We initiate the study of 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 $r \leq n$, 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. We study the problem of computing a maximum $r$-multipacking for 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 NP complete. Further, we provide approximation and parameterized solutions to the $2$-multipacking problem.

翻译：我们首次针对几何点集基于其欧几里得距离的多重装填问题展开研究。给定一个包含 $n$ 个点的集合 $P$，定义 $N_s[v]$ 为 $P$ 的子集，其中包含点 $v \in P$ 的 $s$ 个最近邻点及 $v$ 本身。我们假设对于每个 $s \in \{0, 1, 2, \dots , n-1\}$，每个点的\emph{第 $s$ 个邻点}是唯一的。对于满足 $r \leq n$ 的自然数 $r$，$r$-多重装填是指集合 $ M \subseteq P $，使得对于任意点 $ v \in P $ 和每个整数 $ 1\leq s \leq r $，均有 $|N_s[v]\cap M|\leq (s+1)/2$。$P$ 的 $r$-多重装填数是指 $P$ 的 $r$-多重装填的最大基数，记作 $ \MP_{r}(P) $。当 $r=n-1$ 时，$r$-多重装填称为多重装填，其对应的装填数称为多重装填数。我们研究了在 $\mathbb{R}^2$ 中计算点集的最大 $r$-多重装填的问题。我们证明最大 $1$-多重装填可在多项式时间内计算，但计算最大 $2$-多重装填是 NP 完全问题。此外，我们针对 $2$-多重装填问题提出了近似算法与参数化解决方案。