds of the competitive ratio for similarly sized $\alpha$-fat objects in $\mathbb{R}^2$ and homothetic hypercubes in $\mathbb{R}^d$. Note that piercing translated copies of a convex object is equivalent to the unit covering problem, which is well-studied in the online setup. Surprisingly, no upper bound of the competitive ratio was known for the unit covering problem when the corresponding object is anything other than a ball or a hypercube. Our result yields an upper bound of the competitive ratio for the unit covering problem when the corresponding object is any convex object in $\mathbb{R}^d$.

翻译：本文研究刺穿集问题的在线版本，其中几何对象逐个到达，在线算法必须通过不可撤销的决策为已到达对象维护一个有效的刺穿集。容易观察到，对于$\mathbb{R}$上的区间，任何求解该问题的确定性算法都具有至少$\Omega(n)$的竞争比。本文研究相似尺寸对象的刺穿集问题。我们针对$\mathbb{R}^d$中相似尺寸的胖对象提出了一种确定性在线算法。对于边长在$[1,k]$范围内的$\mathbb{R}^d$中相似超立方体，我们提出的确定性算法竞争比至多为$3^d\lceil\log_2 k\rceil+2^d$。最后，我们给出了$\mathbb{R}^2$中相似尺寸$\alpha$-胖对象与$\mathbb{R}^d$中相似超立方体的竞争比确定性下界。值得注意的是，刺穿凸对象的平移副本等价于单位覆盖问题，该问题在线设置中已有深入研究。令人惊讶的是，当对应对象是球体或超立方体之外的任意凸对象时，单位覆盖问题的竞争比上界此前一直未知。我们的研究结果为$\mathbb{R}^d$中任意凸对象对应的单位覆盖问题提供了竞争比上界。