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Algorithmic strategies for a fast exploration of the TSP $$4$$ 4 -OPT neighborhood

Author

Listed:
  • Giuseppe Lancia

    (University of Udine)

  • Marcello Dalpasso

    (University of Padova)

Abstract

We describe an effective algorithm for exploring the $$4$$ 4 -OPT neighborhood for the Traveling Salesman Problem. $$4$$ 4 -OPT moves change a tour into another by replacing four of its edges. The best move can be found by a $$\Theta (n^4)$$ Θ ( n 4 ) algorithm by complete enumeration, but a $$\Theta (n^3)$$ Θ ( n 3 ) dynamic programming algorithm exists in the literature. Furthermore a $$\Theta (n^2)$$ Θ ( n 2 ) algorithm also exists for a particular subset of symmetric $$4$$ 4 -OPT moves. In this work we describe a new procedure which behaves, on average, slightly worse than a quadratic algorithm over all moves (estimated at $$O(n^{2.5})$$ O ( n 2.5 ) ) and like a quadratic algorithm on the symmetric moves. Computational results are reported which show the effectiveness of our strategy compared to other algorithms for finding the best $$4$$ 4 -OPT move, and discuss the strength of the $$4$$ 4 -OPT neighborhood compared to 2- and $$3$$ 3 -OPT.

Suggested Citation

  • Giuseppe Lancia & Marcello Dalpasso, 2024. "Algorithmic strategies for a fast exploration of the TSP $$4$$ 4 -OPT neighborhood," Journal of Heuristics, Springer, vol. 30(3), pages 109-144, August.
    • Handle: RePEc:spr:joheur:v:30:y:2024:i:3:d:10.1007_s10732-023-09523-w
    DOI: 10.1007/s10732-023-09523-w
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