In this paper, we propose new algorithms for evacuation problems defined on dynamic flow networks. A dynamic flow network is a directed graph in which source nodes are given supplies (i.e., the number of evacuees) and a single sink node is given a demand (i.e., the maximum number of acceptable evacuees). The evacuation problem seeks a dynamic flow that sends all supplies from sources to the sink such that its demand is satisfied in the minimum feasible time horizon. For this problem, the current best algorithms are developed by Schl\"oter (2018) and Kamiyama (2019), which run in strongly polynomial time but with highorder polynomial time complexity because they use submodular function minimization as a subroutine. In this paper, we propose new algorithms that do not explicitly execute submodular function minimization, and we prove that they are faster than those by Schl\"oter (2018) and Kamiyama (2019) when an input network is restricted such that the sink has a small in-degree and every edge has the same capacity.

翻译：本文针对动态流网络中的疏散问题提出了新算法。动态流网络是一个有向图，其中源节点具有供给量（即疏散人数），单个汇点节点具有需求容量（即最大可接纳疏散人数）。疏散问题旨在寻找一种动态流，使得在最小可行时间范围内，所有源节点的供给量都能输送至汇点并满足其需求容量。针对该问题，当前最优算法由Schlöter（2018年）和Kamiyama（2019年）提出，这些算法具有强多项式时间复杂性，但由于使用了子模函数最小化作为子程序，其多项式阶数较高。本文提出的新算法不显式执行子模函数最小化，并证明当输入网络限制为汇点入度较小且所有边容量相等时，该算法比Schlöter（2018年）和Kamiyama（2019年）的算法更快。