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

Fast Iterative Solution of the Optimal Transport Problem on Graphs