Preconditioning for the Geometric Transportation Problem

In the geometric transportation problem, we are given a collection of points $P$ in $d$-dimensional Euclidean space, and each point is given a supply of $μ(p)$ units of mass, where $μ(p)$ could be a positive or a negative integer, and the total sum of the supplies is $0$. The goal is to find a flow (called a transportation map) that transports $μ(p)$ units from any point $p$ with $μ(p) > 0$, and transports $-μ(p)$ units into any point $p$ with $μ(p) < 0$. Moreover, the flow should minimize the total distance traveled by the transported mass. The optimal value is known as the transportation cost, or the Earth Mover's Distance (from the points with positive supply to those with negative supply). This problem has been widely studied in many fields of computer science: from theoretical work in computational geometry, to applications in computer vision, graphics, and machine learning. In this work we study approximation algorithms for the geometric transportation problem. We give an algorithm which, for any fixed dimension $d$, finds a $(1+\varepsilon)$-approximate transportation map in time nearly-linear in $n$, and polynomial in $\varepsilon^{-1}$ and in the logarithm of the total supply. This is the first approximation scheme for the problem whose running time depends on $n$ as $n\cdot \mathrm{polylog}(n)$. Our techniques combine the generalized preconditioning framework of Sherman, which is grounded in continuous optimization, with simple geometric arguments to first reduce the problem to a minimum cost flow problem on a sparse graph, and then to design a good preconditioner for this latter problem.