A probabilistic analysis of the switching algorithm for the Euclidean TSP
Mathematical Programming: Series A and B
Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems
Journal of the ACM (JACM)
New Results on the Old k-opt Algorithm for the Traveling Salesman Problem
SIAM Journal on Computing
Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time
Journal of the ACM (JACM)
The NP-completeness column: Finding needles in haystacks
ACM Transactions on Algorithms (TALG)
The Number of Flips Required to Obtain Non-crossing Convex Cycles
Computational Geometry and Graph Theory
Exact Solutions to the Traveling Salesperson Problem by a Population-Based Evolutionary Algorithm
EvoCOP '09 Proceedings of the 9th European Conference on Evolutionary Computation in Combinatorial Optimization
The impact of parametrization in memetic evolutionary algorithms
Theoretical Computer Science
Smoothed analysis: an attempt to explain the behavior of algorithms in practice
Communications of the ACM - A View of Parallel Computing
Customer Assistance Services for Simulated Shopping Scenarios
KES-AMSTA '09 Proceedings of the Third KES International Symposium on Agent and Multi-Agent Systems: Technologies and Applications
Theoretical properties of two ACO approaches for the traveling salesman problem
ANTS'10 Proceedings of the 7th international conference on Swarm intelligence
Proceedings of the 13th annual conference companion on Genetic and evolutionary computation
Settling the complexity of local max-cut (almost) completely
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Smoothed Analysis of the k-Means Method
Journal of the ACM (JACM)
Smoothed analysis of partitioning algorithms for Euclidean functionals
WADS'11 Proceedings of the 12th international conference on Algorithms and data structures
Smoothed performance guarantees for local search
ESA'11 Proceedings of the 19th European conference on Algorithms
On the power of nodes of degree four in the local max-cut problem
CIAC'10 Proceedings of the 7th international conference on Algorithms and Complexity
Average-case approximation ratio of the 2-opt algorithm for the TSP
Operations Research Letters
MFCS'12 Proceedings of the 37th international conference on Mathematical Foundations of Computer Science
LION'12 Proceedings of the 6th international conference on Learning and Intelligent Optimization
Annals of Mathematics and Artificial Intelligence
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2-Opt is probably the most basic and widely used local search heuristic for the TSP. This heuristic achieves amazingly good results on "real world" Euclidean instances both with respect to running time and approximation ratio. There are numerous experimental studies on the performance of 2-Opt. However, the theoretical knowledge about this heuristic is still very limited. Not even its worst case running time on Euclidean instances was known so far. In this paper, we clarify this issue by presenting a family of Euclidean instances on which 2-Opt can take an exponential number of steps. Previous probabilistic analyses were restricted to instances in which n points are placed uniformly at random in the unit square [0, 1]2, where it was shown that the expected number of steps is bounded by Õ(n10) for Euclidean instances. We consider a more advanced model of probabilistic instances in which the points can be placed according to general distributions on [0, 1]2. In particular, we allow different distributions for different points. We study the expected running time in terms of the number n of points and the maximal density &phis; of the probability distributions. We show an upper bound on the expected length of any 2-Opt improvement path of Õ(n4+1/3 · &phis;8/3). When starting with an initial tour computed by an insertion heuristic, the upper bound on the expected number of steps improves even to Õ(n3+5/6 · &phis;8/3). If the distances are measured according to the Manhattan metric, then the expected number of steps is bounded by Õ(n3+1/2 · &phis;). In addition, we prove an upper bound of O(√&phis;) on the expected approximation factor with respect to both of these metrics. Let us remark that our probabilistic analysis covers as special cases the uniform input model with &phis; = 1 and a smoothed analysis with Gaussian perturbations of standard deviation σ with &phis; ~ 1/σ2. Besides random metric instances, we also consider an alternative random input model in which an adversary specifies a graph and distributions for the edge lengths in this graph. In this model, we achieve even better results on the expected running time of 2-Opt.