Minimum Scan Cover with Angular Transition Costs
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We provide a comprehensive study of a natural geometric optimization problem motivated by questions in the context of satellite communication and astrophysics. In the problem Minimum Scan Cover with Angular Costs (MSC), 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 takes some time proportional to the corresponding turn angle. Our goal is to minimize the time until all scans are completed, i.e., to compute a schedule of minimum makespan. We show that MSC is closely related to both graph coloring and the minimum (directed and undirected) cut cover problem; in particular, we show that the minimum scan time for instances in 1D and 2D lies in Θ(logχ (G)), while for 3D the minimum scan time is not upper bounded by χ (G). We use this relationship to prove that the existence of a constant-factor approximation implies P=NP, even for one-dimensional instances. In 2D, we show that it is NP-hard to approximate a minimum scan cover within less than a factor of 3/2, even for bipartite graphs; conversely, we present a 9/2-approximation algorithm for this scenario. Generally, we give an O(c)-approximation for k-colored graphs with k≤χ(G)^c. For general metric cost functions, we provide approximation algorithms whose performance guarantee depend on the arboricity of the graph.
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