We show a deterministic constant-time local algorithm for constructing an approximately maximum flow and minimum fractional cut in multisource-multitarget networks with bounded degrees and bounded edge capacities. Locality means that the decision we make about each edge only depends on its constant radius neighborhood. We show two applications of the algorithms: one is related to the Aldous-Lyons Conjecture, and the other is about approximating the neighborhood distribution of graphs by bounded-size graphs. The scope of our results can be extended to unimodular random graphs and networks. As a corollary, we generalize the Maximum Flow Minimum Cut Theorem to unimodular random flow networks.

翻译：我们提出了一种确定性常数时间局部算法，用于在具有有界度数与有界边容量的多源多目标网络中构造近似最大流与最小分数割。局部性意味着我们对每条边做出的决策仅依赖于其常数半径邻域。我们展示了该算法的两个应用：其一是与Aldous-Lyons猜想相关，其二是关于用有界规模图近似图的邻域分布。我们的结果适用范围可扩展至单模随机图与网络。作为推论，我们将最大流最小割定理推广到了单模随机流网络。