The Approximation Ratio of the k-Opt Heuristic for the Euclidean Traveling Salesman Problem
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The k-Opt heuristic is a simple improvement heuristic for the Traveling Salesman Problem. It starts with an arbitrary tour and then repeatedly replaces k edges of the tour by k other edges, as long as this yields a shorter tour. We will prove that for 2-dimensional Euclidean Traveling Salesman Problems with n cities the approximation ratio of the k-Opt heuristic is Θ(log n / loglog n). This improves the upper bound of O(log n) given by Chandra, Karloff, and Tovey in 1999 and provides for the first time a non-trivial lower bound for the case k≥ 3. Our results not only hold for the Euclidean norm but extend to arbitrary p-norms.
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