Discriminating Codes in Geometric Setups

We study two geometric variations of the discriminating code problem. In the \emph{discrete version}, a finite set of points $P$ and a finite set of objects $S$ are given in $\mathbb{R}^d$. The objective is to choose a subset $S^* \subseteq S$ of minimum cardinality such that the subsets $S_i^* \subseteq S^*$ covering $p_i$, satisfy $S_i^*\neq \emptyset$ for each $i=1,2,\ldots, n$, and $S_i^* \neq S_j^*$ for each pair $(i,j)$, $i \neq j$. In the \emph{continuous version}, the solution set $S^*$ can be chosen freely among a (potentially infinite) class of allowed geometric objects. In the 1-dimensional case ($d=1$), the points are placed on some fixed-line $L$, and the objects in $S$ and $S^*$ are finite sub-segments of $L$ (called intervals). We show that the discrete version of this problem is NP-complete. This is somewhat surprising as the continuous version is known to be polynomial-time solvable. This is also in contrast with most geometric covering problems, which are usually polynomial-time solvable in 1D. We then design a polynomial-time $2$-approximation algorithm for the 1-dimensional discrete case. We also design a PTAS for both discrete and continuous cases when the intervals are all required to have the same length. We then study the 2-dimensional case ($d=2$) for axis-parallel unit square objects. We show that both continuous and discrete versions are NP-hard, and design polynomial-time approximation algorithms with factors $4+\epsilon$ and $32+\epsilon$, respectively (for every fixed $\epsilon>0$).