On the Approximability of the Traveling Salesman Problem with Line Neighborhoods
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2020
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| _version_ | 1866911846590578688 |
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| author | Antoniadis, Antonios Kisfaludi-Bak, Sándor Laekhanukit, Bundit Vaz, Daniel |
| author_facet | Antoniadis, Antonios Kisfaludi-Bak, Sándor Laekhanukit, Bundit Vaz, Daniel |
| contents | We study the variant of the Euclidean Traveling Salesman problem where instead of a set of points, we are given a set of lines as input, and the goal is to find the shortest tour that visits each line. The best known upper and lower bounds for the problem in $\mathbb{R}^d$, with $d\ge 3$, are $\mathrm{NP}$-hardness and an $O(\log^3 n)$-approximation algorithm which is based on a reduction to the group Steiner tree problem.
We show that TSP with lines in $\mathbb{R}^d$ is APX-hard for any $d\ge 3$. More generally, this implies that TSP with $k$-dimensional flats does not admit a PTAS for any $1\le k \leq d-2$ unless $\mathrm{P}=\mathrm{NP}$, which gives a complete classification of the approximability of these problems, as there are known PTASes for $k=0$ (i.e., points) and $k=d-1$ (hyperplanes). We are able to give a stronger inapproximability factor for $d=O(\log n)$ by showing that TSP with lines does not admit a $(2-ε)$-approximation in $d$ dimensions under the unique games conjecture. On the positive side, we leverage recent results on restricted variants of the group Steiner tree problem in order to give an $O(\log^2 n)$-approximation algorithm for the problem, albeit with a running time of $n^{O(\log\log n)}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2008_12075 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On the Approximability of the Traveling Salesman Problem with Line Neighborhoods Antoniadis, Antonios Kisfaludi-Bak, Sándor Laekhanukit, Bundit Vaz, Daniel Data Structures and Algorithms We study the variant of the Euclidean Traveling Salesman problem where instead of a set of points, we are given a set of lines as input, and the goal is to find the shortest tour that visits each line. The best known upper and lower bounds for the problem in $\mathbb{R}^d$, with $d\ge 3$, are $\mathrm{NP}$-hardness and an $O(\log^3 n)$-approximation algorithm which is based on a reduction to the group Steiner tree problem. We show that TSP with lines in $\mathbb{R}^d$ is APX-hard for any $d\ge 3$. More generally, this implies that TSP with $k$-dimensional flats does not admit a PTAS for any $1\le k \leq d-2$ unless $\mathrm{P}=\mathrm{NP}$, which gives a complete classification of the approximability of these problems, as there are known PTASes for $k=0$ (i.e., points) and $k=d-1$ (hyperplanes). We are able to give a stronger inapproximability factor for $d=O(\log n)$ by showing that TSP with lines does not admit a $(2-ε)$-approximation in $d$ dimensions under the unique games conjecture. On the positive side, we leverage recent results on restricted variants of the group Steiner tree problem in order to give an $O(\log^2 n)$-approximation algorithm for the problem, albeit with a running time of $n^{O(\log\log n)}$. |
| title | On the Approximability of the Traveling Salesman Problem with Line Neighborhoods |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2008.12075 |