Fast Iterative Solution of the Optimal Transport Problem on Graphs

TL;DR

This paper presents a robust and efficient numerical method for solving the Optimal Transport Problem on graphs using a backward Euler scheme and algebraic multigrid methods, achieving scalable performance based on the number of edges.

Contribution

It introduces a novel iterative approach combining backward Euler and inexact Newton-Raphson methods for optimal transport on graphs, with proven efficiency and scalability.

Findings

Requires solving between O(1) and O(M^0.36) linear systems

Uses algebraic multigrid for efficient linear system solutions

Achieves robust and accurate solutions for large graphs

Abstract

In this paper, we address the numerical solution of the Optimal Transport Problem on undirected weighted graphs, taking the shortest path distance as transport cost. The optimal solution is obtained from the long-time limit of the gradient descent dynamics. Among different time stepping procedures for the discretization of this dynamics, a backward Euler time stepping scheme combined with the inexact Newton-Raphson method results in a robust and accurate approach for the solution of the Optimal Transport Problem on graphs. It is found experimentally that the algorithm requires solving between $\mathcal{O}(1)$ and $\mathcal{O}(M^{0.36})$ linear systems involving weighted Laplacian matrices, where $M$ is the number of edges. These linear systems are solved via algebraic multigrid methods, resulting in an efficient solver for the Optimal Transport Problem on graphs.