Motivated by the critical need for unmanned aerial vehicles (UAVs) to patrol grid systems in hazardous and dynamically changing environments, this study addresses a routing problem aimed at minimizing the time-average Age of Information (AoI) for edges in general graphs. We establish a lower bound for all feasible patrol policies and demonstrate that this bound is tight when the graph contains an Eulerian cycle. For graphs without Eulerian cycles, it becomes challenging to identify the optimal patrol strategy due to the extensive range of feasible options. Our analysis shows that restricting the strategy to periodic sequences still results in an exponentially large number of possible strategies. To address this complexity, we introduce two polynomial-time approximation schemes, each involving a two-step process: constructing multigraphs first and then embedding Eulerian cycles within these multigraphs. We prove that both schemes achieve an approximation ratio of 2. Further, both analytical and numerical results suggest that evenly and sparsely distributing edge visits within a periodic route significantly reduces the average AoI compared to strategies that merely minimize the route travel distance. Building on this insight, we propose a heuristic method that not only maintains the approximation ratio of 2 but also ensures robust performance across varying random graphs.

翻译：本研究针对无人机在危险且动态变化环境中巡逻网格系统的关键需求，探讨了旨在最小化一般图中边的时间平均信息年龄（AoI）的路径规划问题。我们建立了所有可行巡逻策略的下界，并证明当图中存在欧拉回路时该下界是紧的。对于不含欧拉回路的图，由于可行策略范围广泛，确定最优巡逻策略变得极具挑战性。分析表明，即使将策略限制为周期序列，仍会面临策略数量指数级增长的问题。为应对这一复杂性，我们提出了两种多项式时间近似方案，均采用两步流程：首先构建多重图，随后在这些多重图中嵌入欧拉回路。我们证明两种方案均能达到2的近似比。进一步的分析与数值结果表明，相较于仅最小化路径行程的策略，在周期路径中均匀稀疏地分布边访问能显著降低平均AoI。基于此发现，我们提出一种启发式方法，该方法不仅能保持2的近似比，还能确保在不同随机图环境下具有鲁棒性能。