AI 中文总结
研究沿组织层次结构的多种异构资源分配及转换问题，将其表述为市场均衡问题，证明可行解存在并给出算法，通过新颖两步过程在真实数据集上测试，能有效解决并逼近该资源分配问题。
AI 中文摘要
我们考虑将多种异构资源分配给按组织层次结构组织的代理。在公司中，这些对应于业务部门、部门和工程团队；在政府中，对应于联邦、州和市各级以及其中的各个机构；在大学中，对应于学院、系和研究小组。资源也沿相同的组织树分布。我们考虑存在资源对之间转换可能性的多种类型资源的分配问题。将此分配问题表述为市场均衡问题并推导找到可行解的必要条件，证明可行解总是存在并提供算法来计算它。最后在谷歌计算资源（如TPU和GPU）分配的真实数据集上测试算法。结果依赖新颖的两步过程，与先前方法不同。首先展示如何解决满足特定结构属性的‘简单’（调和）实例，然后展示如何通过一系列更简单的调和实例有效逼近一般实例，其解有效收敛到原始问题的解。
英文摘要
We consider the allocation of multiple heterogeneous resources to agents who are organized according to an organizational hierarchy. In a company those correspond to business units, departments, and engineering teams. In government it corresponds to federal, state, and municipal levels as well as various agencies within each. In a university it corresponds to schools, departments, and research groups. The resources are also distributed along the same organizational tree: part of the supply is available only to certain sub-trees since it is purchased for the exclusive use of certain departments or units, and some supply is available to the entire tree. We consider the allocation with multiple types of resources where there is a possibility of converting between certain pairs of resources. We formulate this allocation problem as a market equilibrium problem and derive the necessary conditions to find a feasible solution. We prove that such a feasible solution always exists and provide an algorithm to compute it. Finally, we test our algorithm on a real dataset of the allocation of computing resources, such as TPUs and GPUs at Google. Our results rely on a novel two-step process, differentiating it from previous approaches. First, we show how to solve ``easy'' (harmonic) instances of our problem that satisfy certain structural properties. Then, we show how to efficiently approximate general instances by a series of easier harmonic instances, whose solutions converge efficiently to the solution of the original problem.
