
Operations, Information and Decisions Papers
Title
Equidistribution in All Dimensions of Worst-Case Point Sets for the Traveling Salesman Problem
Document Type
Journal Article
Date of this Version
11-1995
Publication Source
SIAM Journal on Discrete Mathematics
Volume
8
Issue
4
Start Page
678
Last Page
683
DOI
10.1137/S0895480194262710
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 \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 )} $
Recommended Citation
Snyder, T. L., & Steele, J. M. (1995). Equidistribution in All Dimensions of Worst-Case Point Sets for the Traveling Salesman Problem. SIAM Journal on Discrete Mathematics, 8 (4), 678-683. http://dx.doi.org/10.1137/S0895480194262710
Date Posted: 27 November 2017