We consider a spectrum of geometric optimization problems motivated by contexts such as satellite communication and astrophysics. In the problem Minimum Scan Cover with Angular Costs, we are given a graph $G$ that is embedded in Euclidean space. The edges of $G$ need to be scanned, i.e., probed from both of their vertices. In order to scan their edge, two vertices need to face each other; changing the heading of a vertex incurs some cost in terms of energy or rotation time that is proportional to the corresponding rotation angle. Our goal is to compute schedules that minimize the following objective functions: (i) in Minimum Makespan Scan Cover (MSC-MS), this is the time until all edges are scanned; (ii) in Minimum Total Energy Scan Cover (MSC-TE), the sum of all rotation angles; (iii) in Minimum Bottleneck Energy Scan Cover (MSC-BE), the maximum total rotation angle at one vertex. Previous theoretical work on MSC-MS revealed a close connection to graph coloring and the cut cover problem, leading to hardness and approximability results. In this paper, we present polynomial-time algorithms for 1D instances of MSC-TE and MSC-BE, but NP-hardness proofs for bipartite 2D instances. For bipartite graphs in 2D, we also give 2-approximation algorithms for both MSC-TE and MSC-BE. Most importantly, we provide a comprehensive study of practical methods for all three problems. We compare three different mixed-integer programming and two constraint programming approaches, and show how to compute provably optimal solutions for geometric instances with up to 300 edges. Additionally, we compare the performance of different meta-heuristics for even larger instances.
翻译:我们考虑的是由卫星通信和天体物理学等环境引发的一系列几何优化问题。 在问题“最小扫描覆盖”中,我们得到一个嵌入 Euclidean 空间的图形$G$。 需要扫描$G$的边缘, 即从两个顶端进行探测。 为了扫描它们的边缘, 需要用两个顶端来对彼此进行对比; 改变一个顶端的标题在能量或旋转时间方面引起一定的成本, 与相应的旋转角度成比例。 我们的目标是计算一个尽可能减少以下目标功能的表:(一) 在最小木板扫描(MSC-MS)中, 这是所有边缘都被扫描之前的时间;(二) 最低总能源扫描(MSC-TE), 所有旋转角度的总和;(三) 在最低博特勒克能源扫描仪(MSC-B-BE)中, 一个最大总旋转角度的旋转角度是所有垂直角度。 MSC-MSMSB 之前的理论实验显示一个与图形颜色的密切连接, 和缩小封面的颜色(MSCSIC-D) 2 O- d 的运行结果。