Counting triangulations and other crossing-free structures approximately

We consider the problem of counting straight-edge triangulations of a given set P of npoints 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∗(2ⁿ) time [9], which is less than the lower bound of Ω(2.43ⁿ) 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 Λ the output of our algorithm and by cn the exact number of triangulations of P, for some positive constant c, we prove that cⁿ ≤ Λ ≤ cⁿ· 2°⁽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 ≤ Λ_n-¹≤ (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.