We describe a simple deterministic near-linear time approximation scheme for uncapacitated minimum cost flow in undirected graphs with real edge weights, a problem also known as transshipment. Specifically, our algorithm takes as input a (connected) undirected graph $G = (V, E)$, vertex demands $b \in \mathbb{R}^V$ such that $\sum_{v \in V} b(v) = 0$, positive edge costs $c \in \mathbb{R}_{>0}^E$, and a parameter $\varepsilon > 0$. In $O(\varepsilon^{-2} m \log^{O(1)} n)$ time, it returns a flow $f$ such that the net flow out of each vertex is equal to the vertex's demand and the cost of the flow is within a $(1 + \varepsilon)$ factor of optimal. Our algorithm is combinatorial and has no running time dependency on the demands or edge costs. With the exception of a recent result presented at STOC 2022 for polynomially bounded edge weights, all almost- and near-linear time approximation schemes for transshipment relied on randomization in two main ways: 1) to embed the problem instance into low-dimensional space and 2) to randomly pick target locations to send flow so nearby opposing demands can be satisfied. Our algorithm instead deterministically approximates the cost of routing decisions that would be made if the input were subject to a random tree embedding. To avoid computing the $\Omega(n^2)$ vertex-vertex distances that an approximation of this kind suggests, we also limit the available routing decisions using distances explicitly stored in the well-known Thorup-Zwick distance oracle.

翻译：我们描述了一个针对无容量限制的最小费用流问题的简单确定性近线性时间近似方案，该问题在无向图中具有实值边权，也称为转运问题。具体而言，我们的算法输入一个（连通）无向图 $G = (V, E)$，顶点需求 $b \in \mathbb{R}^V$ 满足 $\sum_{v \in V} b(v) = 0$，正边费用 $c \in \mathbb{R}_{>0}^E$，以及参数 $\varepsilon > 0$。在 $O(\varepsilon^{-2} m \log^{O(1)} n)$ 时间内，它返回一个流 $f$，使得每个顶点的净流出量等于该顶点的需求，且流的费用在最优值的 $(1 + \varepsilon)$ 因子内。我们的算法是组合的，且运行时间不依赖于需求或边费用。除了最近在 STOC 2022 上呈现的一个针对多项式有界边权的结果外，几乎所有近线性时间近似方案都依赖随机化，主要体现在两个方面：1) 将问题实例嵌入低维空间，以及 2) 随机选择目标位置发送流以就近满足相反需求。而我们的算法则确定性近似了在输入受随机树嵌入时可能做出的路由决策的费用。为避免计算此类近似所隐含的 $\Omega(n^2)$ 个顶点间距离，我们还利用著名的 Thorup-Zwick 距离预言机中显式存储的距离来限制可用的路由决策。