On a Traveling Salesman Problem for Points in the Unit Cube

Abstract

AbstractLet X be an n-element point set in the k-dimensional unit cube $$[0,1]^k$$ [ 0 , 1 ] k where $$k \ge 2$$ k ≥ 2 . According to an old result of Bollobás and Meir (Oper Res Lett 11:19–21, 1992) , there exists a cycle (tour) $$x_1, x_2, \ldots , x_n$$ x 1 , x 2 , … , x n through the n points, such that $$\left( \sum _{i=1}^n |x_i - x_{i+1}|^k \right) ^{1/k} \le c_k$$ ∑ i = 1 n | x i - x i + 1 | k 1 / k ≤ c k , where $$|x-y|$$ | x - y | is the Euclidean distance between x and y, and $$c_k$$ c k is an absolute constant that depends only on k, where $$x_{n+1} \equiv x_1$$ x n + 1 ≡ x 1 . From the other direction, for every $$k \ge 2$$ k ≥ 2 and $$n \ge 2$$ n ≥ 2 , there exist n points in $$[0,1]^k$$ [ 0 , 1 ] k , such that their shortest tour satisfies $$\left( \sum _{i=1}^n |x_i - x_{i+1}|^k \right) ^{1/k} = 2^{1/k} \cdot \sqrt{k}$$ ∑ i = 1 n | x i - x i + 1 | k 1 / k = 2 1 / k · k . For the plane, the best constant is $$c_2=2$$ c 2 = 2 and this is the only exact value known. Bollobás and Meir showed that one can take $$c_k = 9 \left( \frac{2}{3} \right) ^{1/k} \cdot \sqrt{k}$$ c k = 9 2 3 1 / k · k for every $$k \ge 3$$ k ≥ 3 and conjectured that the best constant is $$c_k = 2^{1/k} \cdot \sqrt{k}$$ c k = 2 1 / k · k , for every $$k \ge 2$$ k ≥ 2 . Here we significantly improve the upper bound and show that one can take $$c_k = 3 \sqrt{5} \left( \frac{2}{3} \right) ^{1/k} \cdot \sqrt{k}$$ c k = 3 5 2 3 1 / k · k or $$c_k = 2.91 \sqrt{k} \ (1+o_k(1))$$ c k = 2.91 k ( 1 + o k ( 1 ) ) . Our bounds are constructive. We also show that $$c_3 \ge 2^{7/6}$$ c 3 ≥ 2 7 / 6 , which disproves the conjecture for $$k=3$$ k = 3 . Connections to matching problems, power assignment problems, related problems, including algorithms, are discussed in this context. A slightly revised version of the Bollobás–Meir conjecture is proposed.