A deterministic near-linear time approximation scheme for geometric transportation
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Given a set of points P = (P^+ ⊔ P^-) ⊂ℝ^d for some constant d and a supply function μ:P→ℝ such that μ(p) > 0 ∀ p ∈ P^+, μ(p) < 0 ∀ p ∈ P^-, and ∑_p∈ Pμ(p) = 0, the geometric transportation problem asks one to find a transportation map τ: P^+× P^-→ℝ_≥ 0 such that ∑_q∈ P^-τ(p, q) = μ(p) ∀ p ∈ P^+, ∑_p∈ P^+τ(p, q) = -μ(q) ∀ q ∈ P^-, and the weighted sum of Euclidean distances for the pairs ∑_(p,q)∈ P^+× P^-τ(p, q)· ||q-p||_2 is minimized. We present the first deterministic algorithm that computes, in near-linear time, a transportation map whose cost is within a (1 + ε) factor of optimal. More precisely, our algorithm runs in O(nε^-(d+2)log^5nloglogn) time for any constant ε > 0. Surprisingly, our result is not only a generalization of a bipartite matching one to arbitrary instances of geometric transportation, but it also reduces the running time for all previously known (1 + ε)-approximation algorithms, randomized or deterministic, even for geometric bipartite matching. In particular, we give the first (1 + ε)-approximate deterministic algorithm for geometric bipartite matching and the first (1 + ε)-approximate deterministic or randomized algorithm for geometric transportation with no dependence on d in the exponent of the running time's polylog. As an additional application of our main ideas, we also give the first randomized near-linear O(poly(1 / ε) m log^O(1) n) time (1 + ε)-approximation algorithm for the uncapacitated minimum cost flow (transshipment) problem in undirected graphs with arbitrary real edge costs.
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