Bollob\'as-Meir TSP Conjecture Holds Asymptotically

Abstract

In 1992, Bollob\'as and Meir showed that for every $k \geq 1$ there exists a constant $c_k$ such that, for any $n$ points in the $k$-dimensional unit cube $[0, 1]^k$, one can find a tour $x_1, \dots, x_n$ through these $n$ points with $\sum_{i = 1}^n |x_i - x_{i + 1}|^k \leq c_k$, where $x_{n + 1} = x_1$ and $|x - y|$ is the Euclidean distance between $x$ and $y$. Remarkably, this bound does not depend on $n$, the number of points. They conjectured that the optimal constant is $c_k = 2 \cdot k^{k / 2}$ and showed that it cannot be taken lower than that. This conjecture was recently revised for $k = 3$ by Balogh, Clemen and Dumitrescu, who showed that $c_3 \geq 2^{7/2} > 2 \cdot 3^{3/2}$. It remains open for all $k > 2$, with the best known upper bound $c_k \leq 2.65^k \cdot k^{k / 2} \cdot (1 + o_k(1))$. We significantly narrow the gap between lower and upper bounds on $c_k$, reducing it from exponential to linear. Specifically, we prove that $c_k \leq 2\mathrm{e}(k + 1) \cdot k^{k / 2}$ and $c_k = k^{k / 2} \cdot (2 + o_k(1))$, the latter establishing the conjecture asymptotically. We also obtain analogous results for related problems on Hamiltonian paths, spanning trees and perfect matchings in the unit cube. Our main tool is a new generalization of the ball packing argument used in earlier works.