Multifacility location problems arise in many real world applications. Often, the facilities can only be placed in feasible regions such as development or industrial areas. In this paper we show the existence of a finite dominating set (FDS) for the planar multifacility location problem with polyhedral gauges as distance functions, and polyhedral feasible regions, if the interacting facilities form a tree. As application we show how to solve the planar 2-hub location problem in polynomial time. This approach will yield an $\epsilon$-approximation for the Euclidean norm case which is polynomial in the input data and $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\epsilon $}\right.$.

中文翻译：

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具有树结构和有限支配集的平面多设施位置问题

在许多实际应用中会出现多设备位置问题。通常，设施只能放置在可行的区域，例如开发区或工业区。在本文中，我们说明了存在多面体规作为距离函数的平面多设施位置问题和多面体可行区域（如果相互作用的设施形成一棵树）的有限控制集（FDS）的存在。作为应用程序，我们展示了如何在多项式时间内解决平面2-hub定位问题。这种方法将产生$\epsilon$欧几里德范数情况下的-逼近是输入数据中的多项式 $\raisebox{1ex}{$\mathrm{1个}$}\!\left/ \!\raisebox{-1ex}{$\epsilon $}\right.$。