Minimum Scan Cover with Angular Transition Costs

We provide a comprehensive study of a natural graph optimization problem that arises from transition costs between incident edges. 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. A real-world motivation arises in the context of satellite communication and astrophysics. 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 $\Theta(\log \chi (G))$, while for 3D the minimum scan time is not upper bounded by $\chi (G)$. 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 $\nicefrac{3}{2}$, 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 $k\leq \chi(G)^{c}$. For general metric cost functions, we provide approximation algorithms whose performance guarantees depend on the arboricity of the graph.