We introduce the Unsplittable Transshipment Problem in directed graphs with multiple sources and sinks. An unsplittable transshipment routes given supplies and demands using at most one path for each source-sink pair. Although they are a natural generalization of single source unsplittable flows, unsplittable transshipments raise interesting new challenges and require novel algorithmic techniques. As our main contribution, we give a nontrivial generalization of a seminal result of Dinitz, Garg, and Goemans (1999) by showing how to efficiently turn a given transshipment $x$ into an unsplittable transshipment $y$ with $y_a<x_a+d_{\max}$ for all arcs $a$, where $d_{\max}$ is the maximum demand (or supply) value. Further results include bounds on the number of rounds required to satisfy all demands, where each round consists of an unsplittable transshipment that routes a subset of the demands while respecting arc capacity constraints.

翻译：本文在有向图中引入具有多个源点和汇点的不可分割转运问题。不可分割转运通过为每个源-汇对至多分配一条路径来输送给定的供应量与需求量。尽管该问题是单源不可分割流的自然推广，但不可分割转运引发了新的理论挑战，并需要创新的算法技术。我们的主要贡献是对Dinitz、Garg和Goemans（1999）的开创性结果进行了非平凡推广，证明了如何将给定转运方案$x$高效转化为不可分割转运方案$y$，使得对于所有弧$a$满足$y_a<x_a+d_{\max}$，其中$d_{\max}$为最大需求（或供应）值。其他结果包括满足所有需求所需轮数的界值，其中每轮由满足弧容量约束的不可分割转运方案构成，该方案输送需求量的某个子集。