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√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.
- Barun Chandra, Howard J. Karloff, and Craig A. Tovey. 1999. New results on the old k-opt algorithm for the traveling salesman problem. SIAM J. Comput. 28, 6, 1998--2029. Google ScholarDigital Library
- Matthias Englert, Heiko Röglin, and Berthold Vöcking. 2014. Worst case and probabilistic analysis of the 2-opt algorithm for the TSP. Algorithmica 68, 1, 190--264. DOI:http://dx.doi.org/10.1007/s00453-013-9801-4 Google ScholarCross Ref
- David S. Johnson and Lyle A. McGeoch. 1997. The traveling salesman problem: A case study in local optimization. In Local Search in Combinatorial Optimization, E. H. L. Aarts and J. K. Lenstra (Eds.). John Wiley & Sons, New York, NY.Google Scholar
- Walter Kern. 1989. A probabilistic analysis of the switching algorithm for the Euclidean TSP. Math. Program. 44, 2, 213--219. Google ScholarDigital Library
- George S. Lueker. 1975. Unpublished Manuscript. Princeton University.Google Scholar
- Bodo Manthey and Rianne Veenstra. 2013. Smoothed analysis of the 2-opt heuristic for the TSP. In Proceedings of the 24th International Symposium on Algorithms and Computation (ISAAC). Springer, Berlin, 579--589.Google Scholar
- Daniel A. Spielman and Shang-Hua Teng. 2004. Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time. J. ACM 51, 3, 385--463. Google ScholarDigital Library
Index Terms
- Smoothed Analysis of the 2-Opt Algorithm for the General TSP
Recommendations
Smoothed Analysis of Local Search for the Maximum-Cut Problem
Special Issue on SODA'15 and Regular PapersEven though local search heuristics are the method of choice in practice for many well-studied optimization problems, most of them behave poorly in the worst case. This is, in particular, the case for the Maximum-Cut Problem, for which local search can ...
Exploring Variants of 2-Opt and 3-Opt for the General Routing Problem
The general routing problem (GRP) is the problem of finding a minimum length tour, visiting a number of specified vertices and edges in an undirected graph. In this paper, we describe how the well-known 2-opt and 3-opt local search procedures for node ...
Accelerating 2-opt and 3-opt Local Search Using GPU in the Travelling Salesman Problem
CCGRID '12: Proceedings of the 2012 12th IEEE/ACM International Symposium on Cluster, Cloud and Grid Computing (ccgrid 2012)We are presenting a high-performance GPU implementation of a 2-opt and 3-opt algorithms used to solve the Traveling Salesman Problem. The main idea behind it is to take a route that crosses over itself and reorder it so that it does not. It is a very ...
Comments