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 \subset [ 0,1 ]^d $, where $| S | = n$. 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 $S^{( n )} \cap R$ 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 )} $