On Minimum Stars and Maximum Matchings

We discuss worst-case bounds on the ratio of maximum matching and minimum median values for finite point sets. In particular, we consider “minimum stars,” which are defined 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 L M of a maximum matching is a lower bound for the length L SS of a minimum Steiner star, which makes the worst-case value ρ(SS,M) of the value L SS/L M 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 finite set that consists of an even number of points in the plane and Euclidean distances, the worst-case ratio ρ(S,M) cannot exceed 2/ √ 3. This proves a conjecture of Suri, who gave an example where this bound is achieved. For the case of Euclidean distances in two and three dimensions, we also prove upper and lower bounds for the worst-case value ρ(S, SS) of the ratio L S/L SS , and for the worst-case value ρ(S,M) of the ratio L S/L M . We give tight upper bounds for the case where distances are measured according to the Manhattan metric: we show that in three-dimensional space, ρ(SS,M) is bounded by 32 , while in two-dimensional space L SS = L M , extending some independent observations by Tamir and Mitchell. Finally, we show that ρ(S, SS) is 32 in the two-dimensional case, and 53 in the three-dimensional case. ∗ Parts of this work were done while the first author was visiting Queen’s University, partially supported by the Deutsche Forschungsgemeinschaft, FE 407/3-1. Parts of this work were done while the second author was visiting Universität zu Köln, partially supported by NSERC. A preliminary extended abstract of this article, titled “On minimum stars, minimum Steiner stars, and maximum matchings,” appears in the Proceedings of the 15th ACM Symposium on Computational Geometry [9]. 390 S. P. Fekete and H. Meijer