### 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 had 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 barrier 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 paper can be applied to other optimization problems as well: these applications include improved approximation algorithms and improved LP integrality gap upper bounds for the prize-collecting s-t path problem and the unit-weight graphical metric s-t path TSP.

Original language | English |
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Title of host publication | STOC '12 - Proceedings of the 2012 ACM Symposium on Theory of Computing |

Pages | 875-885 |

Number of pages | 11 |

DOIs | |

Publication status | Published - 2012 Jun 26 |

Event | 44th Annual ACM Symposium on Theory of Computing, STOC '12 - New York, NY, United States Duration: 2012 May 19 → 2012 May 22 |

### Other

Other | 44th Annual ACM Symposium on Theory of Computing, STOC '12 |
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Country | United States |

City | New York, NY |

Period | 12/5/19 → 12/5/22 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Software

### Cite this

*STOC '12 - Proceedings of the 2012 ACM Symposium on Theory of Computing*(pp. 875-885) https://doi.org/10.1145/2213977.2214055

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*STOC '12 - Proceedings of the 2012 ACM Symposium on Theory of Computing.*pp. 875-885, 44th Annual ACM Symposium on Theory of Computing, STOC '12, New York, NY, United States, 12/5/19. https://doi.org/10.1145/2213977.2214055

**Improving Christofides' algorithm for the s-t path TSP.** / An, Hyung Chan; Kleinberg, Robert; Shmoys, David B.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Improving Christofides' algorithm for the s-t path TSP

AU - An, Hyung Chan

AU - Kleinberg, Robert

AU - Shmoys, David B.

PY - 2012/6/26

Y1 - 2012/6/26

N2 - 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 had 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 barrier 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 paper can be applied to other optimization problems as well: these applications include improved approximation algorithms and improved LP integrality gap upper bounds for the prize-collecting s-t path problem and the unit-weight graphical metric s-t path TSP.

AB - 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 had 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 barrier 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 paper can be applied to other optimization problems as well: these applications include improved approximation algorithms and improved LP integrality gap upper bounds for the prize-collecting s-t path problem and the unit-weight graphical metric s-t path TSP.

UR - http://www.scopus.com/inward/record.url?scp=84862601434&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84862601434&partnerID=8YFLogxK

U2 - 10.1145/2213977.2214055

DO - 10.1145/2213977.2214055

M3 - Conference contribution

SN - 9781450312455

SP - 875

EP - 885

BT - STOC '12 - Proceedings of the 2012 ACM Symposium on Theory of Computing

ER -