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arXiv · 2402.07061

The $k$-Opt algorithm for the Traveling Salesman Problem has exponential running time for $k \ge 5$

Sophia Heimann·Hung P. Hoang·Stefan Hougardy
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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 $k \gg 1000$ 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 $k \geq 5$.

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Sophia Heimann, Hung P. Hoang, Stefan Hougardy. 2024-02-10. The $k$-Opt algorithm for the Traveling Salesman Problem has exponential running time for $k \ge 5$. https://arxiv.org/abs/2402.07061

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