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 |
|---|---|
| Official URL: | http://epubs.siam.org/sicomp/ |
| Additional Information: | © 2007 Society for Industrial and Applied Mathematics |
| Uncontrolled Keywords: | assignment problem, asymmetric traveling salesman problem, average‐case analysis of algorithms, random assignment problem, matching, alternating path, patching heuristic, cycle cover, permutation digraph, near‐permutation digraph |
| 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/rights/LSERO.htm |
| URL: | http://eprints.lse.ac.uk/35445/ |
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