The Roamn domination problem is one important combinatorial optimization problem that is derived from an old story of defending the Roman Empire and now regains new significance in cyber space security, considering backups in the face of a dynamic network security requirement. In this paper, firstly, we propose a Roman domination game (RDG) and prove that every Nash equilibrium (NE) of the game corresponds to a strong minimal Roman dominating function (S-RDF), as well as a Pareto-optimal solution. Secondly, we show that RDG is an exact potential game, which guarantees the existence of an NE. Thirdly, we design a game-based synchronous algorithm (GSA), which can be implemented distributively and converge to an NE in $O(n)$ rounds, where $n$ is the number of vertices. In GSA, all players make decisions depending on the local information. Furthermore, we enhance GSA to be enhanced GSA (EGSA), which converges to a better NE in $O(n^2)$ rounds. Finally, we present numerical simulations to demonstrate that EGSA can obtain a better approximate solution in promising computation time compared with state-of-the-art algorithms.

翻译：罗马支配问题是一个重要的组合优化问题，源自保卫罗马帝国的古老故事。鉴于动态网络安全需求下的备份问题，该问题在网络空间安全领域重新获得了重要意义。本文首先提出了一种罗马支配博弈（RDG），并证明该博弈的每个纳什均衡（NE）都对应一个强最小罗马支配函数（S-RDF）及一个帕累托最优解。其次，我们证明了RDG是一个精确势博弈，确保了NE的存在性。第三，我们设计了一种基于博弈的同步算法（GSA），该算法可分布式实现，并在$O(n)$轮内收敛至NE，其中$n$为顶点数。在GSA中，所有玩家根据局部信息做出决策。此外，我们将GSA改进为增强型GSA（EGSA），该算法在$O(n^2)$轮内收敛至更优的NE。最后，通过数值模拟表明，与现有最优算法相比，EGSA能在可接受的计算时间内获得更优的近似解。