Multipacking in Euclidean Metric Space

Here we study the multipacking problems for geometric point sets with respect to their Euclidean distances. We consider a set of $n$ points $P$ and define $N_s[v]$ as the subset of $P$ that includes the $s$ nearest points of $v \in P$ and the point $v$ itself. We assume that the \emph{$s$-th neighbor} of each point is unique, for every $s \in \{0, 1, 2, \dots , n-1\}$. For a natural number $r \leq n-1$, an $r$-multipacking is a set $M \subseteq P$ such that for each point $v \in P$ and for every integer $1\leq s \leq r$, $|N_s[v]\cap M|\leq (s+1)/2$. The $r$-multipacking number of $P$ is the maximum cardinality of an $r$-multipacking of $P$ and is denoted by $\MP_{r}(P)$. For $r=n-1$, an $r$-multipacking is called a multipacking and $r$-multipacking number is called as multipacking number. For $r=1 \text{ and } 2$, we study the problem of computing a maximum $r$-multipacking of the point sets in $\mathbb{R}^2$. We show that a maximum $1$-multipacking can be computed in polynomial time but computing a maximum $2$-multipacking is \textsc{NP-hard}. Further, we provide approximation and parameterized solutions to the $2$-multipacking problem.