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The probabilistic relationship between the assignment and travelling salesman problems

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

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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
Official URL: http://epubs.siam.org/sicomp/
Additional Information: © 2007 Society for Industrial and Applied Mathematics
Library of Congress subject classification: Q Science > QA Mathematics
Sets: Research centres and groups > Management Science Group
Departments > Management
Rights: http://www.lse.ac.uk/library/usingTheLibrary/academicSupport/OA/depositYourResearch.aspx
Date Deposited: 13 Apr 2011 13:45
URL: http://eprints.lse.ac.uk/35445/

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