New bounds for the traveling salesman constant

Abstract

Let X₁, X₂, . . . , X_n be independent and uniformly distributed random variables in the unit square [0, 1]², and let L(X₁, . . . , X_n) 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 lim_n→∞n ^-1/2L(X₁, . . . , X_n) = 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.