A deterministic near-linear time approximation scheme for geometric transportation (2211.03891v3)
Abstract: Given a set of points $P = (P+ \sqcup P-) \subset \mathbb{R}d$ for some constant $d$ and a supply function $\mu:P\to \mathbb{R}$ such that $\mu(p) > 0~\forall p \in P+$, $\mu(p) < 0~\forall p \in P-$, and $\sum_{p\in P}{\mu(p)} = 0$, the geometric transportation problem asks one to find a transportation map $\tau: P+\times P-\to \mathbb{R}{\ge 0}$ such that $\sum{q\in P-}{\tau(p, q)} = \mu(p)~\forall p \in P+$, $\sum_{p\in P+}{\tau(p, q)} = -\mu(q)~ \forall q \in P-$, and the weighted sum of Euclidean distances for the pairs $\sum_{(p,q)\in P+\times P-}\tau(p, q)\cdot ||q-p||_2$ is minimized. We present the first deterministic algorithm that computes, in near-linear time, a transportation map whose cost is within a $(1 + \varepsilon)$ factor of optimal. More precisely, our algorithm runs in $O(n\varepsilon{-(d+2)}\log5{n}\log{\log{n}})$ time for any constant $\varepsilon > 0$. 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 $(1 + \varepsilon)$-approximation algorithms, randomized or deterministic, even for geometric bipartite matching. In particular, we give the first $(1 + \varepsilon)$-approximate deterministic algorithm for geometric bipartite matching and the first $(1 + \varepsilon)$-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 $O(\varepsilon{-2} m \log{O(1)} n)$ time $(1 + \varepsilon)$-approximation algorithm for the uncapacitated minimum cost flow (transshipment) problem in undirected graphs with arbitrary real edge costs.