Online Hitting of Unit Balls and Hypercubes in $\mathbb{R}^d$ using Points from $\mathbb{Z}^d$

We consider the online hitting set problem for the range space $\Sigma=(\cal X,\cal R)$, where the point set $\cal X$ is known beforehand, but the set $\cal R$ of geometric objects is not known in advance. Here, geometric objects arrive one by one, the objective is to maintain a hitting set of minimum cardinality by taking irrevocable decisions. In this paper, we have considered the problem when the objects are unit balls or unit hypercubes in $\mathbb{R}^d$, and the points from $\mathbb{Z}^d$ are used for hitting them. First, we consider the problem for objects (unit balls and unit hypercubes) in lower dimensions. We obtain $4$ and $8$-competitive deterministic online algorithms for hitting unit hypercubes in $\mathbb{R}^2$ and $\mathbb{R}^3$, respectively. On the other hand, we present $4$ and $14$-competitive deterministic online algorithms for hitting unit balls in $\mathbb{R}^2$ and $\mathbb{R}^3$, respectively. Next, we consider the problem for objects (unit balls and unit hypercubes) in the higher dimension. For hitting unit hypercubes in $\mathbb{R}^d$, we present a $O(d^2)$-competitive randomized online algorithm for $d\geq 3$ and prove the competitive ratio of any deterministic algorithm for the problem is at least $d+1$ for any $d\in\mathbb{N}$. Then, for hitting unit balls in $\mathbb{R}^d$, we propose a $O(d^4)$-competitive deterministic algorithm, and for $d<4$, we establish that the competitive ratio of any deterministic algorithm is at least $d+1$.