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 $\gamma_R(G)$ 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 $m\geq 10, n\not\equiv 0\pmod 5$.

翻译：图的罗马控制是经典控制问题的一个变体，其通过所谓的罗马控制函数$f\colon V(G)\to \{0,1,2\}$定义，满足若$f(v)=0$，则顶点$v$至少与一个满足$f(w)=2$的顶点$w$相邻。图$G$的罗马控制函数$f$的权值$f(G)$为$G$中所有顶点权值之和，即$f(G)=\sum_{u\in V(G)}f(u)$。罗马控制数$\gamma_R(G)$是$G$的所有罗马控制函数中的最小权值。本文提出了计算该参数的算法，其涉及具有高计算需求的大规模矩阵的$(\min,+)$幂运算，而GPU（图形处理单元）使我们能够加速此类运算。我们开发了特定例程，以充分利用GPU架构的计算能力，高效计算$(\min ,+)$乘积。这些算法使我们能够计算柱面图$P_m\Box C_n$（即一条路径与一个环的笛卡尔积）的罗马控制数，具体针对$m=7,8,9$，$ n\geq 3$，以及$m\geq 10$，$n\equiv 0\pmod 5$的情形。此外，我们为其余情形$m\geq 10, n\not\equiv 0\pmod 5$提供了一个下界。