On a Traveling Salesman Problem for Points in the Unit Cube
József Balogh, Felix Christian Clemen, Adrian Dumitrescu
TL;DR
This paper analyzes the Euclidean TSP for n points in the unit cube [0,1]^k by studying the k-th power sum of edge lengths, S_k, and the normalized tour cost s_k. It significantly improves previous upper bounds on the optimal constants c_k bounding s_k^{HC}(n), showing c_k ≤ 3√5(2/3)^{1/k}√k or c_k ≤ 2.91√k(1+o_k(1)), and proves a new lower bound c_3 ≥ 2^{7/6}, disproving the BM conjecture for k=3. The authors provide constructive algorithms based on MST traversals and greedy path-building, with asymptotic results: s_k^{ST} ≤ √k(1+o(1)) and s_k^{HC} ≤ 2.91√k(1+o_k(1)) for large k, and exact-or-near-exact bounds for fixed n. They also investigate Hamiltonian cycles on cube-vertex subsets via binary codes, and discuss implications for related matching and power-cost problems, proposing a slightly revised conjecture and avenues for future work.
Abstract
Let $X$ be an $n$-element point set in the $k$-dimensional unit cube $[0,1]^k$ where $k \geq 2$. According to an old result of Bollobás and Meir (1992), there exists a cycle (tour) $x_1, x_2, \ldots, x_n$ through the $n$ points, such that $\left(\sum_{i=1}^n |x_i - x_{i+1}|^k \right)^{1/k} \leq c_k$, where $|x-y|$ is the Euclidean distance between $x$ and $y$, and $c_k$ is an absolute constant that depends only on $k$, where $x_{n+1} \equiv x_1$. From the other direction, for every $k \geq 2$ and $n \geq 2$, there exist $n$ points in $[0,1]^k$, such that their shortest tour satisfies $\left(\sum_{i=1}^n |x_i - x_{i+1}|^k \right)^{1/k} = 2^{1/k} \cdot \sqrt{k}$. For the plane, the best constant is $c_2=2$ and this is the only exact value known. Bollob{á}s and Meir showed that one can take $c_k = 9 \left(\frac23 \right)^{1/k} \cdot \sqrt{k}$ for every $k \geq 3$ and conjectured that the best constant is $c_k = 2^{1/k} \cdot \sqrt{k}$, for every $k \geq 2$. Here we significantly improve the upper bound and show that one can take $c_k = 3 \sqrt5 \left(\frac23 \right)^{1/k} \cdot \sqrt{k}$ or $c_k = 2.91 \sqrt{k} \ (1+o_k(1))$. Our bounds are constructive. We also show that $c_3 \geq 2^{7/6}$, which disproves the conjecture for $k=3$. 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.