发表机构
McCombs School of Business, The University of Texas at Austin（德克萨斯大学奥斯汀分校麦库姆斯商学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文改进了欧几里得旅行商问题中渐近长度常数$\beta$的上下界，从$0.6277\leq\beta\leq0.90367$提升至$0.6421\leq\beta\leq0.8810$，并利用重要性采样以高概率进一步缩小至$0.6536\leq\beta\leq0.8749$。
AI 中文摘要
Beardwood-Halton-Hammersley定理刻画了单位正方形内独立且均匀分布的随机点$X_1,\ldots,X_n$的最优欧几里得旅行商回路长度的渐近行为。该定理指出，存在一个普适常数$\beta$，使得最短回路的长度几乎必然渐近于$\beta \sqrt{n}$。迄今已确立的最佳界为$0.6277\leq \beta \leq 0.90367$。在本文中，我们将这些界改进为$0.6421\leq\beta\leq0.8810$。利用重要性采样，我们进一步证明$0.6536\leq \beta\leq 0.8749$以至少$1-2\times 10^{-4}$的概率成立。这里的概率是关于采样过程的随机性而言的。
英文摘要
The Beardwood-Halton-Hammersley theorem characterizes the asymptotic length of the optimal Euclidean traveling salesman tour through independent and uniformly distributed random points $X_1,\ldots,X_n$ in the unit square. It states that there exists a universal constant $β$ such that the length of the shortest tour is asymptotic to $β\sqrt{n}$ almost surely. The best bounds established to date are $0.6277\leq β\leq 0.90367$. In this paper, we improve these bounds to $0.6421\leqβ\leq0.8810$. Using importance sampling, we further show that $0.6536\leq β\leq 0.8749$ holds with probability at least $1-2\times 10^{-4}$. Here the probability is taken with respect to the randomness of the sampling procedure.
