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 $\mu : P \to \mathbb{R}$. The goal is to find a transportation map, a non-negative assignment $\tau : P \times P \to \mathbb{R}_{\geq 0}$ to pairs of points, so the total assignment leaving each point is equal to its supply, i.e., $\sum_{r \in P} \tau(q, r) - \sum_{p \in P} \tau(p, q) = \mu(q)$ for all points $q \in P$. The goal is to minimize the weighted sum of Euclidean distances for the pairs, $\sum_{(p, q) \in P \times P} \tau(p, q) \cdot ||q - p||_2$. We describe the first algorithm for this problem that returns, with high probability, a $(1 + \epsilon)$-approximation to the optimal transportation map in $O(n\:\text{poly}(1 / \epsilon)\:\text{polylog}{n})$ 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.