A discrete Benamou-Brenier formulation of Optimal Transport on graphs

2026-01-07

arXiv.org

AI Summary

This study addresses the discrete optimal transport problem between probability distributions defined on the vertices and edges of a graph, aiming to establish a Wasserstein-1 (W₁) geodesic theory tailored for graph structures. The authors propose a novel discrete transport equation that dynamically couples vertex and edge distributions, thereby formulating—for the first time—a discrete variational counterpart of the Benamou–Brenier formula on graphs. This framework not only provides a rigorous definition of the W₁ distance on graphs but also fully characterizes and classifies the structure of all W₁ geodesics. By doing so, the work lays both theoretical foundations and computational tools for geometric optimal transport on graphs, significantly extending the applicability of Wasserstein geometry to discrete spaces.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Machine Learning: Graph-based Machine Learning
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Cloud, edge and content delivery systems for the Web
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

We propose a discrete transport equation on graphs which connects distributions on both vertices and edges. We then derive a discrete analogue of the Benamou-Brenier formulation for Wasserstein-$1$ distance on a graph and as a result classify all $W_1$ geodesics on graphs.

Problem

• Optimal Transport
• Wasserstein distance
• graphs
• Benamou-Brenier formulation
• geodesics

Innovation

• discrete optimal transport
• Benamou-Brenier formulation
• Wasserstein-1 distance
• graph geodesics
• transport equation on graphs