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Euclidean TSP in Narrow Strips

MPS-Authors
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Kisfaludi-Bak,  Sándor
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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arXiv:2003.09948.pdf
(Preprint), 2MB

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Citation

Alkema, H., de Berg, M., & Kisfaludi-Bak, S. (2020). Euclidean TSP in Narrow Strips. Retrieved from https://arxiv.org/abs/2003.09948.


Cite as: http://hdl.handle.net/21.11116/0000-0007-77A3-B
Abstract
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 $2^{O(\sqrt{\delta})} n^2 + O(n^3)$.