A deterministic near-linear time approximation scheme for geometric transportation - Fox, Emily; Lu, Jiashuai (Proceedings / annual Symposium on Foundations of Computer Science, 2023)
Given a set of points P=\left(P^{+} \sqcup P^{-}\right) \subset \mathbb{R}^{d} for some constant d and a supply function μ: P → \mathbb{R} such that μ(p)\gt 0 \forall p \in P^{+}, μ(p)\lt 0 \forall p \in P^{-}, and \sum_{p \in P} μ(p)=0, the geometric transportation problem asks one to find a transportation map τ: P^{+} × P^{-} → \mathbb{R}_{≥ 0} such that \sum_{q \in P^{-}} τ(p, q)=μ(p) \forall p \in P^{+}, \sum_{p \in P^{+}} τ(p, q)=-μ(q) \forall q \in P^{-}, and the weighted sum of Euclidean distances for the pairs \sum_{(p, q) \in 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\left(n ε^{-(d+2)} \log ^{5} n \log \log n\right) time for any constant ε>0. While a randomized n ε^{-O(d)} \log ^{O(d)} n time algorithm for this problem was discovered in the last few years, all previously known deterministic (1+ε)-approximation algorithms run in Ω\left(n^{3 / 2}\right) time. A similar situation existed for geometric bipartite matching, the special case of geometric transportation where all supplies are unit, until a deterministic n ε^{-O(d)} \log ^{O(d)} n time (1+ε)-approximation algorithm was presented at STOC 2022. Surprisingly, our result is not only a generalization of the 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\left(ε^{-2} m \log ^{O(1)} n\right) time (1+ε)-approximation algorithm for the uncapacitated minimum cost flow (transshipment) problem in undirected graphs with arbitrary real edge costs.