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New Bounds for the Traveling Salesman Constant

Part of: Limit theorems Geometric probability and stochastic geometry

Published online by Cambridge University Press:  04 January 2016

Stefan Steinerberger
Stefan Steinerberger*
Affiliation:
Universität Bonn
*
∗ Current address: Department of Mathematics, Yale University, 10 Hillhouse Avenue, New Haven, CT 06520, USA. Email address: stefan.steinerberger@yale.edu
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Abstract

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Let X1, X2, …, Xn be independent and uniformly distributed random variables in the unit square [0, 1]2, and let L(X1, …, Xn) be the length of the shortest traveling salesman path through these points. In 1959, Beardwood, Halton and Hammersley proved the existence of a universal constant β such that limn→∞n−1/2L(X1, …, Xn) = β almost surely. The best bounds for β are still those originally established by Beardwood, Halton and Hammersley, namely 0.625 ≤ β ≤ 0.922. We slightly improve both upper and lower bounds.

Keywords

Traveling salesman constantBeardwood-Halton-Hammersley theorem

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60F17: Functional limit theorems; invariance principles
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 47 , Issue 1 , March 2015 , pp. 27 - 36
Copyright
© Applied Probability Trust 

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