A travelling salesman problem in the k-dimensional unit cube
Answering a question of the second author in Operations Research Letters 6 (1987) 289-291, we show that for every k ≥ 1 there is a constant ck with the following property. Let x1,...,xn be points in a k-dimensional cube. Then there is a tour xi1xi2...xin of these points such that, with xin+1 = xi1, we have (Σjn=1|xij - xij+1|k) 1 k ≤ck where |x - y | denotes the Euclidean distance between the points x and y. For k ≥ 3, it is not known what is the best constant ck* one can take, but we show that it satisfies 2 1 k√k≤ck*≤( 3 2) (k-1) k√k. © 1992.
Operations Research Letters
Bollobás, B., & Meir, A. (1992). A travelling salesman problem in the k-dimensional unit cube. Operations Research Letters, 11 (1), 19-21. https://doi.org/10.1016/0167-6377(92)90057-A