压力下概率中心问题的困难性与逼近
Hardness and approximation of the Probabilistic p-Center problem under Pressure
摘要：
压力下的概率p中心问题（Min PpCP）是我们最近在野火管理的背景下引入的通常p中心问题的变体。问题是找到p个避难所，使人们在发生火灾时必须到达最近的避难所的最大距离最小化。景观被划分为区域，并被建模为一个边缘加权图与顶点对应的区域和边缘对应的两个相邻区域之间的直接连接。与火灾爆发相关的风险使用火灾场景的有限集建模。每个场景对应于单个区域上的火灾爆发（即，在顶点上），其主要结果是以两种方式修改疏散路径。首先，疏散路径不能通过着火的顶点。其次，事实上，接近火灾的人可能不会采取合理的决定时，选择一个方向逃生是使用新的疏散路径。本文给出了由边权图G=（V，E，L）和整数p定义的Min PpCP的可行解集的特征.我们证明，最小PpCP不能近似的比例小于56/55的次网格（子图的网格）的程度最多为3。然后，我们提出了一些近似结果的最小PpCP。这些结果需要近似结果的两个变种的（确定性）最小p-中心的问题，称为最小MAC p-中心和最小部分p-中心。
英文摘要：
The Probabilistic p-Center problem under Pressure (Min PpCP) is a variant of the usual p-Center problem we recently introduced in the context of wildfire management. The problem is to locate p shelters minimizing the maximum distance people will have to cover to reach the closest accessible shelter in case of fire. The landscape is divided into zones and is modeled as an edge-weighted graph with vertices corresponding to zones and edges corresponding to direct connections between two adjacent zones. The risk associated with fire outbreaks is modeled using a finite set of fire scenarios. Each scenario corresponds to a fire outbreak on a single zone (i.e., on a vertex) with the main consequence of modifying evacuation paths in two ways. First, an evacuation path cannot pass through the vertex on fire. Second, the fact that someone close to the fire may not take rational decisions when selecting a direction to escape is modeled using new kinds of evacuation paths. In this paper, for a given instance of Min PpCP defined by an edge-weighted graph G=(V,E,L) and an integer p, we characterize the set of feasible solutions of Min PpCP. We prove that Min PpCP cannot be approximated with a ratio less than 56/55 on subgrids (subgraphs of grids) of degree at most 3. Then, we propose some approximation results for Min PpCP. These results require approximation results for two variants of the (deterministic) Min p-Center problem called Min MAC p-Center and Min Partial p-Center.
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