Minimum Scan Cover with Angular Transition Costs

We provide a comprehensive study of natural graph optimization problem that arises from transition costs between incident edges. In the Minimum Scan Cover with Angular Costs (MSC), we are given $G$ is embedded in Euclidean space. The edges need to be scanned, i.e., probed both their vertices. order scan edge, two vertices face each other; changing heading vertex takes some time proportional corresponding turn angle. Our goal minimize until all scans completed, compute schedule minimum makespan. A real-world motivation context satellite communication and astrophysics. show MSC closely related coloring (directed undirected) cut cover problem; particular, for instances 1D 2D lies $\Theta(\log \chi (G))$, while 3D not upper bounded by $\chi (G)$. use this relationship prove existence constant-factor approximation implies P $=$ NP, even one-dimensional instances. 2D, it NP-hard approximate within less than factor $\nicefrac{3}{2}$, bipartite graphs; conversely, present 9/2-approximation algorithm scenario. Generally, give an $O(c)$-approximation $k$-colored graphs $k\leq \chi(G)^{c}$. For general metric cost functions, algorithms whose performance guarantees depend on arboricity graph.