An Asymptotic Determination of the Minimum Spanning Tree and Minimum Matching Constants in Geometrical Probability

Given n uniformly and independently distributed points in a ball of unit volume in dimension d, it is well established that the length of several combinatorial optimization problems (including the minimum spanning tree (MST), the minimum matching (M), the traveling salesman problem (TSP), etc.) on these n points is asymptotic to P(d) n(d-1)/d, where the constant B(d) depends on the dimension d and the problem solved. It has been a long open problem to determine the constants 38(d) for these problems. In this paper we make progress in establishing the constants PMST(d), 3M(d) for the MST and the matching problem. By applying Crofton's method, an old method in geometrical probability, we prove that PMST(d) 6 /, IM(d) as d tends to infinity. Moreover, our method corresponds to heuristics for these problems, which are asymptotically exact as the dimension increases. Finally, we examine the asymptotics for the TSP constant and improve the best known bounds for d= 3,4. · Dimitris Bertsimas, Sloan School of Management and Operations Research Center, MIT, Rm E53-359, Cambridge, MA 02139. tGarrett van Ryzin, Operations Research Center, MIT, Cambridge, MA 02139.