Frieze, Alan and Sorkin, Gregory B.
(2006)
*The probabilistic relationship between the assignment and travelling salesman problems.*
SIAM Journal on Computing, 36 (5).
pp. 1435-1452.
ISSN 0097-5397

## Abstract

We consider the gap between the cost of an optimal assignment in a complete bipartite graph with random edge weights, and the cost of an optimal traveling salesman tour in a complete directed graph with the same edge weights. Using an improved “patching” heuristic, we show that with high probability the gap is $O((\ln n)^2/n)$, and that its expectation is $\Omega(1/n)$. One of the underpinnings of this result is that the largest edge weight in an optimal assignment has expectation $\Theta(\ln n / n)$. A consequence of the small assignment–TSP gap is an $e^{\tilde{O}(\sqrt{n})}$‐time algorithm which, with high probability, exactly solves a random asymmetric traveling salesman instance. In addition to the assignment–TSP gap, we also consider the expected gap between the optimal and second‐best assignments; it is at least $\Omega(1/n^2)$ and at most $O(\ln n/n^2)$.

Item Type: | Article |
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Official URL: | http://epubs.siam.org/sicomp/ |

Additional Information: | © 2007 Society for Industrial and Applied Mathematics |

Divisions: | Management |

Subjects: | Q Science > QA Mathematics |

Sets: | Research centres and groups > Management Science Group Departments > Management |

Date Deposited: | 13 Apr 2011 13:45 |

Last Modified: | 20 Feb 2019 08:31 |

URI: | http://eprints.lse.ac.uk/id/eprint/35445 |

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