Angewandte Mathematik Und Informatik Universit at Zu K Oln on Minimum Stars, Minimum Steiner Stars, and Maximum Matchings Ss Andor P. Fekete Center for Applied Computer Science Universitt at Zu Kk Oln D{50923 Kk Oln Germany

We discuss properties and values of maximum matchings and minimum median problems for nite point sets. In particular, we consider \minimum stars", which are deened by a center chosen from the given point set, such that the total geometric distance min kStk 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 min kStStk. As a consequence of triangle inequality, the total length max kMatk of any maximum matching is a lower bound for the length min kStStk of a minimum Steiner star, which makes the ratio min kStStk maxkMatk 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 an even set of points in the plane and Euclidean distances, the ratio min kStStk maxkMatk cannot exceed 2= p 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 maximal value of the ratios min kStk min kStStk and min kStk max kMatk. We give tight upper bounds for the case where distances are measured according to the Manhattan metric: we show that in three-dimensional space, min kStStk max kMatk is bounded by 3=2, while in two-dimensional space min kStStk = max kMatk, extending some independent observations by Tamir and Mitchell. Finally, we show that min kStk min kStStk is bounded by 3=2 in the two-dimensional case, and by 5=3 in the three-dimensional case. 1 Introduction The problem of nding a maximum weight matching for a given set of vertices in a weighted graph is to nd a set of disjoint edges, such that the total weight of all the edges is maximized. Determining an optimal matching is a classical algorithmic problem, and Edmonds' famous polynomial algorithm 6] is one of the milestones of combinatorial optimization. On the other hand, it has been known for quite a while 9] that the task of nding a minimum weight Steiner tree is an NP-hard problem: nd a network of smallest total length