Balancing resource efficiency and fairness is critical in networked systems that support modern learning applications. We introduce the Fair Minimum Labeling (FML) problem: the task of designing a minimum-cost temporal edge activation plan that ensures each group of nodes in a network has sufficient access to a designated target set, according to specified coverage requirements. FML captures key trade-offs in systems where edge activations incur resource costs and equitable access is essential, such as distributed data collection, update dissemination in edge-cloud systems, and fair service restoration in critical infrastructure. We show that FML is NP-hard and $\Omega(\log |V|)$-hard to approximate, where $V$ is the set of nodes, and we present probabilistic approximation algorithms that match this bound, achieving the best possible guarantee for the activation cost. We demonstrate the practical utility of FML in a fair multi-source data aggregation task for training a shared model. Empirical results show that FML enforces group-level fairness with substantially lower activation cost than baseline heuristics, underscoring its potential for building resource-efficient, equitable temporal reachability in learning-integrated networks.

翻译：在支持现代学习应用的网络系统中，平衡资源效率与公平性至关重要。本文提出公平最小标记问题：其任务在于设计一种最小成本的时序边激活方案，以确保网络中每个节点组根据指定的覆盖要求，能够充分访问指定的目标集合。FML 捕捉了在边激活产生资源成本且公平访问至关重要的系统中的关键权衡，例如分布式数据收集、边缘-云系统中的更新传播以及关键基础设施中的公平服务恢复。我们证明 FML 是 NP 难的，且近似难度为 $\Omega(\log |V|)$，其中 $V$ 是节点集合，并提出了匹配此界限的概率近似算法，实现了激活成本的最佳可能保证。我们在训练共享模型的公平多源数据聚合任务中展示了 FML 的实际效用。实验结果表明，与基线启发式方法相比，FML 能以显著更低的激活成本强制执行组级公平性，凸显了其在学习集成网络中构建资源高效、公平的时序可达性方面的潜力。