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.

中文翻译：

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最小扫描范围和变体-理论和实验

我们考虑了由诸如卫星通信和天体物理学之类的背景所激发的一系列几何优化问题。在带有角成本的最小扫描覆盖率问题中，我们得到了一个嵌在欧几里得空间中的图形$ G $。$ G $的边缘需要进行扫描，即从其两个顶点进行探测。为了扫描其边缘，两个顶点需要彼此面对。改变顶点的前进方向会导致能量或旋转时间方面的成本，这些成本与相应的旋转角度成比例。我们的目标是计算计划表，以最大程度地减少以下目标功能：（i）在“最小makespan扫描范围”（MSC-MS）中，这是直到扫描完所有边缘为止的时间；（ii）在“最小总能量扫描范围”（MSC-TE）中，所有旋转角度的总和；（iii）在“最小瓶颈能量扫描封面”（MSC-BE）中，一个顶点处的最大总旋转角度。先前有关MSC-MS的理论工作表明，图形着色和切口覆盖问题密切相关，从而导致硬度和近似性结果。在本文中，我们提出了针对MSC-TE和MSC-BE的一维实例的多项式时间算法，但针对了二分二维实例的NP硬度证明。对于二维二维图，我们还给出了MSC-TE和MSC-BE的2逼近算法。最重要的是，我们提供了针对所有三个问题的实用方法的综合研究。我们比较了三种不同的混合整数编程和两种约束编程方法，并展示了如何为多达300条边的几何实例计算可证明的最优解。此外，我们比较了更大实例下不同元启发式算法的性能。