Statistics Papers

Document Type

Journal Article

Date of this Version

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 ⊂ [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) 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).

Keywords

Equidistribution, worst-case, non-linear growth, traveling salesman, rectilinear Steiner tree, minimum spanning tree, minimum-weight matching

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Date Posted: 27 November 2017

This document has been peer reviewed.