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：动态流网络由一个带权有向图构成，其中称为源点的节点代表撤离人员位置，称为汇点的节点代表疏散设施位置。每个源点具有表示撤离人数的供给量，每个汇点具有表示最大可接纳撤离人数的需求量。每条边具有容量和通行时间属性：容量限制单位时间内可进入该边的最大撤离速率，通行时间表示撤离人员通过该边所需时间。撤离完成时间定义为所有撤离人员到达任一疏散设施所需的最小时间。给定一个不含汇点的动态流网络，当在某些节点或边上设置汇点后，即可确定该汇点布局对应的撤离完成时间。我们进而研究以最小化撤离完成时间为目标的汇点布局问题，即汇点定位问题。该问题目前仅对路径、环、树等有限网络存在多项式时间算法，对更复杂网络类别尚未已知多项式时间解法。本文证明：当输入网络为具有均匀边容量和通行时间的网格时，单汇点定位问题可在多项式时间内求解。