The $k$-Opt algorithm for the Traveling Salesman Problem has exponential running time for $k \ge 5$
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 $k \gg 1000$ 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 $k \geq 5$.
- Publication:
-
arXiv e-prints
- Pub Date:
- February 2024
- DOI:
- 10.48550/arXiv.2402.07061
- arXiv:
- arXiv:2402.07061
- Bibcode:
- 2024arXiv240207061H
- Keywords:
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- Computer Science - Data Structures and Algorithms;
- Computer Science - Discrete Mathematics;
- Mathematics - Combinatorics;
- 68W25;
- 68W40;
- 68Q25;
- 90C27;
- F.2.2;
- G.2.1;
- G.2.2
- E-Print:
- Appeared in ICALP 2024