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 online algorithm that solves this problem has a competitive ratio of at least $\Omega(n)$, which even holds when the objects are intervals. This paper considers the piercing set problem when objects are bounded scaled. We propose deterministic algorithms for bounded scaled fat objects. Piercing translated copies of an 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 unit objects are anything other than balls and hypercubes. Our result gives an upper bound of the competitive ratio for the unit covering problem for various unit objects. For fixed-oriented hypercubes in $\mathbb{R}^d$ with the scaling factor in the range $[1,k]$, we propose an algorithm having a competitive ratio of at most~$3^d\log_2 k+2^d$. In the end, we show a lower bound of the competitive ratio for bounded scaled objects of various types like $\alpha$-fat objects in $\mathbb{R}^2$, axis-aligned hypercubes in $\mathbb{R}^d$, and balls in $\mathbb{R}^2$ and~$\mathbb{R}^3$.

翻译：我们考虑穿刺集问题的在线版本，其中几何对象逐个到达，在线算法必须通过做出不可撤销的决策来维护已到达对象的有效穿刺集。容易观察到，任何解决该问题的确定性在线算法，其竞争比至少为$\Omega(n)$，即使对象是区间时该结论依然成立。本文研究了对象具有有界缩放性质时的穿刺集问题。我们针对有界缩放的胖对象提出了确定性算法。对某个对象的平移副本进行穿刺等价于单位覆盖问题，后者在在线场景下已被充分研究。令人惊讶的是，当单位对象并非仅仅局限于球体和超立方体时，单位覆盖问题的竞争比上界此前尚不明确。我们的结果为多种单位对象的单位覆盖问题给出了竞争比上界。对于$\mathbb{R}^d$中缩放因子范围在$[1,k]$内的定向固定超立方体，我们提出了一种竞争比至多为$3^d\log_2 k+2^d$的算法。最后，我们展示了多种有界缩放对象（如$\mathbb{R}^2$中的$\alpha$-胖对象、$\mathbb{R}^d$中的轴对齐超立方体、以及$\mathbb{R}^2$和$\mathbb{R}^3$中的球体）的竞争比下界。