### 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 |
---|---|

Article number | 34 |

Journal | Journal of the ACM |

Volume | 62 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2015 Oct |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

*Journal of the ACM*,

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

}

*Journal of the ACM*, vol. 62, no. 5, 34. https://doi.org/10.1145/2818310

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

Research output: Contribution to journal › Article

TY - JOUR

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

AU - An, Hyung Chan

AU - Kleinberg, Robert

AU - Shmoys, David B.

PY - 2015/10

Y1 - 2015/10

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 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.

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 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.

UR - http://www.scopus.com/inward/record.url?scp=84946573162&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84946573162&partnerID=8YFLogxK

U2 - 10.1145/2818310

DO - 10.1145/2818310

M3 - Article

AN - SCOPUS:84946573162

VL - 62

JO - Journal of the ACM

JF - Journal of the ACM

SN - 0004-5411

IS - 5

M1 - 34

ER -