  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.
