Certain special cases are known (e.g., for
computing the number of
monotone simple polygonalizations [ZSSM96]),
but the general
problem remains open. The problem is closely related to
that of generating a ``random'' instance of a simple polygon
on a given set of vertices, with each instance
being generated with probability 1/k, where k is the total number of
simple polygonalizations.
Heuristic methods are known and implemented [AH96].
See [CHUZ01] and [HMO+09] for related topics
and references to relevant papers.