Generalized Schr\"odinger Bridge on Graphs

2026-02-04

AI Summary

Existing optimal transport methods on graphs are hindered by strong assumptions, poor generalization to sparse or complex topologies, and limited scalability, making it difficult to produce executable policies. This work proposes a scalable, data-driven framework based on continuous-time Markov chains that learns controllable transport policies by optimizing trajectory-level likelihoods, subject to endpoint marginal constraints and minimizing state-dependent running costs. For the first time, this approach enables efficient learning of continuous-time transport strategies modulated by state costs on arbitrary graph structures, circumventing the need for global dense solvers. Extensive experiments on multiple real-world complex graphs demonstrate the method’s effectiveness and broad applicability in terms of accuracy, topological compatibility, and cost optimization.

Technology Category

• Planning, Routing, and Scheduling: Planning with Markov Models (MDPs, POMDPs)
• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

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 Schr\"odinger 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.

Problem

• graph transport
• scalability
• topological constraints
• executable policies
• sparse topologies

Innovation

• Generalized Schrödinger Bridge
• continuous-time Markov chain
• graph transport
• trajectory-level policy
• cost-aware dynamics