FAST ITERATIVE SOLUTION OF THE OPTIMAL TRANSPORT PROBLEM ON GRAPHS

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 O(1) and 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.

Automated summary

This paper discusses the numerical solution of the optimal transport problem on undirected weighted graphs, proposing an effective method that combines backward Euler time stepping scheme with inexact Newton-Raphson method. Experimental results show that solving linear systems involving weighted Laplacian matrices efficiently using algebraic multigrid methods leads to an accurate solver for the optimal transport problem on graphs.