An ADMM-Based Approach to Quadratically-Regularized Distributed Optimal Transport on Graphs

TL;DR

This paper presents a distributed ADMM-based algorithm for solving quadratically-regularized optimal transport problems on directed graphs, extending previous methods to more general graph structures and demonstrating convergence and robustness.

Contribution

It introduces a novel distributed algorithm for optimal transport on directed graphs with quadratic regularization, ensuring convergence without convexity constraints.

Findings

Algorithm converges with and without quadratic regularization.

Quadratic regularization influences convergence speed and solution sparsity.

The method is robust to changes in graph topology.

Abstract

Optimal transport on a graph focuses on finding the most efficient way to transfer resources from one distribution to another while considering the graph's structure. This paper introduces a new distributed algorithm that solves the optimal transport problem on directed, strongly connected graphs, unlike previous approaches which were limited to bipartite graphs. Our algorithm incorporates quadratic regularization and guarantees convergence using the Alternating Direction Method of Multipliers (ADMM). Notably, it proves convergence not only with quadratic regularization but also in cases without it, whereas earlier works required strictly convex objective functions. In this approach, nodes are treated as agents that collaborate through local interactions to optimize the total transportation cost, relying only on information from their neighbors. Through numerical experiments, we show how quadratic regularization affects both convergence behavior and solution sparsity under different graph structures. Additionally, we provide a practical example that highlights the algorithm's robustness through its ability to adjust to topological changes in the graph.