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I have a weighted directed graph. I'm defining a "spanning walk" here as a walk through the graph that visits each vertex at least once, with no limitations on what edges need to be traversed. I'm trying to find the shortest of such walks, if one exists. Here's an image of an example graph, whose shortest spanning walk is c->d->b->a, with a length of 6

I tried looking into Hamiltonian paths and the Traveling Salesman Problem and such, but as I'm allowed to visit a vertex multiple times in my problem, I don't know if I can make much use of it. I'm almost completely convinced the problem is NP-hard, but wondering if there's anything to make a search not take centuries for larger graphs.

Alyrii
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  • This looks like a version of the [Travelling Salesman Problem](https://en.m.wikipedia.org/wiki/Travelling_salesman_problem). – Tangentially Perpendicular Jul 02 '21 at 09:53
  • the TSP states you can visit each city only exactly once, which isn't a restriction here, but I'm sure it's very related, making an edit, thanks c: – Alyrii Jul 02 '21 at 09:59

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