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.
Here is one efficient approach, performing a random walk with a rapid mixing time, that has been implemented for a particular class of polytopes, but which might well be adaptable to a more general setting: Random Walks on the Vertices of Transportation Polytopes (2008).