Are you aware that

• you are¹ describing the well known travelling salesman problem?

• "perfect" solutions for non-trivial cases of NP-complete problems are unknown?

• there are very good solutions for real world problems though?

Please check if you really need all criteria!

Pick at most 2 of 3 and we might be able to help:

• Fast (i.e. polynomial²) computation

• Proven minimal path

• unknown number of nodes > n (n ~ 20 ?)

If it's just a theoretical question and you can't limit the requirements, then I suppose a branch and bound algorithm might be the best approach, but it won't be faster than brute force in some edge cases.

HTH! =)

update

had a short glance at the WP article and it describes much better what I wanted to tell.

footnotes

¹) almost

²) corrected, educated foo++

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