EQUIDISTRIBUTION IN ALL DIMENSIONS OF WORST-CASE POINT SETS FOR THE TRAVELING SALESMAN PROBLEM

Given a set S of n points in the unit square [0,1] d, an optimal traveling salesman tour of S is a tour of S that is of minimum length. A worst-case point set for the traveling salesman problem in the unit square is a point set S (n) whose optimal traveling salesman tour achieves the maximum possible length among all point sets S C [0, 1] d, where IS n. An open problem is to determine the structure of S(n). We show that for any rectangular parallelepiped R contained in [0, 1] d, the number of points in S (n) N R is asymptotic to n times the volume of R. Analogous results are proved for the minimum spanning tree, minimum-weight matching, and rectilinear Steiner minimum tree. These equidistribution theorems are the first results concerning the structure of worst-case point sets like S(n)