Traveling Salesman Constants

Let  be the smallest tour length for  points in a -D hypercube. Then there exists a smallest constant  such that for all optimal tours in the hypercube,

(1)

and a constant  such that for almost all optimal tours in the hypercube,

(2)

These constants satisfy the inequalities

			(3)
			(4)
			(5)
			(6)
			(7)
			(8)
			(9)

(Fejes Tóth 1940, Verblunsky 1951, Few 1955, Beardwood et al. 1959), where

(10)

 is the gamma function,  is an expression involving Struve functions and Bessel functions of the second kind,

(11)

(OEIS A086306; Karloff 1989), and

(12)

(OEIS A086307; Goddyn 1990).

In the limit ,

			(13)
			(14)
			(15)

and

(16)

where

(17)

and  is the best sphere packing density in -D space (Goddyn 1990, Moran 1984, Kabatyanskii and Levenshtein 1978). Steele and Snyder (1989) proved that the limit  exists.

Now consider the constant

(18)

so

(19)

Nonrigorous numerical estimates give  (Johnson et al. 1996) and  (Percus and Martin 1996).

A certain self-avoiding space-filling function is an optimal tour through a set of  points, where  can be arbitrarily large. It has length

(20)

(OEIS A073008), where  is the length of the curve at the th iteration and  is the point-set size (Norman and Moscato 1995).

REFERENCES

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Moscato, P. "Fractal Instances of the Traveling Salesman Constant." http://www.ing.unlp.edu.ar/cetad/mos/FRACTAL_TSP_home.html.Norman, M. G. and Moscato, P. "The Euclidean Traveling Salesman Problem and a Space-Filling Curve." Chaos Solitons Fractals 6, 389-397, 1995.

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