A dynamic flow network consists of a directed graph, where nodes called sources represent locations of evacuees, and nodes called sinks represent locations of evacuation facilities. Each source and each sink are given supply representing the number of evacuees and demand representing the maximum number of acceptable evacuees, respectively. Each edge is given capacity and transit time. Here, the capacity of an edge bounds the rate at which evacuees can enter the edge per unit time, and the transit time represents the time which evacuees take to travel across the edge. The evacuation completion time is the minimum time at which each evacuees can arrive at one of the evacuation facilities. Given a dynamic flow network without sinks, once sinks are located on some nodes or edges, the evacuation completion time for this sink location is determined. We then consider the problem of locating sinks to minimize the evacuation completion time, called the sink location problem. The problems have been given polynomial-time algorithms only for limited networks such as paths, cycles, and trees, but no polynomial-time algorithms are known for more complex network classes. In this paper, we prove that the 1-sink location problem can be solved in polynomial-time when an input network is a grid with uniform edge capacity and transit time.

翻译：动态流网络由一个有向图构成，其中称为源的节点代表疏散人员的位置，称为汇的节点代表疏散设施的位置。每个源和每个汇分别被赋予供应量（表示疏散人员数量）和需求量（表示可接受的最大疏散人员数量）。每条边被赋予容量和通行时间。其中，边的容量限制了单位时间内疏散人员进入该边的速率，通行时间表示疏散人员穿越该边所需的时间。疏散完成时间是指每位疏散人员能够到达某个疏散设施的最短时间。给定一个没有汇的动态流网络，一旦汇被定位在某些节点或边上，该汇定位对应的疏散完成时间便得以确定。我们进而考虑通过定位汇来最小化疏散完成时间的问题，称为汇点定位问题。该问题仅在路径、环和树等有限网络上存在多项式时间算法，而对于更复杂的网络类别尚无已知的多项式时间算法。在本文中，我们证明当输入网络为具有均匀边容量和通行时间的网格时，1-汇点定位问题可在多项式时间内求解。