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Smoothed analysis of the 2-Opt algorithm for the general TSP

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Englert, Matthias, Röglin, Heiko and Vocking, Berthold (2016) Smoothed analysis of the 2-Opt algorithm for the general TSP. ACM Transactions on Algorithms , 13 (1). 10. doi:10.1145/2972953

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Official URL: http://dx.doi.org/10.1145/2972953

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Abstract

2-Opt is a simple local search heuristic for the traveling salesperson problem that performs very well in experiments with respect to both running time and solution quality. In contrast to this, there are instances on which 2-Opt may need an exponential number of steps to reach a local optimum. To understand why 2-Opt usually finds local optima quickly in experiments, we study its expected running time in the model of smoothed analysis, which can be considered as a less-pessimistic variant of worst-case analysis in which the adversarial input is subject to a small amount of random noise.

In our probabilistic input model, an adversary chooses an arbitrary graph G and a probability density function for each edge according to which its length is chosen. We prove that in this model the expected number of local improvements is O(mnφ ċ 16&sqrt;ln m)=m1+o(1)nφ, where n and m denote the number of vertices and edges of G, respectively, and φ denotes an upper bound on the density functions.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science, Engineering and Medicine > Science > Computer Science
Library of Congress Subject Headings (LCSH): Graph algorithms
Journal or Publication Title: ACM Transactions on Algorithms
Publisher: Association for Computing Machinery, Inc.
ISSN: 1549-6325
Official Date: 1 October 2016
Dates:
DateEvent
1 October 2016Published
12 July 2016Accepted
Volume: 13
Number: 1
Article Number: 10
DOI: 10.1145/2972953
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access
Funder: Sixth Framework Programme (European Commission) (FP6)
Grant number: Contract 001907 (DELIS), Grants VO889/2, WE2842/1

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