We investigate how the complexity of Euclidean TSP for point sets $P$ inside the strip $(-\infty,+\infty)\times [0,\delta]$ depends on the strip width $\delta$. 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 $\delta\leq 2\sqrt{2}$, a bound which is best possible. Second, we present an algorithm that is fixed-parameter tractable with respect to $\delta$. More precisely, our algorithm has running time $2^{O(\sqrt{\delta})} n^2$ for sparse point sets, where each $1\times\delta$ rectangle inside the strip contains $O(1)$ points. For random point sets, where the points are chosen uniformly at random from the rectangle~$[0,n]\times [0,\delta]$, it has an expected running time of $2^{O(\sqrt{\delta})} n^2 + O(n^3)$.

中文翻译：

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窄带中的欧几里得 TSP

我们研究了带 $(-\infty,+\infty)\times [0,\delta]$ 内点集 $P$ 的欧几里得 TSP 的复杂性如何取决于带宽度 $\delta$。我们得到两个主要结果。首先，对于点具有不同整数 $x$ 坐标的情况，我们证明最短的双音游（可以使用现有算法在 $O(n\log^2 n)$ 时间内计算）保证当 $\delta\leq 2\sqrt{2}$ 是一个最短的游览，这是一个最好的界限。其次，我们提出了一个关于 $\delta$ 的固定参数易处理的算法。更准确地说，对于稀疏点集，我们的算法的运行时间为 $2^{O(\sqrt{\delta})} n^2$，其中条带内的每个 $1\times\delta$ 矩形包含 $O(1)$ 个点. 对于随机点集，从矩形~$[0,n]\times [0,