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 to embed the problem instance into low-dimensional space. 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 take advantage of the clustering method used in the well-known Thorup-Zwick distance oracle.

翻译：我们描述了一种简单的确定性近线性时间近似方案，用于解决具有实边权重的无向图中无容量最小成本流问题，该问题也称为转运问题。具体而言，我们的算法以（连通）无向图 $G = (V, E)$、顶点需求 $b \in \mathbb{R}^V$（满足 $\sum_{v \in V} b(v) = 0$）、正边成本 $c \in \mathbb{R}_{>0}^E$ 以及参数 $\varepsilon > 0$ 作为输入。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等于该顶点的需求，且流的成本在最优解的 $(1 + \varepsilon)$ 因子范围内。我们的算法是组合式的，其运行时间不依赖于需求或边成本。除了STOC 2022上提出的针对多项式有界边权重的最新结果外，所有几乎线性和近线性时间的转运问题近似方案都依赖于随机化将问题实例嵌入低维空间。相反，我们的算法确定性地近似了如果输入受到随机树嵌入影响时将做出的路由决策成本。为了避免计算此类近似所暗示的 $\Omega(n^2)$ 顶点-顶点距离，我们还利用了著名的Thorup-Zwick距离预言机中使用的聚类方法。