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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2411.12351 |
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| _version_ | 1866913941575172096 |
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| author | Das, Arun Kumar Das, Sandip Islam, Sk Samim Mitra, Ritam M Roy, Bodhayan |
| author_facet | Das, Arun Kumar Das, Sandip Islam, Sk Samim Mitra, Ritam M Roy, Bodhayan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_12351 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multipacking in Euclidean Metric Space Das, Arun Kumar Das, Sandip Islam, Sk Samim Mitra, Ritam M Roy, Bodhayan Computational Geometry Combinatorics 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. |
| title | Multipacking in Euclidean Metric Space |
| topic | Computational Geometry Combinatorics |
| url | https://arxiv.org/abs/2411.12351 |