Consider $n$ points generated uniformly at random in the unit square, and let $L_n$ be the length of their optimal traveling salesman tour. Beardwood, Halton, and Hammersley (1959) showed $L_n / \sqrt n \to β$ almost surely as $n\to \infty$ for some constant $β$. The exact value of $β$ is unknown but estimated to be approximately $0.71$ (Applegate, Bixby, Chvátal, Cook 2011). Beardwood et al. further showed that $0.625 \leq β\leq 0.92116.$ Currently, the best known bounds are $0.6277 \leq β\leq 0.90380$, due to Gaudio and Jaillet (2019) and Carlsson and Yu (2023), respectively. The upper bound was derived using a computer-aided approach that is amenable to lower bounds with improved computation speed. In this paper, we show via simulation and concentration analysis that future improvement of the $0.90380$ is limited to $\sim0.88$. Moreover, we provide an alternative tour-constructing heuristic that, via simulation, could potentially improve the upper bound to $\sim0.85$. Our approach builds on a prior \emph{band-traversal} strategy, initially proposed by Beardwood et al. (1959) and subsequently refined by Carlsson and Yu (2023): divide the unit square into bands of height $Θ(1/\sqrt{n})$, construct paths within each band, and then connect the paths to create a TSP tour. Our approach allows paths to cross bands, and takes advantage of pairs of points in adjacent bands which are close to each other. A rigorous numerical analysis improves the upper bound to $0.90367$.

翻译：考虑在单位正方形内均匀随机生成的 $n$ 个点，并令 $L_n$ 表示其最优旅行商路径的长度。Beardwood、Halton 和 Hammersley (1959) 证明了当 $n\to \infty$ 时，$L_n / \sqrt n$ 几乎必然收敛于某个常数 $β$。$β$ 的确切值未知，但估计约为 $0.71$ (Applegate, Bixby, Chvátal, Cook 2011)。Beardwood 等人进一步证明了 $0.625 \leq β\leq 0.92116$。目前，已知的最佳上下界分别为 $0.6277 \leq β\leq 0.90380$，分别由 Gaudio 和 Jaillet (2019) 以及 Carlsson 和 Yu (2023) 得出。其上界的推导采用了计算机辅助方法，该方法有助于以更高的计算速度获得下界。本文通过模拟与集中性分析表明，未来对 $0.90380$ 这一上界的改进将限于 $\sim0.88$ 左右。此外，我们提出了一种替代的路径构造启发式方法，通过模拟显示，该方法可能将上界改进至 $\sim0.85$ 左右。我们的方法建立在先前的 \emph{带状遍历} 策略之上，该策略最初由 Beardwood 等人 (1959) 提出，随后由 Carlsson 和 Yu (2023) 改进：将单位正方形划分为高度为 $Θ(1/\sqrt{n})$ 的带状区域，在每个带内构造路径，然后连接这些路径以形成 TSP 环游。我们的方法允许路径跨越不同的带，并利用相邻带中彼此接近的点对。严格的数值分析将上界改进至 $0.90367$。