Counting triangulations and other crossing-free structures approximately

We consider the problem of counting straight-edge triangulations of a given set P of n points in the plane. Until very recently it was not known whether the exact number of triangulations of P can be computed asymptotically faster than by enumerating all triangulations. We now know that the number of triangulations of P can be computed in O * ( 2 n ) time 9, which is less than the lower bound of ? ( 2.43 n ) on the number of triangulations of any point set 30. In this paper we address the question of whether one can approximately count triangulations in sub-exponential time. We present an algorithm with sub-exponential running time and sub-exponential approximation ratio, that is, denoting by ? the output of our algorithm and by c n the exact number of triangulations of P, for some positive constant c, we prove that c n ? ? ? c n ? 2 o ( n ) . This is the first algorithm that in sub-exponential time computes a ( 1 + o ( 1 ) ) -approximation of the base of the number of triangulations, more precisely, c ? ? 1 n ? ( 1 + o ( 1 ) ) c . Our algorithm can be adapted to approximately count other crossing-free structures on P, keeping the quality of approximation and running time intact. In this paper we show how to do this for matchings and spanning trees.