动态汇流下极大极小树设施选址与汇疏散
Minmax Tree Facility Location and Sink Evacuation with Dynamic Confluent Flows
摘要：
假设$G=(V,E)$是一个图形，它模拟了一个建筑物或道路网络，其中的边具有与它们相关联的旅行时间（长度）和容量。边缘的容量是单位时间内可以进入该边缘的人数。在紧急情况下，人们向出口疏散。如果太多的人试图通过同一个边缘疏散，拥挤就会加剧并减缓疏散速度。具有长度和容量的图被称为动态流网络。$G$的疏散计划包括选择出口位置和将顶点处的人员划分为组，每组人员疏散到同一出口。计划的疏散时间是直到最后一个人疏散所需的时间。$k$ -汇疏散问题是提供一个疏散计划与$k$出口位置，使疏散时间最小化。众所周知，这个问题是NP-困难的一般图，但没有多项式时间的算法是以前已知的情况下，甚至$G$树。针对树上的$k$ -sink疏散问题，提出了一种$O(n k^2 \log^5 n)$算法.我们的算法也适用于更一般的一类问题，我们称之为极大极小树设施的位置。
英文摘要：
Let $G=(V,E)$ be a graph modelling a building or road network in which edges have-both travel times (lengths) and capacities associated with them. An edge's capacity is the number of people that can enter that edge in a unit of time. In emergencies, people evacuate towards the exits. If too many people try to evacuate through the same edge, congestion builds up and slows down the evacuation. Graphs with both lengths and capacities are known as Dynamic Flow networks. An evacuation plan for $G$ consists of a choice of exit locations and a partition of the people at the vertices into groups, with each group evacuating to the same exit. The evacuation time of a plan is the time it takes until the last person evacuates. The $k$-sink evacuation problem is to provide an evacuation plan with $k$ exit locations that minimizes the evacuation time. It is known that this problem is NP-Hard for general graphs but no polynomial time algorithm was previously known even for the case of $G$ a tree. This paper presents an $O(n k^2 \log^5 n)$ algorithm for the $k$-sink evacuation problem on trees. Our algorithms also apply to a more general class of problems, which we call minmax tree facility location.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
