Online Geometric Covering and Piercing

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~$\mathbb{R}^3$.