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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2020
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2003.09948 |
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| _version_ | 1866909161173811200 |
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| author | Alkema, Henk de Berg, Mark van der Hofstad, Remco Kisfaludi-Bak, Sándor |
| author_facet | Alkema, Henk de Berg, Mark van der Hofstad, Remco Kisfaludi-Bak, Sándor |
| contents | 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 $2^{O(δ^{1-1/d})}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2003_09948 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Euclidean TSP in Narrow Strips Alkema, Henk de Berg, Mark van der Hofstad, Remco Kisfaludi-Bak, Sándor Computational Geometry 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 $2^{O(δ^{1-1/d})}$. |
| title | Euclidean TSP in Narrow Strips |
| topic | Computational Geometry |
| url | https://arxiv.org/abs/2003.09948 |