The k-Opt Algorithm for the Traveling Salesman Problem Has Exponential Running Time for k ≥ 5

Abstract

The k-Opt algorithm is a local search algorithm for the Traveling Salesman Problem. Starting with an initial tour, it iteratively replaces at most k edges in the tour with the same number of edges to obtain a better tour. Krentel (FOCS 1989) showed that the Traveling Salesman Problem with the k-Opt neighborhood is complete for the class PLS (polynomial time local search) and that the k-Opt algorithm can have exponential running time for any pivot rule. However, his proof requires k ≫ 1000 and has a substantial gap. We show the two properties above for a much smaller value of k, addressing an open question by Monien, Dumrauf, and Tscheuschner (ICALP 2010). In particular, we prove the PLS-completeness for k ≥ 17 and the exponential running time for k ≥ 5.