# Sampling uniformly from the vertices of a polytope

I'm looking for a reference on how to sample uniformly (and preferably efficiently, elegantly, etc.) from the vertices of a polytope. I gather that enumerating vertices is hard. I also note the MO questions Uniformly Sampling from Convex Polytopes and Is it possible to sample uniformly on the surface of a high-dimensional polytope?. A bit of poking around Google Scholar hasn't turned anything up.

• I recently asked a question that turned out to be equivalent: cstheory.stackexchange.com/questions/42705/… I found some interesting references, but nothing definitive yet. – Lorenzo Najt Apr 14 '19 at 4:30
• I think I was able to show it is NP-hard. Take a look and see if you believe my argument. :-) – Lorenzo Najt Apr 20 '19 at 5:17
• It is $NP$-hard. See updated answer. – Lorenzo Najt Nov 13 '19 at 0:22
• @LorenzoNajt- Thanks for the update! – Steve Huntsman Nov 13 '19 at 15:37