窄条带中的欧几里得 TSP
Euclidean TSP in Narrow Strips
AI总结：
本文研究窄条带中欧几里得 TSP 随宽度变化的复杂度，证明双调巡游在 $\delta\leq2\sqrt2$ 时最优，并给出关于 $\delta$ 的固定参数可处理算法及随机、稀疏情形复杂度，还推广至超圆柱。
摘要：
我们研究条带 $(-\infty,+\infty)\times[0,\delta]$ 内点集 $P$ 的欧几里得 TSP 复杂度如何依赖于条带宽度 $\delta$。我们得到两个主要结果。首先，对于点具有互不相同整数 $x$ 坐标的情形，我们证明：当 $\delta\leq 2\sqrt{2}$ 时，一条最短双调巡游（可用现有算法在 $O(n\log^2 n)$ 时间内计算）保证是整体最短巡游；该界是最优的。其次，我们提出一种关于 $\delta$ 的固定参数可处理算法。对于稀疏点集，即条带内每个 $1\times\delta$ 矩形包含 $O(1)$ 个点，该算法的运行时间为 $2^{O(\sqrt\delta)}n+O(\delta^2n^2)$。对于随机点集，即点从矩形 $[0,n]\times[0,\delta]$ 中均匀随机选取，其期望运行时间为 $2^{O(\sqrt\delta)}n$。这些结果可推广到宽度为 $\delta$ 的超圆柱内的点集 $P$；在此情形下，因子 $2^{O(\sqrt\delta)}$ 变为 $2^{O(\delta^{1-1/d})}$。
英文摘要：
We investigate how the complexity of Euclidean TSP for point sets $P$ inside the strip $(-\infty,+\infty)\times [0,δ]$ depends on the strip width $δ$. We obtain two main results. First, for the case where the points have distinct integer $x$-coordinates, we prove that a shortest bitonic tour (which can be computed in $O(n\log^2 n)$ time using an existing algorithm) is guaranteed to be a shortest tour overall when $δ\leq 2\sqrt{2}$, a bound which is best possible. Second, we present an algorithm that is fixed-parameter tractable with respect to $δ$. Our algorithm has running time $2^{O(\sqrtδ)} n + O(δ^2 n^2)$ for sparse point sets, where each $1\timesδ$ rectangle inside the strip contains $O(1)$ points. For random point sets, where the points are chosen uniformly at random from the rectangle $[0,n]\times [0,δ]$, it has an expected running time of $2^{O(\sqrtδ)} n$. These results generalise to point sets $P$ inside a hypercylinder of width $δ$. In this case, the factors $2^{O(\sqrtδ)}$ become $2^{O(δ^{1-1/d})}$.
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。
