Research involving computing with mobile agents is a fast-growing field, given the advancement of technology in automated systems, e.g., robots, drones, self-driving cars, etc. Therefore, it is pressing to focus on solving classical network problems using mobile agents. In this paper, we study one such problem -- finding small dominating sets of a graph $G$ using mobile agents. Dominating set is interesting in the field of mobile agents as it opens up a way for solving various robotic problems, e.g., guarding, covering, facility location, transport routing, etc. In this paper, we first present two algorithms for computing a {\em minimal dominating set}: (i) an $O(m)$ time algorithm if the robots start from a single node (i.e., gathered initially), (ii) an $O(\ell\Delta\log(\lambda)+n\ell+m)$ time algorithm, if the robots start from multiple nodes (i.e., positioned arbitrarily), where $m$ is the number of edges and $\Delta$ is the maximum degree of $G$, $\ell$ is the number of clusters of the robot initially and $\lambda$ is the maximum ID-length of the robots. Then we present a $\ln (\Delta)$ approximation algorithm for the {\em minimum} dominating set which takes $O(n\Delta\log (\lambda))$ rounds.

翻译：随着自动化系统（如机器人、无人机、自动驾驶汽车等）技术的进步，涉及移动代理计算的研究正快速发展。因此，聚焦于利用移动代理解决经典网络问题变得尤为迫切。本文研究其中一个问题——利用移动代理寻找图$G$的最小控制集。控制集在移动代理领域颇具意义，因为它为解决多种机器人问题（如巡逻、覆盖、设施选址、运输路径规划等）开辟了途径。本文首先提出两种计算{\em最小控制集}的算法：（i）若机器人从单一节点启动（即初始聚集），则算法时间复杂度为$O(m)$；（ii）若机器人从多节点启动（即任意分布），则算法时间复杂度为$O(\ell\Delta\log(\lambda)+n\ell+m)$，其中$m$为边数，$\Delta$为图$G$的最大度，$\ell$为机器人初始簇数，$\lambda$为机器人最大ID长度。随后，我们提出一种针对{\em最小}控制集的$\ln (\Delta)$近似算法，其运行轮次为$O(n\Delta\log (\lambda))$。