[usamec]

Nope, there are constrains on how the path might look or not. The constrains have form: Is path enters vertex v, then it might only use edges {...}.

[ColinWright]

Yes - it's getting harder and harder to pin down exactly what this problem really is, what an example of it is really like, and what's really going on.

I'm sure the author has something in mind. I'm sure it's not well explained, and it might even prove insufficiently explained to be able to assess it properly.

It's looking like it's locally "OK" but globally nonsense.

[usamec]

Well, he get proving that his problem is NP-hard right (because the proof is really easy). But his algorithm for solving his problem is really messy and hard to follow.