不确定区域的连通图
Connectivity graphs of uncertainty regions
摘要：
我们研究点之间的连接关系，其中每个输入点的精确位置位于一个区域的不确定性。我们区分了两种不确定性产生的基本情景。在有利的最佳情况不确定性（BU）中，可以从给定的集合中选择每个输入点，以产生最佳可能的目标值。在不利的最坏情况不确定性（WU）中，输入集在所有可能的点定位中具有最坏的可能目标值，这些点位置由于例如不精确的数据而不确定。我们考虑这些概念的不确定性的瓶颈生成树问题，从而产生以下的最佳情况下的连接与不确定性（BCU）的问题：给定一个家庭的几何区域，每个区域选择一个点，使相关的几何生成树的最长边长度最小化。我们表明，这个问题是NP-困难的，即使是非常简单的情况下，该地区是线段或广场。另一方面，我们给出了一个精确的解决方案的情况下，有n+k个区域，其中k个区域是线段和n个区域是不动点。然后，我们给近似算法的情况下，无论是所有的线段或所有的单位盘。我们还提供了相应的不确定性最坏情况连通性（WCU）问题的近似方法：给定一组不确定区域，找到最小距离r，使得对于任何点的选择，每个区域一个，在边长最多为r的点之间存在一棵生成树。
英文摘要：
We study connectivity relations among points, where the precise location of each input point lies in a region of uncertainty. We distinguish two fundamental scenarios under which uncertainty arises. In the favorable Best-Case Uncertainty (BU), each input point can be chosen from a given set to yield the best possible objective value. In the unfavorable Worst-Case Uncertainty (WU), the input set has worst possible objective value among all possible point locations, which are uncertain due, for example, to imprecise data. We consider these notions of uncertainty for the bottleneck spanning tree problem, giving rise to the following Best-Case Connectivity with Uncertainty (BCU) problem: Given a family of geometric regions, choose one point per region, such that the longest edge length of an associated geometric spanning tree is minimized. We show that this problem is NP-hard even for very simple scenarios in which the regions are line segments or squares. On the other hand, we give an exact solution for the case in which there are n+k regions, where k of the regions are line segments and n of the regions are fixed points. We then give approximation algorithms for cases where the regions are either all line segments or all unit discs. We also provide approximation methods for the corresponding Worst-Case Connectivity with Uncertainty (WCU) problem: Given a set of uncertainty regions, find the minimal distance r such that for any choice of points, one per region, there is a spanning tree among the points with edge length at most r.
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