On minimum stars, minimum Steiner stars, and maximum matchings

We discuss worst-case bounds on the ratio of maximum matching and minimum median values for nite point sets. In particular, we consider "minimum stars", which are dened by a center chosen from the given point set, such that the total geometric distance L S to all the points in the set is minimized. If the center point is not required to be an element of the set (i. e., the center may be a Steiner point), we get a "minimum Steiner star", of total length L SS. As a consequence of triangle inequality, the total length LM of a maximum matching is a lower bound for the length L SS of a minimum Steiner star, which makes the worst-case value (SS; M) of the value L SS =LM interesting in the context of optimal communication networks. The ratio also appears as the duality gap in an integer programming formulation of a location problem by Tamir and Mitchell. In this paper, we show that for a nite set that consists of an even number of points in the plane and Euclidean distances, the worst-case ratio (S; M) cannot exceed 2=