Faster Approximation Scheme for Euclidean $k$-TSP
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917704877735936 |
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| author | van Wijland, Ernest Zhou, Hang |
| author_facet | van Wijland, Ernest Zhou, Hang |
| contents | In the Euclidean $k$-traveling salesman problem ($k$-TSP), we are given $n$ points in the $d$-dimensional Euclidean space, for some fixed constant $d\geq 2$, and a positive integer $k$. The goal is to find a shortest tour visiting at least $k$ points. We give an approximation scheme for the Euclidean $k$-TSP in time $n\cdot 2^{O(1/\varepsilon^{d-1})} \cdot(\log n)^{2d^2\cdot 2^d}$. This improves Arora's approximation scheme of running time $n\cdot k\cdot (\log n)^{\left(O\left(\sqrt{d}/\varepsilon\right)\right)^{d-1}}$ [J. ACM 1998]. Our algorithm is Gap-ETH tight and can be derandomized by increasing the running time by a factor $O(n^d)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_08069 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Faster Approximation Scheme for Euclidean $k$-TSP van Wijland, Ernest Zhou, Hang Computational Geometry Data Structures and Algorithms In the Euclidean $k$-traveling salesman problem ($k$-TSP), we are given $n$ points in the $d$-dimensional Euclidean space, for some fixed constant $d\geq 2$, and a positive integer $k$. The goal is to find a shortest tour visiting at least $k$ points. We give an approximation scheme for the Euclidean $k$-TSP in time $n\cdot 2^{O(1/\varepsilon^{d-1})} \cdot(\log n)^{2d^2\cdot 2^d}$. This improves Arora's approximation scheme of running time $n\cdot k\cdot (\log n)^{\left(O\left(\sqrt{d}/\varepsilon\right)\right)^{d-1}}$ [J. ACM 1998]. Our algorithm is Gap-ETH tight and can be derandomized by increasing the running time by a factor $O(n^d)$. |
| title | Faster Approximation Scheme for Euclidean $k$-TSP |
| topic | Computational Geometry Data Structures and Algorithms |
| url | https://arxiv.org/abs/2307.08069 |