改进的欧几里得TSP常数上界使用带交叉
An Improved Upper Bound for the Euclidean TSP Constant Using Band Crossovers
AI总结：
本文通过改进的带交叉策略，结合模拟和集中分析，将欧几里得TSP常数上界从0.90380改进至约0.88，并提出可能进一步提高至约0.85的替代启发式方法。
摘要：
考虑在单位正方形中均匀生成的n个点，令L_n为这些点的最优旅行商路线长度。Beardwood、Halton和Hammersley（1959）证明当n趋近于无穷大时，L_n /√n几乎必然收敛于某个常数β。β的确切值尚不清楚，但估计约为0.71（Applegate、Bixby、Chvátal、Cook，2011）。Beardwood等人进一步证明0.625≤β≤0.92116。目前最好的已知界限是0.6277≤β≤0.90380，由Gaudio和Jaillet（2019）和Carlsson和Yu（2023）分别得出。上界是通过计算机辅助方法得出的，这种方法可以用于改进的下界计算。在本文中，我们通过模拟和集中分析表明，未来改进0.90380的上限仅能达到约0.88。此外，我们提供了一种替代的路线构造启发式方法，通过模拟，可能将上限提高到约0.85。我们的方法基于先前的带遍历策略，最初由Beardwood等人（1959）提出，随后由Carlsson和Yu（2023）进一步完善：将单位正方形分成高度为Θ(1/√n)的带，构造每个带内的路径，然后将路径连接起来形成TSP路线。我们的方法允许路径跨越带，并利用相邻带中接近的点对。严谨的数值分析将上限提高到0.90367。
英文摘要：
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 $0.90367$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
