Counting Locally Optimal Tours in the TSP

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.