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算法是求解旅行商问题的一种局部搜索算法。该算法从一个初始回路出发，迭代地替换回路中最多$k$条边，并以相同数量的边替换以获得更优回路。Krentel（FOCS 1989）证明了具有$k$-Opt邻域的旅行商问题是PLS（多项式时间局部搜索）类完备的，且$k$-Opt算法在任何枢轴规则下都可能具有指数级运行时间。然而，其证明要求$k \gg 1000$且存在显著缺陷。我们针对更小的$k$值证明了上述两个性质，从而解决了Monien、Dumrauf和Tscheuschner（ICALP 2010）提出的一个开放性问题。具体而言，我们证明了当$k \geq 17$时该问题是PLS完备的，且当$k \geq 5$时算法具有指数级运行时间。