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 $k \gg 1000$ 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 $k \geq 17$ and the exponential running time for $k \geq 5$.

翻译：$k$-Opt算法是一种用于旅行商问题的局部搜索算法。该算法从初始路线开始，迭代地替换路线中最多$k$条边，用相同数量的边替代以获得更优的路线。Krentel（FOCS 1989）指出，具有$k$-Opt邻域的旅行商问题对于PLS类（多项式时间局部搜索）是完备的，并且$k$-Opt算法对于任何枢轴规则都可能具有指数运行时间。然而，他的证明要求$k \gg 1000$且存在重大漏洞。我们针对更小的$k$值证明了上述两个性质，回应了Monien、Dumrauf和Tscheuschner（ICALP 2010）提出的开放问题。具体而言，我们证明了当$k \geq 17$时的PLS完备性，以及当$k \geq 5$时的指数运行时间。