New Bounds for the Traveling Salesman Constant
Abstract
Let $X_1, X_2, \dots, X_n$ be independent and uniformly distributed random variables in the unit square $[0,1]^2$ and let $L(X_1, \dots, X_n)$ be the length of the shortest traveling salesman path through these points. In 1959, Beardwood, Halton $\&$ Hammersley proved the existence of a universal constant $\beta$ such that $$ \lim_{n \rightarrow \infty}{n^{-1/2}L(X_1, \dots, X_n)} = \beta \qquad \mbox{almost surely.}$$ The best bounds for $\beta$ are still the ones originally established by Beardwood, Halton $\&$ Hammersley $0.625 \leq \beta \leq 0.922$. We slightly improve both upper and lower bounds.
- Publication:
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arXiv e-prints
- Pub Date:
- November 2013
- DOI:
- 10.48550/arXiv.1311.6338
- arXiv:
- arXiv:1311.6338
- Bibcode:
- 2013arXiv1311.6338S
- Keywords:
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- Mathematics - Probability;
- Mathematics - Optimization and Control
- E-Print:
- Advances in Applied Probability 47.1 (March 2015)