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 )} $ Disciplines Other Mathematics This journal article is available at ScholarlyCommons: http://repository.upenn.edu/oid_papers/261 SIAM J. DISC. MATH. Vol. 8, No. 4, pp. 678-683, November 1995 () 1995 Society for Industrial and Applied Mathematics 015 EQUIDISTRIBUTION IN ALL DIMENSIONS OF WORST-CASE POINT SETS FOR THE TRAVELING SALESMAN PROBLEM* TIMOTHY LAW SNYDER AND J. MICHAEL STEELE: Abstract. 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). 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). Key words, equidistribution, worst-case, nonlinear growth, traveling salesman, rectilinear Steiner tree, minimum spanning tree, minimum-weight matching AMS subject classifications. 68R10, 05C45, 90C35, 68U05