A near-linear time approximation scheme for geometric transportation with real supplies
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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., ∑_r ∈ Pτ(q, r) - ∑_p ∈ Pτ(p, q) = μ(q) for all points q ∈ P. The goal is to minimize the weighted sum of Euclidean distances for the pairs, ∑_(p, q) ∈ P × Pτ(p, q) · ||q - p||_2. We describe the first algorithm for this problem that returns, with high probability, a (1 + ϵ)-approximation to the optimal transportation map in O(n poly(1 / ϵ) polylogn) 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.
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