Counting Locally Optimal Tours in the TSP

Bodo Manthey; Jesse van Rhijn

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arXiv · 2410.18650

Counting Locally Optimal Tours in the TSP

Bodo Manthey·Jesse van Rhijn

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Abstract

We show that the problem of counting the number of 2-optimal tours in instances of the Travelling Salesperson Problem (TSP) on complete graphs is #P-complete. In addition, we show that the expected number of 2-optimal tours in random instances of the TSP on complete graphs is $O(1.2098^n \sqrt{n!})$. Based on numerical experiments, we conjecture that the true bound is at most $O(\sqrt{n!})$, which is approximately the square root of the total number of tours.

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Bodo Manthey, Jesse van Rhijn. 2024-10-24. Counting Locally Optimal Tours in the TSP. https://arxiv.org/abs/2410.18650

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