Approximating longest spanning tree with neighborhoods

We study the following maximization problem in the Euclidean plane: Given a collection of neighborhoods (polygonal regions) in the plane, select a point in each neighborhood so that the longest spanning tree on selected points has maximum length. It is not known whether or not this problem is NP-hard. We present an approximation algorithm with ratio 0.548 for this problem. This improves the previous best known ratio of 0.511 due to Chen and Dumitrescu (Discrete Mathematics, Algorithms and Applications, 2020). The presented algorithm takes linear time after computing a diameter of the neighborhoods. Even though our algorithm itself is fairly simple, its analysis is rather involved. In some part we deal with a minimization problem involving multiple parameters. We use a sequence of geometric transformations to reduce the number of parameters and simplify the analysis.