### Abstract

We present a deterministic ( 1+ 5 2 )-approximation algorithm for the s-t path TSP for an arbitrary metric. Given a symmetric metric cost on n vertices including two prespecified endpoints, the problem is to find a shortest Hamiltonian path between the two endpoints; Hoogeveen showed that the natural variant of Christofides' algorithm is a 5/3-approximation algorithm for this problem, and this asymptotically tight bound in fact has been the best approximation ratio known until now. We modify this algorithm so that it chooses the initial spanning tree based on an optimal solution to the Held-Karp relaxation rather than a minimum spanning tree; we prove this simple but crucial modification leads to an improved approximation ratio, surpassing the 20-year-old ratio set by the natural Christofides' algorithm variant. Our algorithm also proves an upper bound of 1+ 5 2 on the integrality gap of the path-variant Held-Karp relaxation. The techniques devised in this article can be applied to other optimization problems as well: these applications include improved approximation algorithms and improved LP integrality gap upper bounds for the prizecollecting s-t path problem and the unit-weight graphical metric s-t path TSP.

Original language | English |
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Article number | 34 |

Journal | Journal of the ACM |

Volume | 62 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2015 Oct |

### All Science Journal Classification (ASJC) codes

- Software
- Control and Systems Engineering
- Information Systems
- Hardware and Architecture
- Artificial Intelligence

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## Cite this

*Journal of the ACM*,

*62*(5), [34]. https://doi.org/10.1145/2818310