Operations, Information and Decisions Papers

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 )} $

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Date Posted: 27 November 2017