  We address here the problem of generating random graphs uniformly from the
set of simple connected graphs having a prescribed degree sequence. Our goal is
to provide an algorithm designed for practical use both because of its ability
to generate very large graphs (efficiency) and because it is easy to implement
(simplicity). We focus on a family of heuristics for which we prove optimality
conditions, and show how this optimality can be reached in practice. We then
propose a different approach, specifically designed for typical real-world
degree distributions, which outperforms the first one. Assuming a conjecture
which we state and argue rigorously, we finally obtain an log-linear algorithm,
which, in spite of being very simple, improves the best known complexity.
