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：图上的输运问题是一个跨多个领域的基础性挑战，其决策必须尊重拓扑与运行约束。尽管需要可执行的策略，现有图输运方法却缺乏这种表达力。它们依赖限制性假设，无法在稀疏拓扑上泛化，且难以随图规模和时间跨度扩展。为解决这些问题，我们提出了图上的广义薛定谔桥（GSBoG），这是一种新颖的可扩展数据驱动框架，用于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，并在状态代价增强动力学框架下运行。值得注意的是，GSBoG学习轨迹级策略，避免了稠密的全局求解器，从而增强了可扩展性。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，同时优化状态依赖运行成本下的中间行为。在具有挑战性的真实世界图拓扑上的大量实验表明，GSBoG能够可靠地学习准确且尊重拓扑的策略，同时优化特定应用场景的中间状态代价，突显了其广泛的适用性，并为一般图上代价感知的动态输运开辟了新途径。