## Abstract

The traveling salesman problem (TSP) is among the most famous NP-hard optimization problems. We design for this problem an algorithm that for any fixed ∈ >0 computes in randomized polynomial time a (1 + ∈)-approximation to the optimal tour in TSP instances that form an arbitrary metric space with bounded intrinsic dimension. The celebrated results of Arora [J. ACM, 45 (1998), pp. 753-782] and Mitchell [SIAM J. Comput., 28 (1999), pp. 1298-1309] prove that the above result holds in the special case of TSP in a fixed-dimensional Euclidean space. Thus, our algorithm demonstrates that the algorithmic tractability of metric TSP depends on the dimensionality of the space and not on its specific geometry. This result resolves a problem that has been open since the quasi-polynomial time algorithm of Talwar [Proceedings of the 36th Annual ACM Symposium on Theory of Computing, 2004, pp. 281-290].

Original language | American English |
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Pages (from-to) | 1563-1581 |

Number of pages | 19 |

Journal | SIAM Journal on Computing |

Volume | 45 |

Issue number | 4 |

DOIs | |

State | Published - 2016 |

### Bibliographical note

Funding Information:This author's work was supported in part by Israel Science Foundation grant 1609/11. This author's work was supported in part by US-Israel BSF grant 2010418 and by the Citi Foundation.

Publisher Copyright:

© 2016 the authors.

## Keywords

- Approximation algorithm
- Hierarchies
- Traveling salesman