Transportation on graphs is a fundamental challenge across many domains, where decisions must respect topological and operational constraints. Despite the need for actionable policies, existing graph-transport methods lack this expressivity. They rely on restrictive assumptions, fail to generalize across sparse topologies, and scale poorly with graph size and time horizon. To address these issues, we introduce Generalized Schrödinger Bridge on Graphs (GSBoG), a novel scalable data-driven framework for learning executable controlled continuous-time Markov chain (CTMC) policies on arbitrary graphs under state cost augmented dynamics. Notably, GSBoG learns trajectory-level policies, avoiding dense global solvers and thereby enhancing scalability. This is achieved via a likelihood optimization approach, satisfying the endpoint marginals, while simultaneously optimizing intermediate behavior under state-dependent running costs. Extensive experimentation on challenging real-world graph topologies shows that GSBoG reliably learns accurate, topology-respecting policies while optimizing application-specific intermediate state costs, highlighting its broad applicability and paving new avenues for cost-aware dynamical transport on general graphs.

翻译：图上的运输是跨多个领域的一个基本挑战，其中决策必须遵循拓扑和操作约束。尽管需要可执行的策略，但现有的图运输方法缺乏这种表达能力。它们依赖于限制性假设，无法在稀疏拓扑结构中泛化，并且随着图规模和时间范围的增加而扩展性差。为解决这些问题，我们引入了图上的广义薛定谔桥（GSBoG），这是一个新颖的、可扩展的数据驱动框架，用于在状态成本增强的动态下，在任意图上学习可执行的受控连续时间马尔可夫链（CTMC）策略。值得注意的是，GSBoG学习轨迹级别的策略，避免了密集的全局求解器，从而提高了可扩展性。这是通过一种似然优化方法实现的，该方法满足端点边际分布，同时优化在状态相关运行成本下的中间行为。在具有挑战性的真实世界图拓扑结构上进行的大量实验表明，GSBoG能够可靠地学习准确、尊重拓扑的策略，同时优化特定应用的中间状态成本，突显了其广泛的适用性，并为一般图上的成本感知动态运输开辟了新途径。