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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
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 > 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: |
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| 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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