Euclidean TSP in Narrow Strip
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 fixedparameter 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 $2^{O(\sqrt{\delta})} n^2 + O(n^3)$.
 Publication:

arXiv eprints
 Pub Date:
 March 2020
 arXiv:
 arXiv:2003.09948
 Bibcode:
 2020arXiv200309948A
 Keywords:

 Computer Science  Computational Geometry
 EPrint:
 To appear in: Proceedings 36th International Symposium on Computational Geometry (SoCG 2020)