The computation of the domination-type parameters is a challenging problem in Cartesian product graphs. We present an algorithmic method to compute the $2$-domination number of the Cartesian product of a path with small order and any cycle, involving the $(\min,+)$ matrix product. We establish some theoretical results that provide the algorithms necessary to compute that parameter, and the main challenge to run such algorithms comes from the large size of the matrices used, which makes it necessary to improve the techniques to handle these objects. We analyze the performance of the algorithms on modern multicore CPUs and on GPUs and we show the advantages over the sequential implementation. The use of these platforms allows us to compute the $2$-domination number of cylinders such that their paths have at most $12$ vertices.

翻译：笛卡尔积图中支配型参数的计算是一个具有挑战性的问题。本文提出一种算法方法，用于计算路径（阶数较小）与任意环的笛卡尔积的2-支配数，该方法涉及(min, +)矩阵乘积运算。我们建立了若干理论结果，为计算该参数提供了必要的算法基础，而运行此类算法的主要挑战源于所用矩阵的庞大尺寸，这使得改进处理这些对象的技术成为必需。我们分析了算法在现代多核CPU和GPU上的性能表现，并展示了其相对于串行实现的优势。利用这些计算平台，我们成功计算了路径顶点数不超过12的柱状图（cylinder）的2-支配数。