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 $O(n\varepsilon^{-(d+2)}\log^5{n}\log{\log{n}})$ 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 $O(poly(1 / \varepsilon) m \log^{O(1)} n)$ time $(1 + \varepsilon)$-approximation algorithm for the uncapacitated minimum cost flow (transshipment) problem in undirected graphs with arbitrary real edge costs.

翻译：给定一组点 $P = (P^+ \sqcup P^-) \subset \mathbb{R}^d$（其中 $d$ 为常数）以及供给函数 $\mu:P\to \mathbb{R}$，满足 $\mu(p) > 0~\forall p \in P^+$，$\mu(p) < 0~\forall p \in P^-$，且 $\sum_{p\in P}{\mu(p)} = 0$。几何运输问题要求寻找一种运输方案 $\tau: P^+\times P^-\to \mathbb{R}_{\ge 0}$，使得 $\sum_{q\in P^-}{\tau(p, q)} = \mu(p)~\forall p \in P^+$，$\sum_{p\in P^+}{\tau(p, q)} = -\mu(q)~ \forall q \in P^-$，并最小化所有配对 $(p,q)$ 的欧几里得距离的加权和 $\sum_{(p,q)\in P^+\times P^-}\tau(p, q)\cdot ||q-p||_2$。我们提出了首个确定性算法，在近线性时间内计算成本在 $(1 + \varepsilon)$ 因子内最优的运输方案。更精确地说，对于任意常数 $\varepsilon > 0$，该算法的时间复杂度为 $O(n\varepsilon^{-(d+2)}\log^5{n}\log{\log{n}})$。令人惊讶的是，我们的结果不仅将二分匹配泛化为任意几何运输实例，还降低了所有已知 $(1 + \varepsilon)$-近似算法（无论是随机化还是确定性）的运行时间，甚至包括几何二分匹配问题。特别地，我们给出了首个几何二分匹配的 $(1 + \varepsilon)$-近似确定性算法，以及首个几何运输的 $(1 + \varepsilon)$-近似确定性或随机化算法——其运行时间多项式对数部分中 $d$ 的指数与 $d$ 无关。作为主要思想的附加应用，我们还针对无向图中具有任意实数边成本的无容量最小费用流（转运）问题，提出了首个随机化近线性 $O(\text{poly}(1 / \varepsilon) m \log^{O(1)} n)$ 时间 $(1 + \varepsilon)$-近似算法。