Graph realization problem
The first problem is the so-called bipartite realization problem.
The second is known as the digraph realization problem.
The problem of constructing a solution for the graph realization problem with the additional constraint that each such solution comes with the same probability was shown to have a polynomial-time approximation scheme for the degree sequences of regular graphs by Cooper, Martin, and Greenhill.
The general problem is still unsolved.
Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Graph realization problem.