A near-linear time approximation scheme for geometric transportation with arbitrary supplies and spread

The geometric transportation problem takes as input a set of points P in d-dimensional Euclidean space and a supply function μ : P → ℝ. The goal is to find a transportation map, a non-negative assignment τ : P × P → ℝ≥ 0 to pairs of points, so the total assignment leaving each point is equal to its supply, i.e., sumr ∈ P τ(q, r) - sump ∈ P τ(p, q) = μ(q) for all points q ∈ P. The goal is to minimize the weighted sum of Euclidean distances for the pairs, sum(p, q) ∈ P × P τ(p, q) · ||q - p||2. We describe the first algorithm for this problem that returns, with high probability, a (1 + varepsilon)-approximation to the optimal transportation map in nvarepsilon-O(d)logO(d)n time. In contrast to the previous best algorithms for this problem, our near-linear running time bound is independent of the spread of P and the magnitude of its real-valued supplies.