Fast Iterative Solution of the Optimal Transport Problem on Graphs

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