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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2409.17658 |
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| _version_ | 1866913519620849664 |
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| author | Martínez, J. A. Garzón, E. M. Puertas, M. L. |
| author_facet | Martínez, J. A. Garzón, E. M. Puertas, M. L. |
| contents | The Roman domination in a graph $G$ is a variant of the classical domination, defined by means of a so-called Roman domination function $f\colon V(G)\to \{0,1,2\}$ such that if $f(v)=0$ then, the vertex $v$ is adjacent to at least one vertex $w$ with $f(w)=2$. The weight $f(G)$ of a Roman dominating function of $G$ is the sum of the weights of all vertices of $G$, that is, $f(G)=\sum_{u\in V(G)}f(u)$. The Roman domination number $γ_R(G)$ is the minimum weight of a Roman dominating function of $G$. In this paper we propose algorithms to compute this parameter involving the $(\min,+)$ powers of large matrices with high computational requirements and the GPU (Graphics Processing Unit) allows us to accelerate such operations. Specific routines have been developed to efficiently compute the $(\min ,+)$ product on GPU architecture, taking advantage of its computational power. These algorithms allow us to compute the Roman domination number of cylindrical graphs $P_m\Box C_n$ i.e., the Cartesian product of a path and a cycle, in cases $m=7,8,9$, $ n\geq 3$ and $m\geq $10$, n\equiv 0\pmod 5$. Moreover, we provide a lower bound for the remaining cases $m\geq 10, n\not\equiv 0\pmod 5$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_17658 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Powers of large matrices on GPU platforms to compute the Roman domination number of cylindrical graphs Martínez, J. A. Garzón, E. M. Puertas, M. L. Combinatorics Discrete Mathematics The Roman domination in a graph $G$ is a variant of the classical domination, defined by means of a so-called Roman domination function $f\colon V(G)\to \{0,1,2\}$ such that if $f(v)=0$ then, the vertex $v$ is adjacent to at least one vertex $w$ with $f(w)=2$. The weight $f(G)$ of a Roman dominating function of $G$ is the sum of the weights of all vertices of $G$, that is, $f(G)=\sum_{u\in V(G)}f(u)$. The Roman domination number $γ_R(G)$ is the minimum weight of a Roman dominating function of $G$. In this paper we propose algorithms to compute this parameter involving the $(\min,+)$ powers of large matrices with high computational requirements and the GPU (Graphics Processing Unit) allows us to accelerate such operations. Specific routines have been developed to efficiently compute the $(\min ,+)$ product on GPU architecture, taking advantage of its computational power. These algorithms allow us to compute the Roman domination number of cylindrical graphs $P_m\Box C_n$ i.e., the Cartesian product of a path and a cycle, in cases $m=7,8,9$, $ n\geq 3$ and $m\geq $10$, n\equiv 0\pmod 5$. Moreover, we provide a lower bound for the remaining cases $m\geq 10, n\not\equiv 0\pmod 5$. |
| title | Powers of large matrices on GPU platforms to compute the Roman domination number of cylindrical graphs |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2409.17658 |