Is there a solution to the TSP problem that is effective and not limited to space?

Question

Is there any method of solving the traveling salesman problem effective and that is not limited by space (as is the case of the Held-Karp algorithm), excepting brute force?

If we use algorithms based on dynamic programming, it solves the problem more efficiently than with brute force. Both manage to obtain the optimal result (in practice they are useless when the size of the problem grows), but to perform dynamic programming, space is needed to store the operations. On the other hand, if we use brute force, it needs practically no storage.

For example, a case with 40 "cities" using brute force to solve it is intractable, and if dynamic programming is used it becomes more efficient but it would need a lot of space to store the operations, so it is also physically limited.

Rephrasing the question, is there any method that is more efficient than brute force and has no problem with data storage no matter the size of the problem?

For example one that the time is O(2^n) and the memory is not an inconvenience.

Thank you!